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| author | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
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| committer | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
| commit | 6b59540a2460633df4e3d8347cb4dfe2fb3a3afb (patch) | |
| tree | 1239c1d5553d51a7d73f2f8b465f6a23178ff8a0 /docs/htmldoc/mathcomp.algebra.mxalgebra.html | |
| parent | dd82aaeae7e9478efc178ce8430986649555b032 (diff) | |
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diff --git a/docs/htmldoc/mathcomp.algebra.mxalgebra.html b/docs/htmldoc/mathcomp.algebra.mxalgebra.html deleted file mode 100644 index cbd9867..0000000 --- a/docs/htmldoc/mathcomp.algebra.mxalgebra.html +++ /dev/null @@ -1,1946 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" -"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<link href="coqdoc.css" rel="stylesheet" type="text/css" /> -<title>mathcomp.algebra.mxalgebra</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.algebra.mxalgebra</h1> - -<div class="code"> -<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><br/> - -<br/> -</div> - -<div class="doc"> - In this file we develop the rank and row space theory of matrices, based - on an extended Gaussian elimination procedure similar to LUP - decomposition. This provides us with a concrete but generic model of - finite dimensional vector spaces and F-algebras, in which vectors, linear - functions, families, bases, subspaces, ideals and subrings are all - represented using matrices. This model can be used as a foundation for - the usual theory of abstract linear algebra, but it can also be used to - develop directly substantial theories, such as the theory of finite group - linear representation. - Here we define the following concepts and notations: - Gaussian_elimination A == a permuted triangular decomposition (L, U, r) - of A, with L a column permutation of a lower triangular - invertible matrix, U a row permutation of an upper - triangular invertible matrix, and r the rank of A, all - satisfying the identity L *m pid_mx r *m U = A. - \rank A == the rank of A. - row_free A <=> the rows of A are linearly free (i.e., the rank and - height of A are equal). - row_full A <=> the row-space of A spans all row-vectors (i.e., the - rank and width of A are equal). - col_ebase A == the extended column basis of A (the first matrix L - returned by Gaussian_elimination A). - row_ebase A == the extended row base of A (the second matrix U - returned by Gaussian_elimination A). - col_base A == a basis for the columns of A: a row-full matrix - consisting of the first \rank A columns of col_ebase A. - row_base A == a basis for the rows of A: a row-free matrix consisting - of the first \rank A rows of row_ebase A. - pinvmx A == a partial inverse for A in its row space (or on its - column space, equivalently). In particular, if u is a - row vector in the row_space of A, then u *m pinvmx A is - the row vector of the coefficients of a decomposition - of u as a sub of rows of A. - kermx A == the row kernel of A : a square matrix whose row space - consists of all u such that u *m A = 0 (it consists of - the inverse of col_ebase A, with the top \rank A rows - zeroed out). Also, kermx A is a partial right inverse - to col_ebase A, in the row space anihilated by A. - cokermx A == the cokernel of A : a square matrix whose column space - consists of all v such that A *m v = 0 (it consists of - the inverse of row_ebase A, with the leftmost \rank A - columns zeroed out). - eigenvalue g a <=> a is an eigenvalue of the square matrix g. - eigenspace g a == a square matrix whose row space is the eigenspace of - the eigenvalue a of g (or 0 if a is not an eigenvalue). - We use a different scope %MS for matrix row-space set-like operations; to - avoid confusion, this scope should not be opened globally. Note that the - the arguments of \rank _ and the operations below have default scope %MS. - (A <= B)%MS <=> the row-space of A is included in the row-space of B. - We test for this by testing if cokermx B anihilates A. - (A < B)%MS <=> the row-space of A is properly included in the - row-space of B. - (A <= B <= C)%MS == (A <= B)%MS && (B <= C)%MS, and similarly for - (A < B <= C)%MS, (A < B <= C)%MS and (A < B < C)%MS. - (A == B)%MS == (A <= B <= A)%MS (A and B have the same row-space). - (A :=: B)%MS == A and B behave identically wrt. \rank and <=. This - triple rewrite rule is the Prop version of (A == B)%MS. - Note that :=: cannot be treated as a setoid-style - Equivalence because its arguments can have different - types: A and B need not have the same number of rows, - and often don't (e.g., in row_base A :=: A). - <tt>A</tt>%MS == a square matrix with the same row-space as A; <tt>A</tt>%MS - is a canonical representation of the subspace generated - by A, viewed as a list of row-vectors: if (A == B)%MS, - then <tt>A</tt>%MS = <tt>B</tt>%MS. - (A + B)%MS == a square matrix whose row-space is the sum of the - row-spaces of A and B; thus (A + B == col_mx A B)%MS. - (\sum_i <expr i>)%MS == the "big" version of (_ + _)%MS; as the latter - has a canonical abelian monoid structure, most generic - bigop lemmas apply (the other bigop indexing notations - are also defined). - (A :&: B)%MS == a square matrix whose row-space is the intersection of - the row-spaces of A and B. - (\bigcap_i <expr i>)%MS == the "big" version of (_ :&: _)%MS, which also - has a canonical abelian monoid structure. - A^C%MS == a square matrix whose row-space is a complement to the - the row-space of A (it consists of row_ebase A with the - top \rank A rows zeroed out). - (A :\: B)%MS == a square matrix whose row-space is a complement of the - the row-space of (A :&: B)%MS in the row-space of A. - We have (A :\: B := A :&: (capmx_gen A B)^C)%MS, where - capmx_gen A B is a rectangular matrix equivalent to - (A :&: B)%MS, i.e., (capmx_gen A B == A :&: B)%MS. - proj_mx A B == a square matrix that projects (A + B)%MS onto A - parallel to B, when (A :&: B)%MS = 0 (A and B must also - be square). - mxdirect S == the sum expression S is a direct sum. This is a NON - EXTENSIONAL notation: the exact boolean expression is - inferred from the syntactic form of S (expanding - definitions, however); both (\sum(i | _) _)%MS and - (_ + _)%MS sums are recognized. This construct uses a - variant of the reflexive ("quote") canonical structure, - mxsum_expr. The structure also recognizes sums of - matrix ranks, so that lemmas concerning the rank of - direct sums can be used bidirectionally. - The next set of definitions let us represent F-algebras using matrices: - 'A[F](m, n) == the type of matrices encoding (sub)algebras of square - n x n matrices, via mxvec; as in the matrix type - notation, m and F can be omitted (m defaults to n ^ 2). - := 'M[F](m, n ^ 2). - (A \in R)%MS <=> the square matrix A belongs to the linear set of - matrices (most often, a sub-algebra) encoded by the - row space of R. This is simply notation, so all the - lemmas and rewrite rules for (_ <= _)%MS can apply. - := (mxvec A <= R)%MS. - (R * S)%MS == a square n^2 x n^2 matrix whose row-space encodes the - linear set of n x n matrices generated by the pointwise - product of the sets of matrices encoded by R and S. - 'C(R)%MS == a square matric encoding the centraliser of the set of - square matrices encoded by R. - 'C_S(R)%MS := (S :&: 'C(R))%MS (the centraliser of R in S). - 'Z(R)%MS == the center of R (i.e., 'C_R(R)%MS). - left_mx_ideal R S <=> S is a left ideal for R (R * S <= S)%MS. - right_mx_ideal R S <=> S is a right ideal for R (S * R <= S)%MS. - mx_ideal R S <=> S is a bilateral ideal for R. - mxring_id R e <-> e is an identity element for R (Prop predicate). - has_mxring_id R <=> R has a nonzero identity element (bool predicate). - mxring R <=> R encodes a nontrivial subring. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GRing.Theory</span>.<br/> -<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Reserved Notation</span> "\rank A" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">A</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "\rank A").<br/> -<span class="id" title="keyword">Reserved Notation</span> "A ^C" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "A ^C").<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">"</span></a>''A_' ( m , n )" := <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''A_' ( m , n )") : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="0ed1e82db53ee1509413a6511cc81fc4"><span class="id" title="notation">"</span></a>''A_' ( n )" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="b1e29251d11cce92985c9439f2515d69"><span class="id" title="notation">"</span></a>''A_' n" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#0ed1e82db53ee1509413a6511cc81fc4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0ed1e82db53ee1509413a6511cc81fc4"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0ed1e82db53ee1509413a6511cc81fc4"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#0ed1e82db53ee1509413a6511cc81fc4"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>, <span class="id" title="var">format</span> "''A_' n") : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">"</span></a>''A' [ F ]_ ( m , n )" := <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><span class="id" title="var">F</span><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">"</span></a>''A' [ F ]_ ( n )" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">[</span></a><span class="id" title="var">F</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="8979683c066fe0b1e230a9041364b121"><span class="id" title="notation">"</span></a>''A' [ F ]_ n" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">[</span></a><span class="id" title="var">F</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#2475ede5ceca5980a34d04718d948c78"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">matrix_set_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">MS</span>.<br/> - -<br/> - -<br/> -</div> - -<div class="doc"> - Rank and row-space theory ***************************** -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory"><span class="id" title="section">RowSpaceTheory</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="RowSpaceTheory.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">r</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/> - -<br/> - -<br/> -</div> - -<div class="doc"> - Decomposition with double pivoting; computes the rank, row and column - images, kernels, and complements of a matrix. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="Gaussian_elimination"><span class="id" title="definition">Gaussian_elimination</span></a> {<span class="id" title="var">m</span> <span class="id" title="var">n</span>} : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">_</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>, <span class="id" title="var">_</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> ⇒<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#17198bb01f8e546f36bb74df399b01c5"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#17198bb01f8e546f36bb74df399b01c5"><span class="id" title="notation">pick</span></a> <span class="id" title="var">ij</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#17198bb01f8e546f36bb74df399b01c5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ij"><span class="id" title="variable">ij</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ij"><span class="id" title="variable">ij</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="mathcomp.ssreflect.fintype.html#17198bb01f8e546f36bb74df399b01c5"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">j</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">then</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> <span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">A1</span> := <a class="idref" href="mathcomp.algebra.matrix.html#xrow"><span class="id" title="definition">xrow</span></a> <span class="id" title="var">i</span> 0 (<a class="idref" href="mathcomp.algebra.matrix.html#xcol"><span class="id" title="definition">xcol</span></a> <span class="id" title="var">j</span> 0 <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">u</span> := <a class="idref" href="mathcomp.algebra.matrix.html#ursubmx"><span class="id" title="definition">ursubmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">v</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#dlsubmx"><span class="id" title="definition">dlsubmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">L</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">U</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#Gaussian_elimination"><span class="id" title="definition">Gaussian_elimination</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#drsubmx"><span class="id" title="definition">drsubmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a>) <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#xrow"><span class="id" title="definition">xrow</span></a> <span class="id" title="var">i</span> 0 (<a class="idref" href="mathcomp.algebra.matrix.html#block_mx"><span class="id" title="definition">block_mx</span></a> 1 0 <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <span class="id" title="var">L</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#xcol"><span class="id" title="definition">xcol</span></a> <span class="id" title="var">j</span> 0 (<a class="idref" href="mathcomp.algebra.matrix.html#block_mx"><span class="id" title="definition">block_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a> 0 <span class="id" title="var">U</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">r</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.Defs"><span class="id" title="section">Defs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="RowSpaceTheory.Defs.m"><span class="id" title="variable">m</span></a> <a name="RowSpaceTheory.Defs.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RowSpaceTheory.Defs.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="Gaussian_elimination_key"><span class="id" title="lemma">Gaussian_elimination_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.Defs.LUr"><span class="id" title="variable">LUr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#Gaussian_elimination_key"><span class="id" title="lemma">Gaussian_elimination_key</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#Gaussian_elimination"><span class="id" title="definition">Gaussian_elimination</span></a>) <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="col_ebase"><span class="id" title="definition">col_ebase</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.LUr"><span class="id" title="variable">LUr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1.1</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="row_ebase"><span class="id" title="definition">row_ebase</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.LUr"><span class="id" title="variable">LUr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="mxrank"><span class="id" title="definition">mxrank</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">[||</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">]</span></a>%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.LUr"><span class="id" title="variable">LUr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="row_free"><span class="id" title="definition">row_free</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.m"><span class="id" title="variable">m</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="row_full"><span class="id" title="definition">row_full</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="row_base"><span class="id" title="definition">row_base</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="col_base"><span class="id" title="definition">col_base</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="complmx"><span class="id" title="definition">complmx</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#copid_mx"><span class="id" title="definition">copid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="kermx"><span class="id" title="definition">kermx</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_m</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#copid_mx"><span class="id" title="definition">copid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#invmx"><span class="id" title="definition">invmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cokermx"><span class="id" title="definition">cokermx</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#invmx"><span class="id" title="definition">invmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#copid_mx"><span class="id" title="definition">copid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="pinvmx"><span class="id" title="definition">pinvmx</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs.m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> :=<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#invmx"><span class="id" title="definition">invmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#invmx"><span class="id" title="definition">invmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Defs"><span class="id" title="section">Defs</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="submx_def"><span class="id" title="definition">submx_def</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) ⇒<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0).<br/> -<span class="id" title="keyword">Fact</span> <a name="submx_key"><span class="id" title="lemma">submx_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="submx"><span class="id" title="definition">submx</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#submx_key"><span class="id" title="lemma">submx_key</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#submx_def"><span class="id" title="definition">submx_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">submx_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="ltmx"><span class="id" title="definition">ltmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :=<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="eqmx"><span class="id" title="definition">eqmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#prod"><span class="id" title="inductive">prod</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)<br/> - (<span class="id" title="keyword">∀</span> <span class="id" title="var">m3</span> (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>),<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.LtmxIdentities"><span class="id" title="section">LtmxIdentities</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="RowSpaceTheory.LtmxIdentities.m1"><span class="id" title="variable">m1</span></a> <a name="RowSpaceTheory.LtmxIdentities.m2"><span class="id" title="variable">m2</span></a> <a name="RowSpaceTheory.LtmxIdentities.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<a name="RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmxE"><span class="id" title="lemma">ltmxE</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmxW"><span class="id" title="lemma">ltmxW</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmxEneq"><span class="id" title="lemma">ltmxEneq</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submxElt"><span class="id" title="lemma">submxElt</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.LtmxIdentities"><span class="id" title="section">LtmxIdentities</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - The definition of the row-space operator is rigged to return the identity - matrix for full matrices. To allow for further tweaks that will make the - row-space intersection operator strictly commutative and monoidal, we - slightly generalize some auxiliary definitions: we parametrize the - "equivalent subspace and identity" choice predicate equivmx by a boolean - determining whether the matrix should be the identity (so for genmx A its - value is row_full A), and introduce a "quasi-identity" predicate qidmx - that selects non-square full matrices along with the identity matrix 1%:M - (this does not affect genmx, which chooses a square matrix). - The choice witness for genmx A is either 1%:M for a row-full A, or else - row_base A padded with null rows. -</div> -<div class="code"> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.equivmx"><span class="id" title="variable">equivmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">idA</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :=<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#idA"><span class="id" title="variable">idA</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.equivmx_spec"><span class="id" title="variable">equivmx_spec</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">idA</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#prod"><span class="id" title="inductive">prod</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#idA"><span class="id" title="variable">idA</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="genmx_witness"><span class="id" title="definition">genmx_witness</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="genmx_def"><span class="id" title="definition">genmx_def</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) ⇒<br/> - <a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.equivmx"><span class="id" title="variable">equivmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#genmx_witness"><span class="id" title="definition">genmx_witness</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="genmx_key"><span class="id" title="lemma">genmx_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="genmx"><span class="id" title="definition">genmx</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#genmx_key"><span class="id" title="lemma">genmx_key</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#genmx_def"><span class="id" title="definition">genmx_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">genmx_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#genmx"><span class="id" title="definition">genmx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - The setwise sum is tweaked so that 0 is a strict identity element for - square matrices, because this lets us use the bigop component. As a result - setwise sum is not quite strictly extensional. -</div> -<div class="code"> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) := <a class="idref" href="mathcomp.algebra.matrix.html#conform_mx"><span class="id" title="definition">conform_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="addsmx_def"><span class="id" title="definition">addsmx_def</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) ⇒<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="addsmx_key"><span class="id" title="lemma">addsmx_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="addsmx"><span class="id" title="definition">addsmx</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx_key"><span class="id" title="lemma">addsmx_key</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx_def"><span class="id" title="definition">addsmx_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">addsmx_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - The set intersection is similarly biased so that the identity matrix is a - strict identity. This is somewhat more delicate than for the sum, because - the test for the identity is non-extensional. This forces us to actually - bias the choice operator so that it does not accidentally map an - intersection of non-identity matrices to 1%:M; this would spoil - associativity: if B :&: C = 1%:M but B and C are not identity, then for a - square matrix A we have A :&: (B :&: C) = A != (A :&: B) :&: C in general. - To complicate matters there may not be a square non-singular matrix - different than 1%:M, since we could be dealing with 'M['F_2]_1. We - sidestep the issue by making all non-square row-full matrices identities, - and choosing a normal representative that preserves the qidmx property. - Thus A :&: B = 1%:M iff A and B are both identities, and this suffices for - showing that associativity is strict. -</div> -<div class="code"> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_witness"><span class="id" title="variable">capmx_witness</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#conform_mx"><span class="id" title="definition">conform_mx</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span>.<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.equivmx"><span class="id" title="variable">equivmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_witness"><span class="id" title="variable">capmx_witness</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_nop"><span class="id" title="variable">capmx_nop</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) := <a class="idref" href="mathcomp.algebra.matrix.html#conform_mx"><span class="id" title="definition">conform_mx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#lsubmx"><span class="id" title="definition">lsubmx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)) <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="capmx_def"><span class="id" title="definition">capmx_def</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) ⇒<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_nop"><span class="id" title="variable">capmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_nop"><span class="id" title="variable">capmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="capmx_key"><span class="id" title="lemma">capmx_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="capmx"><span class="id" title="definition">capmx</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_key"><span class="id" title="lemma">capmx_key</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_def"><span class="id" title="definition">capmx_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">capmx_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="diffmx_def"><span class="id" title="definition">diffmx_def</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) ⇒<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="diffmx_key"><span class="id" title="lemma">diffmx_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="diffmx"><span class="id" title="definition">diffmx</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#diffmx_key"><span class="id" title="lemma">diffmx_key</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#diffmx_def"><span class="id" title="definition">diffmx_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">diffmx_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#diffmx"><span class="id" title="definition">diffmx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="proj_mx"><span class="id" title="definition">proj_mx</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">U</span> <span class="id" title="var">V</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#pinvmx"><span class="id" title="definition">pinvmx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a>) <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> 0.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="mxrankE"><span class="id" title="lemma">mxrankE</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#GaussE"><span class="id" title="abbreviation">GaussE</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">).2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rank_leq_row"><span class="id" title="lemma">rank_leq_row</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_leq_rank"><span class="id" title="lemma">row_leq_rank</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rank_leq_col"><span class="id" title="lemma">rank_leq_col</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="col_leq_rank"><span class="id" title="lemma">col_leq_rank</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.unitmx1F"><span class="id" title="variable">unitmx1F</span></a> := @<a class="idref" href="mathcomp.algebra.matrix.html#unitmx1"><span class="id" title="lemma">unitmx1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a>.<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_ebase_unit"><span class="id" title="lemma">row_ebase_unit</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="col_ebase_unit"><span class="id" title="lemma">col_ebase_unit</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a>.<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">rank_leq_row</span> <span class="id" title="var">rank_leq_col</span> <span class="id" title="var">row_ebase_unit</span> <span class="id" title="var">col_ebase_unit</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx_ebase"><span class="id" title="lemma">mulmx_ebase</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx_base"><span class="id" title="lemma">mulmx_base</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_base"><span class="id" title="definition">col_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_base"><span class="id" title="definition">row_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx1_min_rank"><span class="id" title="lemma">mulmx1_min_rank</span></a> <span class="id" title="var">r</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">M</span> <span class="id" title="var">N</span> :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx_max_rank"><span class="id" title="lemma">mulmx_max_rank</span></a> <span class="id" title="var">r</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">M</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">N</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#N"><span class="id" title="variable">N</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_tr"><span class="id" title="lemma">mxrank_tr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_add"><span class="id" title="lemma">mxrank_add</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrankM_maxl"><span class="id" title="lemma">mxrankM_maxl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrankM_maxr"><span class="id" title="lemma">mxrankM_maxr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_scale"><span class="id" title="lemma">mxrank_scale</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">a</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_scale_nz"><span class="id" title="lemma">mxrank_scale_nz</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">a</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_opp"><span class="id" title="lemma">mxrank_opp</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank0"><span class="id" title="lemma">mxrank0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_eq0"><span class="id" title="lemma">mxrank_eq0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx_coker"><span class="id" title="lemma">mulmx_coker</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submxE"><span class="id" title="lemma">submxE</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmxKpV"><span class="id" title="lemma">mulmxKpV</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#pinvmx"><span class="id" title="definition">pinvmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submxP"><span class="id" title="lemma">submxP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">D</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submx_refl"><span class="id" title="lemma">submx_refl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">submx_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submxMl"><span class="id" title="lemma">submxMl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submxMr"><span class="id" title="lemma">submxMr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx_sub"><span class="id" title="lemma">mulmx_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> <span class="id" title="var">p</span> (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">A</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submx_trans"><span class="id" title="lemma">submx_trans</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmx_sub_trans"><span class="id" title="lemma">ltmx_sub_trans</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_ltmx_trans"><span class="id" title="lemma">sub_ltmx_trans</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmx_trans"><span class="id" title="lemma">ltmx_trans</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmx_irrefl"><span class="id" title="lemma">ltmx_irrefl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#irreflexive"><span class="id" title="definition">irreflexive</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub0mx"><span class="id" title="lemma">sub0mx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submx0null"><span class="id" title="lemma">submx0null</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submx0"><span class="id" title="lemma">submx0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lt0mx"><span class="id" title="lemma">lt0mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmx0"><span class="id" title="lemma">ltmx0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx0P"><span class="id" title="lemma">eqmx0P</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_eq0"><span class="id" title="lemma">eqmx_eq0</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addmx_sub"><span class="id" title="lemma">addmx_sub</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="summx_sub"><span class="id" title="lemma">summx_sub</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>)<br/> - <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> ((<a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="scalemx_sub"><span class="id" title="lemma">scalemx_sub</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">a</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_sub"><span class="id" title="lemma">row_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.matrix.html#row"><span class="id" title="definition">row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_row_sub"><span class="id" title="lemma">eq_row_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.matrix.html#row"><span class="id" title="definition">row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nz_row_sub"><span class="id" title="lemma">nz_row_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.matrix.html#nz_row"><span class="id" title="definition">nz_row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_subP"><span class="id" title="lemma">row_subP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.matrix.html#row"><span class="id" title="definition">row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rV_subP"><span class="id" title="lemma">rV_subP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">v</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_subPn"><span class="id" title="lemma">row_subPn</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#row"><span class="id" title="definition">row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_rVP"><span class="id" title="lemma">sub_rVP</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">u</span> <span class="id" title="var">v</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rank_rV"><span class="id" title="lemma">rank_rV</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">v</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rowV0Pn"><span class="id" title="lemma">rowV0Pn</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rowV0P"><span class="id" title="lemma">rowV0P</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">v</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submx_full"><span class="id" title="lemma">submx_full</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_fullP"><span class="id" title="lemma">row_fullP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_full_inj"><span class="id" title="lemma">row_full_inj</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (@<a class="idref" href="mathcomp.algebra.matrix.html#mulmx"><span class="id" title="definition">mulmx</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_freeP"><span class="id" title="lemma">row_freeP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_free_inj"><span class="id" title="lemma">row_free_inj</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.matrix.html#mulmx"><span class="id" title="definition">mulmx</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_free_unit"><span class="id" title="lemma">row_free_unit</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_full_unit"><span class="id" title="lemma">row_full_unit</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_unit"><span class="id" title="lemma">mxrank_unit</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank1"><span class="id" title="lemma">mxrank1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_delta"><span class="id" title="lemma">mxrank_delta</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#delta_mx"><span class="id" title="definition">delta_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrankS"><span class="id" title="lemma">mxrankS</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submx1"><span class="id" title="lemma">submx1</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub1mx"><span class="id" title="lemma">sub1mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmx1"><span class="id" title="lemma">ltmx1</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lt1mx"><span class="id" title="lemma">lt1mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmxP"><span class="id" title="lemma">eqmxP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rV_eqP"><span class="id" title="lemma">rV_eqP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_refl"><span class="id" title="lemma">eqmx_refl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_sym"><span class="id" title="lemma">eqmx_sym</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_trans"><span class="id" title="lemma">eqmx_trans</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_rank"><span class="id" title="lemma">eqmx_rank</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lt_eqmx"><span class="id" title="lemma">lt_eqmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>, ((<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>)%<span class="id" title="keyword">type</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmxMr"><span class="id" title="lemma">eqmxMr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmxMfull"><span class="id" title="lemma">eqmxMfull</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx0"><span class="id" title="lemma">eqmx0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_scale"><span class="id" title="lemma">eqmx_scale</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">a</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_opp"><span class="id" title="lemma">eqmx_opp</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="submxMfree"><span class="id" title="lemma">submxMfree</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmxMfree"><span class="id" title="lemma">eqmxMfree</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrankMfree"><span class="id" title="lemma">mxrankMfree</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_row_base"><span class="id" title="lemma">eq_row_base</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_base"><span class="id" title="definition">row_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.qidmx_eq1"><span class="id" title="variable">qidmx_eq1</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.genmx_witnessP"><span class="id" title="variable">genmx_witnessP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.equivmx"><span class="id" title="variable">equivmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#genmx_witness"><span class="id" title="definition">genmx_witness</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmxE"><span class="id" title="lemma">genmxE</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_genmx"><span class="id" title="lemma">eq_genmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmxP"><span class="id" title="lemma">genmxP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx0"><span class="id" title="lemma">genmx0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx1"><span class="id" title="lemma">genmx1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_id"><span class="id" title="lemma">genmx_id</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<<<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>>>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_base_free"><span class="id" title="lemma">row_base_free</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_base"><span class="id" title="definition">row_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_gen"><span class="id" title="lemma">mxrank_gen</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="col_base_full"><span class="id" title="lemma">col_base_full</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#col_base"><span class="id" title="definition">col_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">row_base_free</span> <span class="id" title="var">col_base_full</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_leqif_sup"><span class="id" title="lemma">mxrank_leqif_sup</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">iff</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_leqif_eq"><span class="id" title="lemma">mxrank_leqif_eq</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">iff</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltmxErank"><span class="id" title="lemma">ltmxErank</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rank_ltmx"><span class="id" title="lemma">rank_ltmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cf4c296714679a4f3b1759a3b0e27ede"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_cast"><span class="id" title="lemma">eqmx_cast</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">e</span> :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#castmx"><span class="id" title="definition">castmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_conform"><span class="id" title="lemma">eqmx_conform</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#conform_mx"><span class="id" title="definition">conform_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#conform_mx"><span class="id" title="definition">conform_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.eqmx_sum_nop"><span class="id" title="variable">eqmx_sum_nop</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.AddsmxSub"><span class="id" title="section">AddsmxSub</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="RowSpaceTheory.AddsmxSub.m1"><span class="id" title="variable">m1</span></a> <a name="RowSpaceTheory.AddsmxSub.m2"><span class="id" title="variable">m2</span></a> <a name="RowSpaceTheory.AddsmxSub.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<a name="RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="col_mx_sub"><span class="id" title="lemma">col_mx_sub</span></a> <span class="id" title="var">m3</span> (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxE"><span class="id" title="lemma">addsmxE</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_sub"><span class="id" title="lemma">addsmx_sub</span></a> <span class="id" title="var">m3</span> (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxSl"><span class="id" title="lemma">addsmxSl</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxSr"><span class="id" title="lemma">addsmxSr</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_idPr"><span class="id" title="lemma">addsmx_idPr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_idPl"><span class="id" title="lemma">addsmx_idPl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub.A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.AddsmxSub"><span class="id" title="section">AddsmxSub</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="adds0mx"><span class="id" title="lemma">adds0mx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx0"><span class="id" title="lemma">addsmx0</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.addsmx_nop_eq0"><span class="id" title="variable">addsmx_nop_eq0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.addsmx_nop0"><span class="id" title="variable">addsmx_nop0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.addsmx_nop_id"><span class="id" title="variable">addsmx_nop_id</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.addsmx_nop"><span class="id" title="variable">addsmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxC"><span class="id" title="lemma">addsmxC</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="adds0mx_id"><span class="id" title="lemma">adds0mx_id</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">n</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx0_id"><span class="id" title="lemma">addsmx0_id</span></a> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxA"><span class="id" title="lemma">addsmxA</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">addsmx_monoid</span> <span class="id" title="var">n</span> :=<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.Law"><span class="id" title="constructor">Monoid.Law</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmxA"><span class="id" title="lemma">addsmxA</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#adds0mx_id"><span class="id" title="lemma">adds0mx_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx0_id"><span class="id" title="lemma">addsmx0_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">addsmx_comoid</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.ComLaw"><span class="id" title="constructor">Monoid.ComLaw</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmxC"><span class="id" title="lemma">addsmxC</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxMr"><span class="id" title="lemma">addsmxMr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmxS"><span class="id" title="lemma">addsmxS</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">m4</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m4"><span class="id" title="variable">m4</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addmx_sub_adds"><span class="id" title="lemma">addmx_sub_adds</span></a> <span class="id" title="var">m</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_addKl"><span class="id" title="lemma">addsmx_addKl</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_addKr"><span class="id" title="lemma">addsmx_addKr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="adds_eqmx"><span class="id" title="lemma">adds_eqmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">m4</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m4"><span class="id" title="variable">m4</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_adds"><span class="id" title="lemma">genmx_adds</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_addsmxP"><span class="id" title="lemma">sub_addsmxP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="RowSpaceTheory.I"><span class="id" title="variable">I</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_sums"><span class="id" title="lemma">genmx_sums</span></a> <span class="id" title="var">P</span> <span class="id" title="var">n</span> (<span class="id" title="var">B_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumsmx_sup"><span class="id" title="lemma">sumsmx_sup</span></a> <span class="id" title="var">i0</span> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i0"><span class="id" title="variable">i0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i0"><span class="id" title="variable">i0</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumsmx_subP"><span class="id" title="lemma">sumsmx_subP</span></a> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="summx_sub_sums"><span class="id" title="lemma">summx_sub_sums</span></a> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <span class="id" title="var">B</span> :<br/> - (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - ((<a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumsmxS"><span class="id" title="lemma">sumsmxS</span></a> <span class="id" title="var">P</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">_n</span></a>) :<br/> - (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmx_sums"><span class="id" title="lemma">eqmx_sums</span></a> <span class="id" title="var">P</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">_n</span></a>) :<br/> - (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_sumsmxP"><span class="id" title="lemma">sub_sumsmxP</span></a> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#u_"><span class="id" title="variable">u_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumsmxMr_gen"><span class="id" title="lemma">sumsmxMr_gen</span></a> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) :<br/> - ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumsmxMr"><span class="id" title="lemma">sumsmxMr</span></a> <span class="id" title="var">P</span> <span class="id" title="var">n</span> (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">_n</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rank_pid_mx"><span class="id" title="lemma">rank_pid_mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#pid_mx"><span class="id" title="definition">pid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rank_copid_mx"><span class="id" title="lemma">rank_copid_mx</span></a> <span class="id" title="var">n</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#copid_mx"><span class="id" title="definition">copid_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r"><span class="id" title="variable">r</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_compl"><span class="id" title="lemma">mxrank_compl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_ker"><span class="id" title="lemma">mxrank_ker</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="kermx_eq0"><span class="id" title="lemma">kermx_eq0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_coker"><span class="id" title="lemma">mxrank_coker</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cokermx_eq0"><span class="id" title="lemma">cokermx_eq0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx_ker"><span class="id" title="lemma">mulmx_ker</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmxKV_ker"><span class="id" title="lemma">mulmxKV_ker</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_kermxP"><span class="id" title="lemma">sub_kermxP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulmx0_rank_max"><span class="id" title="lemma">mulmx0_rank_max</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_Frobenius"><span class="id" title="lemma">mxrank_Frobenius</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) <span class="id" title="var">B</span> (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_mul_min"><span class="id" title="lemma">mxrank_mul_min</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_compl_full"><span class="id" title="lemma">addsmx_compl_full</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_capmx_gen"><span class="id" title="lemma">sub_capmx_gen</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_witnessP"><span class="id" title="variable">capmx_witnessP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.equivmx"><span class="id" title="variable">equivmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_witness"><span class="id" title="variable">capmx_witness</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_normP"><span class="id" title="variable">capmx_normP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.equivmx_spec"><span class="id" title="variable">equivmx_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_norm_eq"><span class="id" title="variable">capmx_norm_eq</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_nopP"><span class="id" title="variable">capmx_nopP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.equivmx_spec"><span class="id" title="variable">equivmx_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_nop"><span class="id" title="variable">capmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.sub_qidmx"><span class="id" title="variable">sub_qidmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.qidmx_cap"><span class="id" title="variable">qidmx_cap</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_eq_norm"><span class="id" title="variable">capmx_eq_norm</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.qidmx"><span class="id" title="variable">qidmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_norm"><span class="id" title="variable">capmx_norm</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxE"><span class="id" title="lemma">capmxE</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxSl"><span class="id" title="lemma">capmxSl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_capmx"><span class="id" title="lemma">sub_capmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxC"><span class="id" title="lemma">capmxC</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxSr"><span class="id" title="lemma">capmxSr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmx_idPr"><span class="id" title="lemma">capmx_idPr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmx_idPl"><span class="id" title="lemma">capmx_idPl</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxS"><span class="id" title="lemma">capmxS</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">m4</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m4"><span class="id" title="variable">m4</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cap_eqmx"><span class="id" title="lemma">cap_eqmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">m4</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m4"><span class="id" title="variable">m4</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxMr"><span class="id" title="lemma">capmxMr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cap0mx"><span class="id" title="lemma">cap0mx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmx0"><span class="id" title="lemma">capmx0</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxT"><span class="id" title="lemma">capmxT</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capTmx"><span class="id" title="lemma">capTmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.capmx_nop_id"><span class="id" title="variable">capmx_nop_id</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.capmx_nop"><span class="id" title="variable">capmx_nop</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cap1mx"><span class="id" title="lemma">cap1mx</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : (1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmx1"><span class="id" title="lemma">capmx1</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_cap"><span class="id" title="lemma">genmx_cap</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmxA"><span class="id" title="lemma">capmxA</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">capmx_monoid</span> <span class="id" title="var">n</span> :=<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.Law"><span class="id" title="constructor">Monoid.Law</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#capmxA"><span class="id" title="lemma">capmxA</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#cap1mx"><span class="id" title="lemma">cap1mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx1"><span class="id" title="lemma">capmx1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">capmx_comoid</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.ComLaw"><span class="id" title="constructor">Monoid.ComLaw</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#capmxC"><span class="id" title="lemma">capmxC</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="bigcapmx_inf"><span class="id" title="lemma">bigcapmx_inf</span></a> <span class="id" title="var">i0</span> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i0"><span class="id" title="variable">i0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i0"><span class="id" title="variable">i0</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_bigcapmxP"><span class="id" title="lemma">sub_bigcapmxP</span></a> <span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B_"><span class="id" title="variable">B_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_bigcap"><span class="id" title="lemma">genmx_bigcap</span></a> <span class="id" title="var">P</span> <span class="id" title="var">n</span> (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0144fb51819f0f8c24047b11c3b903bc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="matrix_modl"><span class="id" title="lemma">matrix_modl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="matrix_modr"><span class="id" title="lemma">matrix_modr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmx_compl"><span class="id" title="lemma">capmx_compl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_mul_ker"><span class="id" title="lemma">mxrank_mul_ker</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_injP"><span class="id" title="lemma">mxrank_injP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_disjoint_sum"><span class="id" title="lemma">mxrank_disjoint_sum</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="diffmxE"><span class="id" title="lemma">diffmxE</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2bba6607cd5441a3a280283039979ef6"><span class="id" title="notation">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_diff"><span class="id" title="lemma">genmx_diff</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5678c2522c46860f8f34d965719dc347"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="diffmxSl"><span class="id" title="lemma">diffmxSl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="capmx_diff"><span class="id" title="lemma">capmx_diff</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="addsmx_diff_cap_eq"><span class="id" title="lemma">addsmx_diff_cap_eq</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_cap_compl"><span class="id" title="lemma">mxrank_cap_compl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#558e7d7f1cdeac24f358810e8b2a6c90"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_sum_cap"><span class="id" title="lemma">mxrank_sum_cap</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_adds_leqif"><span class="id" title="lemma">mxrank_adds_leqif</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">iff</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -</div> - -<div class="doc"> - Subspace projection matrix -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="proj_mx_sub"><span class="id" title="lemma">proj_mx_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">U</span> <span class="id" title="var">V</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="proj_mx_compl_sub"><span class="id" title="lemma">proj_mx_compl_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">U</span> <span class="id" title="var">V</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="proj_mx_id"><span class="id" title="lemma">proj_mx_id</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">U</span> <span class="id" title="var">V</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="proj_mx_0"><span class="id" title="lemma">proj_mx_0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">U</span> <span class="id" title="var">V</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="add_proj_mx"><span class="id" title="lemma">add_proj_mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">U</span> <span class="id" title="var">V</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="proj_mx_proj"><span class="id" title="lemma">proj_mx_proj</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">U</span> <span class="id" title="var">V</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">P</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#proj_mx"><span class="id" title="definition">proj_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Completing a partially injective matrix to get a unit matrix. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="complete_unitmx"><span class="id" title="lemma">complete_unitmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">U</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">{</span></a><span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#g"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Two matrices with the same shape represent the same subspace - iff they differ only by a change of basis. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqmxMunitP"><span class="id" title="lemma">eqmxMunitP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">U</span> <span class="id" title="var">V</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">P</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b38dda025cba8c04feef5c1eeec2b3bb"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -</div> - -<div class="doc"> - Mapping between two subspaces with the same dimension. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_rank_unitmx"><span class="id" title="lemma">eq_rank_unitmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">U</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">V</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">{</span></a><span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#60865e2842769a434fa149a87581dd5d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">}</span></a>%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.SumExpr"><span class="id" title="section">SumExpr</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is the infrastructure to support the mxdirect predicate. We use a - bespoke canonical structure to decompose a matrix expression into binary - and n-ary products, using some of the "quote" technology. This lets us - characterize direct sums as set sums whose rank is equal to the sum of the - ranks of the individual terms. The mxsum_expr/proper_mxsum_expr structures - below supply both the decomposition and the calculation of the rank sum. - The mxsum_spec dependent predicate family expresses the consistency of - these two decompositions. - The main technical difficulty we need to overcome is the fact that - the "catch-all" case of canonical structures has a priority lower than - constant expansion. However, it is undesireable that local abbreviations - be opaque for the direct-sum predicate, e.g., not be able to handle - let S := (\sum(i | P i) LargeExpression i)%MS in mxdirect S -> ...). - As in "quote", we use the interleaving of constant expansion and - canonical projection matching to achieve our goal: we use a "wrapper" type - (indeed, the wrapped T type defined in ssrfun.v) with a self-inserting - non-primitive constructor to gain finer control over the type and - structure inference process. The innermost, primitive, constructor flags - trivial sums; it is initially hidden by an eta-expansion, which has been - made into a (default) canonical structure -- this lets type inference - automatically insert this outer tag. - In detail, we define three types - mxsum_spec S r <-> There exists a finite list of matrices A1, ..., Ak - such that S is the set sum of the Ai, and r is the sum - of the ranks of the Ai, i.e., S = (A1 + ... + Ak)%MS - and r = \rank A1 + ... + \rank Ak. Note that - mxsum_spec is a recursive dependent predicate family - whose elimination rewrites simultaneaously S, r and - the height of S. - proper_mxsum_expr n == The interface for proper sum expressions; this is - a double-entry interface, keyed on both the matrix sum - value and the rank sum. The matrix value is restricted - to square matrices, as the "+"%MS operator always - returns a square matrix. This interface has two - canonical insances, for binary and n-ary sums. - mxsum_expr m n == The interface for general sum expressions, comprising - both proper sums and trivial sums consisting of a - single matrix. The key values are WRAPPED as this lets - us give priority to the "proper sum" interpretation - (see below). To allow for trivial sums, the matrix key - can have any dimension. The mxsum_expr interface has - two canonical instances, for trivial and proper sums, - keyed to the Wrap and wrap constructors, respectively. - The projections for the two interfaces above are - proper_mxsum_val, mxsum_val : these are respectively coercions to 'M_n - and wrapped 'M(m, n); thus, the matrix sum for an - S : mxsum_expr m n can be written unwrap S. - proper_mxsum_rank, mxsum_rank : projections to the nat and wrapped nat, - respectively; the rank sum for S : mxsum_expr m n is - thus written unwrap (mxsum_rank S). - The mxdirect A predicate actually gets A in a phantom argument, which is - used to infer an (implicit) S : mxsum_expr such that unwrap S = A; the - actual definition is \rank (unwrap S) == unwrap (mxsum_rank S). - Note that the inference of S is inherently ambiguous: ANY matrix can be - viewed as a trivial sum, including one whose description is manifestly a - proper sum. We use the wrapped type and the interaction between delta - reduction and canonical structure inference to resolve this ambiguity in - favor of proper sums, as follows: -<ul class="doclist"> -<li> The phantom type sets up a unification problem of the form - unwrap (mxsum_val ?S) = A - with unknown evar ?S : mxsum_expr m n. - -</li> -<li> As the constructor wrap is also a default Canonical instance for the - wrapped type, so A is immediately replaced with unwrap (wrap A) and - we get the residual unification problem - mxsum_val ?S = wrap A - -</li> -<li> Now Coq tries to apply the proper sum Canonical instance, which has - key projection wrap (proper_mxsum_val ?PS) where ?PS is a fresh evar - (of type proper_mxsum_expr n). This can only succeed if m = n, and if - a solution can be found to the recursive unification problem - proper_mxsum_val ?PS = A - This causes Coq to look for one of the two canonical constants for - proper_mxsum_val (addsmx or bigop) at the head of A, delta-expanding - A as needed, and then inferring recursively mxsum_expr structures for - the last argument(s) of that constant. - -</li> -<li> If the above step fails then the wrap constant is expanded, revealing - the primitive Wrap constructor; the unification problem now becomes - mxsum_val ?S = Wrap A - which fits perfectly the trivial sum canonical structure, whose key - projection is Wrap ?B where ?B is a fresh evar. Thus the inference - succeeds, and returns the trivial sum. - -</li> -</ul> - Note that the rank projections also register canonical values, so that the - same process can be used to infer a sum structure from the rank sum. In - that case, however, there is no ambiguity and the inference can fail, - because the rank sum for a trivial sum is not an arbitrary integer -- it - must be of the form \rank ?B. It is nevertheless necessary to use the - wrapped nat type for the rank sums, because in the non-trivial case the - head constant of the nat expression is determined by the proper_mxsum_expr - canonical structure, so the mxsum_expr structure must use a generic - constant, namely wrap. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Inductive</span> <a name="mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> <span class="id" title="var">n</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">m</span>, <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/> - | <a name="TrivialMxsum"><span class="id" title="constructor">TrivialMxsum</span></a> <span class="id" title="var">m</span> <span class="id" title="var">A</span><br/> - : @<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)<br/> - | <a name="ProperMxsum"><span class="id" title="constructor">ProperMxsum</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> <span class="id" title="var">r1</span> <span class="id" title="var">r2</span> <span class="id" title="keyword">of</span><br/> - @<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r1"><span class="id" title="variable">r1</span></a> & @<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#T2"><span class="id" title="variable">T2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r2"><span class="id" title="variable">r2</span></a><br/> - : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#T2"><span class="id" title="variable">T2</span></a>)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#r1"><span class="id" title="variable">r1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#r2"><span class="id" title="variable">r2</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Structure</span> <a name="mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> := <a name="Mxsum"><span class="id" title="constructor">Mxsum</span></a> {<br/> - <a name="mxsum_val"><span class="id" title="projection">mxsum_val</span></a> :> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#wrapped"><span class="id" title="record">wrapped</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>;<br/> - <a name="mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#wrapped"><span class="id" title="record">wrapped</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>;<br/> - <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_val"><span class="id" title="method">mxsum_val</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="method">mxsum_rank</span></a>)<br/> -}.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">trivial_mxsum</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> :=<br/> - @<a class="idref" href="mathcomp.algebra.mxalgebra.html#Mxsum"><span class="id" title="constructor">Mxsum</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#Wrap"><span class="id" title="constructor">Wrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#Wrap"><span class="id" title="constructor">Wrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#TrivialMxsum"><span class="id" title="constructor">TrivialMxsum</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Structure</span> <a name="proper_mxsum_expr"><span class="id" title="record">proper_mxsum_expr</span></a> <span class="id" title="var">n</span> := <a name="ProperMxsumExpr"><span class="id" title="constructor">ProperMxsumExpr</span></a> {<br/> - <a name="proper_mxsum_val"><span class="id" title="projection">proper_mxsum_val</span></a> :> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>;<br/> - <a name="proper_mxsum_rank"><span class="id" title="projection">proper_mxsum_rank</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>;<br/> - <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_val"><span class="id" title="method">proper_mxsum_val</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_rank"><span class="id" title="method">proper_mxsum_rank</span></a><br/> -}.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="proper_mxsumP"><span class="id" title="definition">proper_mxsumP</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_expr"><span class="id" title="record">proper_mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.mxalgebra.html#ProperMxsumExpr"><span class="id" title="constructor">ProperMxsumExpr</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">termS</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_rank"><span class="id" title="projection">proper_mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>)<br/> - <span class="id" title="tactic">in</span> <span class="id" title="var">termS</span>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sum_mxsum</span> <span class="id" title="var">n</span> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_expr"><span class="id" title="record">proper_mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) :=<br/> - @<a class="idref" href="mathcomp.algebra.mxalgebra.html#Mxsum"><span class="id" title="constructor">Mxsum</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#wrap"><span class="id" title="definition">wrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#wrap"><span class="id" title="definition">wrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_rank"><span class="id" title="projection">proper_mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>)) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsumP"><span class="id" title="definition">proper_mxsumP</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.SumExpr.Binary"><span class="id" title="section">Binary</span></a>.<br/> -<span class="id" title="keyword">Variable</span> (<a name="RowSpaceTheory.SumExpr.Binary.m1"><span class="id" title="variable">m1</span></a> <a name="RowSpaceTheory.SumExpr.Binary.m2"><span class="id" title="variable">m2</span></a> <a name="RowSpaceTheory.SumExpr.Binary.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RowSpaceTheory.SumExpr.Binary.S1"><span class="id" title="variable">S1</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) (<a name="RowSpaceTheory.SumExpr.Binary.S2"><span class="id" title="variable">S2</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="binary_mxsum_proof"><span class="id" title="lemma">binary_mxsum_proof</span></a> :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Binary.S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Binary.S2"><span class="id" title="variable">S2</span></a>)<br/> - (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Binary.S1"><span class="id" title="variable">S1</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Binary.S2"><span class="id" title="variable">S2</span></a>)).<br/> - <span class="id" title="keyword">Canonical</span> <span class="id" title="var">binary_mxsum_expr</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#ProperMxsumExpr"><span class="id" title="constructor">ProperMxsumExpr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#binary_mxsum_proof"><span class="id" title="lemma">binary_mxsum_proof</span></a>.<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Binary"><span class="id" title="section">Binary</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.SumExpr.Nary"><span class="id" title="section">Nary</span></a>.<br/> -<span class="id" title="keyword">Context</span> <span class="id" title="var">J</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#J"><span class="id" title="variable">J</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#J"><span class="id" title="variable">J</span></a>) <span class="id" title="var">n</span> (<span class="id" title="var">S_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#J"><span class="id" title="variable">J</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="nary_mxsum_proof"><span class="id" title="lemma">nary_mxsum_proof</span></a> :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_spec"><span class="id" title="inductive">mxsum_spec</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#2060ffb6da9b0d74639202e23e21e2ba"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2060ffb6da9b0d74639202e23e21e2ba"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2060ffb6da9b0d74639202e23e21e2ba"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2060ffb6da9b0d74639202e23e21e2ba"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary.r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2060ffb6da9b0d74639202e23e21e2ba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2060ffb6da9b0d74639202e23e21e2ba"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary.S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a>))<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#f9660601caed2f1f92279ff134c53c8a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f9660601caed2f1f92279ff134c53c8a"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f9660601caed2f1f92279ff134c53c8a"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f9660601caed2f1f92279ff134c53c8a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary.r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#f9660601caed2f1f92279ff134c53c8a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f9660601caed2f1f92279ff134c53c8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary.S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a>))).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nary_mxsum_expr</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#ProperMxsumExpr"><span class="id" title="constructor">ProperMxsumExpr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#nary_mxsum_proof"><span class="id" title="lemma">nary_mxsum_proof</span></a>.<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr.Nary"><span class="id" title="section">Nary</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mxdirect_def"><span class="id" title="definition">mxdirect_def</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">T</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#phantom"><span class="id" title="inductive">phantom</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_val"><span class="id" title="projection">mxsum_val</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#T"><span class="id" title="variable">T</span></a>)) :=<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#T"><span class="id" title="variable">T</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SumExpr"><span class="id" title="section">SumExpr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> <span class="id" title="var">A</span> := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect_def"><span class="id" title="definition">mxdirect_def</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Phantom"><span class="id" title="constructor">Phantom</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a> <span class="id" title="var">A</span>%<span class="id" title="var">MS</span>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirectP"><span class="id" title="lemma">mxdirectP</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_expr"><span class="id" title="record">proper_mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#proper_mxsum_rank"><span class="id" title="projection">proper_mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_trivial"><span class="id" title="lemma">mxdirect_trivial</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (@<a class="idref" href="mathcomp.algebra.mxalgebra.html#trivial_mxsum"><span class="id" title="definition">trivial_mxsum</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_sum_leqif"><span class="id" title="lemma">mxrank_sum_leqif</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirectE"><span class="id" title="lemma">mxdirectE</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirectEgeq"><span class="id" title="lemma">mxdirectEgeq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c9e4b28fe217104569228ddbd881731d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#73030c22bc0b1fa771c65aa5414c65f9"><span class="id" title="notation">≥</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_rank"><span class="id" title="projection">mxsum_rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S"><span class="id" title="variable">S</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.BinaryDirect"><span class="id" title="section">BinaryDirect</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="RowSpaceTheory.BinaryDirect.m1"><span class="id" title="variable">m1</span></a> <a name="RowSpaceTheory.BinaryDirect.m2"><span class="id" title="variable">m2</span></a> <a name="RowSpaceTheory.BinaryDirect.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_addsE"><span class="id" title="lemma">mxdirect_addsE</span></a> (<span class="id" title="var">S1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.n"><span class="id" title="variable">n</span></a>) (<span class="id" title="var">S2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S2"><span class="id" title="variable">S2</span></a>)<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S1"><span class="id" title="variable">S1</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S2"><span class="id" title="variable">S2</span></a>)<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S2"><span class="id" title="variable">S2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_addsP"><span class="id" title="lemma">mxdirect_addsP</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.BinaryDirect"><span class="id" title="section">BinaryDirect</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.NaryDirect"><span class="id" title="section">NaryDirect</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a>) (<a name="RowSpaceTheory.NaryDirect.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.NaryDirect.TIsum"><span class="id" title="variable">TIsum</span></a> <span class="id" title="var">A_</span> <span class="id" title="var">i</span> := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="RowSpaceTheory.NaryDirect.mxdirect_sums_recP"><span class="id" title="variable">mxdirect_sums_recP</span></a> (<span class="id" title="var">S_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.TIsum"><span class="id" title="variable">TIsum</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#S_"><span class="id" title="variable">S_</span></a>) <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_sumsP"><span class="id" title="lemma">mxdirect_sumsP</span></a> (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span><br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_sumsE"><span class="id" title="lemma">mxdirect_sumsE</span></a> (<span class="id" title="var">S_</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxsum_expr"><span class="id" title="record">mxsum_expr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.n"><span class="id" title="variable">n</span></a>) (<span class="id" title="var">xunwrap</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#and"><span class="id" title="inductive">and</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)))<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#xunwrap"><span class="id" title="variable">xunwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a>)))<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#S_"><span class="id" title="variable">S_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.NaryDirect"><span class="id" title="section">NaryDirect</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.SubDaddsmx"><span class="id" title="section">SubDaddsmx</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="RowSpaceTheory.SubDaddsmx.m"><span class="id" title="variable">m</span></a> <a name="RowSpaceTheory.SubDaddsmx.m1"><span class="id" title="variable">m1</span></a> <a name="RowSpaceTheory.SubDaddsmx.m2"><span class="id" title="variable">m2</span></a> <a name="RowSpaceTheory.SubDaddsmx.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/> -<span class="id" title="keyword">Variables</span> (<a name="RowSpaceTheory.SubDaddsmx.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<a name="RowSpaceTheory.SubDaddsmx.B1"><span class="id" title="variable">B1</span></a> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<a name="RowSpaceTheory.SubDaddsmx.B2"><span class="id" title="variable">B2</span></a> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="sub_daddsmx_spec"><span class="id" title="inductive">sub_daddsmx_spec</span></a> : <span class="id" title="keyword">Prop</span> :=<br/> - <a name="SubDaddsmxSpec"><span class="id" title="constructor">SubDaddsmxSpec</span></a> <span class="id" title="var">A1</span> <span class="id" title="var">A2</span> <span class="id" title="keyword">of</span> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B1"><span class="id" title="variable">B1</span></a>)%<span class="id" title="var">MS</span> & (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B2"><span class="id" title="variable">B2</span></a>)%<span class="id" title="var">MS</span> & <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a><br/> - & <span class="id" title="keyword">∀</span> <span class="id" title="var">C1</span> <span class="id" title="var">C2</span>, (<a class="idref" href="mathcomp.algebra.mxalgebra.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B1"><span class="id" title="variable">B1</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#C2"><span class="id" title="variable">C2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B2"><span class="id" title="variable">B2</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C2"><span class="id" title="variable">C2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C2"><span class="id" title="variable">C2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_daddsmx"><span class="id" title="lemma">sub_daddsmx</span></a> : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9b2c1db5d2375a5b8d3548a4a995de32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B2"><span class="id" title="variable">B2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fc376d0ef5d57f2ced00029c73bfae3e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx.B2"><span class="id" title="variable">B2</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#sub_daddsmx_spec"><span class="id" title="inductive">sub_daddsmx_spec</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDaddsmx"><span class="id" title="section">SubDaddsmx</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.SubDsumsmx"><span class="id" title="section">SubDsumsmx</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a>) (<a name="RowSpaceTheory.SubDsumsmx.m"><span class="id" title="variable">m</span></a> <a name="RowSpaceTheory.SubDsumsmx.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RowSpaceTheory.SubDsumsmx.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<a name="RowSpaceTheory.SubDsumsmx.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">_n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="sub_dsumsmx_spec"><span class="id" title="inductive">sub_dsumsmx_spec</span></a> : <span class="id" title="keyword">Prop</span> :=<br/> - <a name="SubDsumsmxSpec"><span class="id" title="constructor">SubDsumsmxSpec</span></a> <span class="id" title="var">A_</span> <span class="id" title="keyword">of</span> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span><br/> - & <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><br/> - & <span class="id" title="keyword">∀</span> <span class="id" title="var">C</span>, (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#SimplPred"><span class="id" title="definition">SimplPred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_dsumsmx"><span class="id" title="lemma">sub_dsumsmx</span></a> :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#sub_dsumsmx_spec"><span class="id" title="inductive">sub_dsumsmx_spec</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.SubDsumsmx"><span class="id" title="section">SubDsumsmx</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RowSpaceTheory.Eigenspace"><span class="id" title="section">Eigenspace</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="RowSpaceTheory.Eigenspace.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RowSpaceTheory.Eigenspace.g"><span class="id" title="variable">g</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5f150f70ed272d7494c263afd1ab08fa"><span class="id" title="notation">M_n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="eigenspace"><span class="id" title="definition">eigenspace</span></a> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Eigenspace.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="eigenvalue"><span class="id" title="definition">eigenvalue</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.F"><span class="id" title="variable">F</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">a</span> ⇒ <a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenspace"><span class="id" title="definition">eigenspace</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eigenspaceP"><span class="id" title="lemma">eigenspaceP</span></a> <span class="id" title="var">a</span> <span class="id" title="var">m</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Eigenspace.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#86216ed4d3483e6fad890671e5a52e3b"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Eigenspace.g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aa19237bd672944c59049593f9cf504"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenspace"><span class="id" title="definition">eigenspace</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eigenvalueP"><span class="id" title="lemma">eigenvalueP</span></a> <span class="id" title="var">a</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Eigenspace.g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenvalue"><span class="id" title="definition">eigenvalue</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_sum_eigenspace"><span class="id" title="lemma">mxdirect_sum_eigenspace</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">a_</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a_"><span class="id" title="variable">a_</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e3eadb2258a983802424aabe03eb3fa5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenspace"><span class="id" title="definition">eigenspace</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#a_"><span class="id" title="variable">a_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory.Eigenspace"><span class="id" title="section">Eigenspace</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#RowSpaceTheory"><span class="id" title="section">RowSpaceTheory</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">submx_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">"</span></a>\rank A" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank"><span class="id" title="definition">mxrank</span></a> <span class="id" title="var">A</span>) : <span class="id" title="var">nat_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="3962b76563fd8a8f45948950a775860e"><span class="id" title="notation">"</span></a><< A >>" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#genmx"><span class="id" title="definition">genmx</span></a> <span class="id" title="var">A</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="7772cb4a238f5fc3b7cf2f735c00df9d"><span class="id" title="notation">"</span></a>A ^C" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#complmx"><span class="id" title="definition">complmx</span></a> <span class="id" title="var">A</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">"</span></a>A <= B" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="74f1d33aea43cd94f177c35b7a221cde"><span class="id" title="notation">"</span></a>A < B" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="af8133c7d0b5241165f1b5224efdf81a"><span class="id" title="notation">"</span></a>A <= B <= C" := (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">B</span> <span class="id" title="var">C</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="814e4349e6a02e2206615dfc9811ea85"><span class="id" title="notation">"</span></a>A < B <= C" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">B</span> <span class="id" title="var">C</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="14e1c0c3409d8e7d5db755e0de9d5941"><span class="id" title="notation">"</span></a>A <= B < C" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <span class="id" title="var">B</span> <span class="id" title="var">C</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="e18aecf4b1cc2ebc6d460286590d05be"><span class="id" title="notation">"</span></a>A < B < C" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ltmx"><span class="id" title="definition">ltmx</span></a> <span class="id" title="var">B</span> <span class="id" title="var">C</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="2face00c9cbc11f22bacfabff84e3b9a"><span class="id" title="notation">"</span></a>A == B" := (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> <span class="id" title="var">B</span> <span class="id" title="var">A</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">"</span></a>A :=: B" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#eqmx"><span class="id" title="definition">eqmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">"</span></a>A + B" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="92683a3ca3b0c0704351ce117beaffe3"><span class="id" title="notation">"</span></a>A :&: B" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="f7be369075cd7f5f2c01766717911ece"><span class="id" title="notation">"</span></a>A :\: B" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#diffmx"><span class="id" title="definition">diffmx</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> <span class="id" title="var">S</span> := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect_def"><span class="id" title="definition">mxdirect_def</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Phantom"><span class="id" title="constructor">Phantom</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a> <span class="id" title="var">S</span>%<span class="id" title="var">MS</span>)).<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="994c9f44fcb3e626f86425e0ec6ef6f1"><span class="id" title="notation">"</span></a>\sum_ ( i <- r | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="e05e2317c6b2ac68de112ee705a9b280"><span class="id" title="notation">"</span></a>\sum_ ( i <- r ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="82a87f9516816e1a6b3f8cade728fcc0"><span class="id" title="notation">"</span></a>\sum_ ( m <= i < n | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="35d7e7799cbf2a1653ade992df002ad2"><span class="id" title="notation">"</span></a>\sum_ ( m <= i < n ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">"</span></a>\sum_ ( i | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">"</span></a>\sum_ i B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">_i</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="83c6f00b5e6d1ad22616b0c10916b08d"><span class="id" title="notation">"</span></a>\sum_ ( i : t | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="4cc20c6ab533394b2a577ee2dd2a6a4f"><span class="id" title="notation">"</span></a>\sum_ ( i : t ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="c53dc748ea4f286b76de1b0acf1d0bfc"><span class="id" title="notation">"</span></a>\sum_ ( i < n | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="016ed28d62a97bafcd23b7249aedbd2b"><span class="id" title="notation">"</span></a>\sum_ ( i < n ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="9d0e110bd76910c4581a165c9a723da9"><span class="id" title="notation">"</span></a>\sum_ ( i 'in' A | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="82c1a6a5184deaa3ae19991e126caeb4"><span class="id" title="notation">"</span></a>\sum_ ( i 'in' A ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#addsmx"><span class="id" title="definition">addsmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="f50f3b23c8e3019caf5cf4a7815105e5"><span class="id" title="notation">"</span></a>\bigcap_ ( i <- r | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="5f44d8d24964f9a02a6b848b509cff3e"><span class="id" title="notation">"</span></a>\bigcap_ ( i <- r ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="2c6b95db3ac95b89c931db8dab04d58e"><span class="id" title="notation">"</span></a>\bigcap_ ( m <= i < n | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#61699f7bcd958ceac6b63f7e240f7ee7"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="229c33015793707dec532c8fa8ac0f7c"><span class="id" title="notation">"</span></a>\bigcap_ ( m <= i < n ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#a0f72dd08c3295710348031e7df784a3"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="6494fa7ca7263193f211f3687fd72488"><span class="id" title="notation">"</span></a>\bigcap_ ( i | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="bf4de16af4da3045156e8c028c958850"><span class="id" title="notation">"</span></a>\bigcap_ i B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">_i</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="992ac586203bc0b4d9b5a3a0898328f9"><span class="id" title="notation">"</span></a>\bigcap_ ( i : t | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d9e61c197d846298f3e26b588e67e5ec"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="26e3a701eeaf5a78b9f982e694c376f0"><span class="id" title="notation">"</span></a>\bigcap_ ( i : t ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.bigop.html#0a668c1f377e113a6f68dd824f1c2031"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="1f08524fdead87ac5c4cadd463584433"><span class="id" title="notation">"</span></a>\bigcap_ ( i < n | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#0b83d32979b1fdd5833879356bbfd57b"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="53c79654dc2d114aa8ca2ea224ce5286"><span class="id" title="notation">"</span></a>\bigcap_ ( i < n ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="85e048d74ef4236fdb8acdb27fd773b6"><span class="id" title="notation">"</span></a>\bigcap_ ( i 'in' A | P ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">|</span></a> <span class="id" title="var">P</span>%<span class="id" title="var">B</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d82d692725683c4eeb1ed4ba22855f30"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="4533cc85b0b8cf320c43837d45e76595"><span class="id" title="notation">"</span></a>\bigcap_ ( i 'in' A ) B" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx"><span class="id" title="definition">capmx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">)</span></a> <span class="id" title="var">B</span>%<span class="id" title="var">MS</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="DirectSums"><span class="id" title="section">DirectSums</span></a>.<br/> -<span class="id" title="keyword">Variables</span> (<a name="DirectSums.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="DirectSums.I"><span class="id" title="variable">I</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="DirectSums.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#I"><span class="id" title="variable">I</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_delta"><span class="id" title="lemma">mxdirect_delta</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#DirectSums.P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#DirectSums.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.matrix.html#delta_mx"><span class="id" title="definition">delta_mx</span></a> 0 (<a class="idref" href="mathcomp.algebra.mxalgebra.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">rV</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#DirectSums.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation">>></span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#DirectSums"><span class="id" title="section">DirectSums</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="CardGL"><span class="id" title="section">CardGL</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="CardGL.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_GL"><span class="id" title="lemma">card_GL</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">GL_n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">,</span></a> 2<a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> 1<a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -</div> - -<div class="doc"> - An alternate, somewhat more elementary proof, that does not rely on the - row-space theory, but directly performs the LUP decomposition. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="LUP_card_GL"><span class="id" title="lemma">LUP_card_GL</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">GL_n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">,</span></a> 2<a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> 1<a class="idref" href="mathcomp.ssreflect.bigop.html#b46f4e825b398c0eb860f658bff447f6"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_GL_1"><span class="id" title="lemma">card_GL_1</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">GL_1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">.-1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_GL_2"><span class="id" title="lemma">card_GL_2</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">GL_2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e4fbb9440521cdeb861c5b6e5cc78252"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">.-1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#CardGL"><span class="id" title="section">CardGL</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="logn_card_GL_p"><span class="id" title="lemma">logn_card_GL_p</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#c7dadf81985821165b2931d70aca7d91"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#c7dadf81985821165b2931d70aca7d91"><span class="id" title="notation">GL_n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#c7dadf81985821165b2931d70aca7d91"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.matrix.html#c7dadf81985821165b2931d70aca7d91"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">,</span></a> 2<a class="idref" href="mathcomp.ssreflect.binomial.html#95129fba25b300adee91ca933fa2fcdc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MatrixAlgebra"><span class="id" title="section">MatrixAlgebra</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="MatrixAlgebra.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem0mx"><span class="id" title="lemma">mem0mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : 0 <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx0"><span class="id" title="lemma">memmx0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b1e29251d11cce92985c9439f2515d69"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b1e29251d11cce92985c9439f2515d69"><span class="id" title="notation">A_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx1"><span class="id" title="lemma">memmx1</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#mxvec"><span class="id" title="definition">mxvec</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#is_scalar_mx"><span class="id" title="definition">is_scalar_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx_subP"><span class="id" title="lemma">memmx_subP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx_eqP"><span class="id" title="lemma">memmx_eqP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2face00c9cbc11f22bacfabff84e3b9a"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx_addsP"><span class="id" title="lemma">memmx_addsP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">D</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx_sumsP"><span class="id" title="lemma">memmx_sumsP</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_n</span></a>) <span class="id" title="var">R_</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R_"><span class="id" title="variable">R_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R_"><span class="id" title="variable">R_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_non_scalar_mxP"><span class="id" title="lemma">has_non_scalar_mxP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - (1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#is_scalar_mx"><span class="id" title="definition">is_scalar_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">MS</span> (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mulsmx"><span class="id" title="definition">mulsmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :=<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#lin_mx"><span class="id" title="definition">lin_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#mulmxr"><span class="id" title="abbreviation">mulmxr</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#vec_mx"><span class="id" title="definition">vec_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#row"><span class="id" title="definition">row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)))<a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="genmx_muls"><span class="id" title="lemma">genmx_muls</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation"><<</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_mulsmx"><span class="id" title="lemma">mem_mulsmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) <span class="id" title="var">A1</span> <span class="id" title="var">A2</span> :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmx_subP"><span class="id" title="lemma">mulsmx_subP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">A1</span> <span class="id" title="var">A2</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmxS"><span class="id" title="lemma">mulsmxS</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">m4</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R4</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m4"><span class="id" title="variable">m4</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R4"><span class="id" title="variable">R4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R4"><span class="id" title="variable">R4</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="muls_eqmx"><span class="id" title="lemma">muls_eqmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">m4</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>)<br/> - (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R4</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m4"><span class="id" title="variable">m4</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R4"><span class="id" title="variable">R4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R4"><span class="id" title="variable">R4</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmxP"><span class="id" title="lemma">mulsmxP</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#784f0af919f467115774be372bf0dbd7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#784f0af919f467115774be372bf0dbd7"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#784f0af919f467115774be372bf0dbd7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#784f0af919f467115774be372bf0dbd7"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#784f0af919f467115774be372bf0dbd7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmxA"><span class="id" title="lemma">mulsmxA</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmx_addl"><span class="id" title="lemma">mulsmx_addl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmx_addr"><span class="id" title="lemma">mulsmx_addr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">m3</span> <span class="id" title="var">n</span><br/> - (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m3"><span class="id" title="variable">m3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R3"><span class="id" title="variable">R3</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulsmx0"><span class="id" title="lemma">mulsmx0</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="muls0mx"><span class="id" title="lemma">muls0mx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="left_mx_ideal"><span class="id" title="definition">left_mx_ideal</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :=<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="right_mx_ideal"><span class="id" title="definition">right_mx_ideal</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :=<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d438ddb283d0b7a7b8496794bbb81b"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mx_ideal"><span class="id" title="definition">mx_ideal</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#left_mx_ideal"><span class="id" title="definition">left_mx_ideal</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#right_mx_ideal"><span class="id" title="definition">right_mx_ideal</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mxring_id"><span class="id" title="definition">mxring_id</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) <span class="id" title="var">e</span> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">A</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">A</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a>%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="has_mxring_id"><span class="id" title="definition">has_mxring_id</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a><br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#row_mx"><span class="id" title="definition">row_mx</span></a> 0 (<a class="idref" href="mathcomp.algebra.matrix.html#row_mx"><span class="id" title="definition">row_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#mxvec"><span class="id" title="definition">mxvec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>) (<a class="idref" href="mathcomp.algebra.matrix.html#mxvec"><span class="id" title="definition">mxvec</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>))<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#row_mx"><span class="id" title="definition">row_mx</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>) (<a class="idref" href="mathcomp.algebra.matrix.html#row_mx"><span class="id" title="definition">row_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#lin_mx"><span class="id" title="definition">lin_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#mulmx"><span class="id" title="definition">mulmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#lin_mulmx"><span class="id" title="definition">lin_mulmx</span></a>))<br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#lin_mx"><span class="id" title="definition">lin_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#mulmx"><span class="id" title="definition">mulmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#lin_mulmxr"><span class="id" title="definition">lin_mulmxr</span></a>))))%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mxring"><span class="id" title="definition">mxring</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :=<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#left_mx_ideal"><span class="id" title="definition">left_mx_ideal</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#has_mxring_id"><span class="id" title="definition">has_mxring_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxring_idP"><span class="id" title="lemma">mxring_idP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxring_id"><span class="id" title="definition">mxring_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e"><span class="id" title="variable">e</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#has_mxring_id"><span class="id" title="definition">has_mxring_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MatrixAlgebra.CentMxDef"><span class="id" title="section">CentMxDef</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MatrixAlgebra.CentMxDef.m"><span class="id" title="variable">m</span></a> <a name="MatrixAlgebra.CentMxDef.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="MatrixAlgebra.CentMxDef.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e2f1837d9e295cb2494b108fba00dc94"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cent_mx_fun"><span class="id" title="definition">cent_mx_fun</span></a> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">_n</span></a>) := <a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.CentMxDef.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#lin_mx"><span class="id" title="definition">lin_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#mulmxr"><span class="id" title="abbreviation">mulmxr</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#0bb5d11fe8761db1e6494d75cb45a3cb"><span class="id" title="notation">\-</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#mulmx"><span class="id" title="definition">mulmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cent_mx_fun_is_linear"><span class="id" title="lemma">cent_mx_fun_is_linear</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cent_mx_fun"><span class="id" title="definition">cent_mx_fun</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cent_mx_fun_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cent_mx_fun_is_linear"><span class="id" title="lemma">cent_mx_fun_is_linear</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cent_mx_fun_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cent_mx_fun_is_linear"><span class="id" title="lemma">cent_mx_fun_is_linear</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cent_mx"><span class="id" title="definition">cent_mx</span></a> := <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#lin_mx"><span class="id" title="definition">lin_mx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cent_mx_fun"><span class="id" title="definition">cent_mx_fun</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="center_mx"><span class="id" title="definition">center_mx</span></a> := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.CentMxDef.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92683a3ca3b0c0704351ce117beaffe3"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cent_mx"><span class="id" title="definition">cent_mx</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra.CentMxDef"><span class="id" title="section">CentMxDef</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cent_rowP"><span class="id" title="lemma">cent_rowP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">B</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> (<span class="id" title="var">A</span> := <a class="idref" href="mathcomp.algebra.matrix.html#vec_mx"><span class="id" title="definition">vec_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#row"><span class="id" title="definition">row</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>)), <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cent_mxP"><span class="id" title="lemma">cent_mxP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">B</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="scalar_mx_cent"><span class="id" title="lemma">scalar_mx_cent</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">a</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="center_mx_sub"><span class="id" title="lemma">center_mx_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="center_mxP"><span class="id" title="lemma">center_mxP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">B</span>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7aed6abca956bf1cb53b7f7024bbdd8c"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxring_id_uniq"><span class="id" title="lemma">mxring_id_uniq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) <span class="id" title="var">e1</span> <span class="id" title="var">e2</span> :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxring_id"><span class="id" title="definition">mxring_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxring_id"><span class="id" title="definition">mxring_id</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e2"><span class="id" title="variable">e2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cent_mx_ideal"><span class="id" title="lemma">cent_mx_ideal</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#left_mx_ideal"><span class="id" title="definition">left_mx_ideal</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cent_mx_ring"><span class="id" title="lemma">cent_mx_ring</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxring"><span class="id" title="definition">mxring</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5580b74451c106276b6637b4c3d0e16a"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_adds_center"><span class="id" title="lemma">mxdirect_adds_center</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">R1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mx_ideal"><span class="id" title="definition">mx_ideal</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mx_ideal"><span class="id" title="definition">mx_ideal</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R1"><span class="id" title="variable">R1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R2"><span class="id" title="variable">R2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxdirect_sums_center"><span class="id" title="lemma">mxdirect_sums_center</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) <span class="id" title="var">R_</span> :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R_"><span class="id" title="variable">R_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R_"><span class="id" title="variable">R_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#I"><span class="id" title="variable">I</span></a>, <a class="idref" href="mathcomp.algebra.mxalgebra.html#mx_ideal"><span class="id" title="definition">mx_ideal</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#R_"><span class="id" title="variable">R_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c8f30cdc06d84b3164901828b8ce3cb3"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#R_"><span class="id" title="variable">R_</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#59d3f372b35ea6f62351d739453bd3ac"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#MatrixAlgebra"><span class="id" title="section">MatrixAlgebra</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">"</span></a>A \in R" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#submx"><span class="id" title="definition">submx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#mxvec"><span class="id" title="definition">mxvec</span></a> <span class="id" title="var">A</span>) <span class="id" title="var">R</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="17486d1fe34aeecf54f5debb0e4245b6"><span class="id" title="notation">"</span></a>R * S" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#mulsmx"><span class="id" title="definition">mulsmx</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="450fad4e10028541ec558897fa67947d"><span class="id" title="notation">"</span></a>''C' ( R )" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#cent_mx"><span class="id" title="definition">cent_mx</span></a> <span class="id" title="var">R</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="e53b97cf292dbfc23c05100fe9f26e33"><span class="id" title="notation">"</span></a>''C_' R ( S )" := (<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92683a3ca3b0c0704351ce117beaffe3"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="c052b90c6e924804e88f863f45bb47ec"><span class="id" title="notation">"</span></a>''C_' ( R ) ( S )" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#e53b97cf292dbfc23c05100fe9f26e33"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e53b97cf292dbfc23c05100fe9f26e33"><span class="id" title="notation">C_R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e53b97cf292dbfc23c05100fe9f26e33"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#e53b97cf292dbfc23c05100fe9f26e33"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">"</span></a>''Z' ( R )" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#center_mx"><span class="id" title="definition">center_mx</span></a> <span class="id" title="var">R</span>) : <span class="id" title="var">matrix_set_scope</span>.<br/> - -<br/> - -<br/> -</div> - -<div class="doc"> - Parametricity for the row-space/F-algebra theory. -</div> -<div class="code"> -<span class="id" title="keyword">Section</span> <a name="MapMatrixSpaces"><span class="id" title="section">MapMatrixSpaces</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MapMatrixSpaces.aF"><span class="id" title="variable">aF</span></a> <a name="MapMatrixSpaces.rF"><span class="id" title="variable">rF</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="MapMatrixSpaces.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#aF"><span class="id" title="variable">aF</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#rF"><span class="id" title="variable">rF</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Gaussian_elimination_map"><span class="id" title="lemma">Gaussian_elimination_map</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#Gaussian_elimination"><span class="id" title="definition">Gaussian_elimination</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxrank_map"><span class="id" title="lemma">mxrank_map</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b8af73c258a533909a2acba13114d67c"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_free_map"><span class="id" title="lemma">row_free_map</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_free"><span class="id" title="definition">row_free</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="row_full_map"><span class="id" title="lemma">row_full_map</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_full"><span class="id" title="definition">row_full</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_row_ebase"><span class="id" title="lemma">map_row_ebase</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#row_ebase"><span class="id" title="definition">row_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_col_ebase"><span class="id" title="lemma">map_col_ebase</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#col_ebase"><span class="id" title="definition">col_ebase</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_row_base"><span class="id" title="lemma">map_row_base</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#row_base"><span class="id" title="definition">row_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#castmx"><span class="id" title="definition">castmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank_map"><span class="id" title="lemma">mxrank_map</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#erefl"><span class="id" title="abbreviation">erefl</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#row_base"><span class="id" title="definition">row_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_col_base"><span class="id" title="lemma">map_col_base</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#col_base"><span class="id" title="definition">col_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#castmx"><span class="id" title="definition">castmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#erefl"><span class="id" title="abbreviation">erefl</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxrank_map"><span class="id" title="lemma">mxrank_map</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#col_base"><span class="id" title="definition">col_base</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_pinvmx"><span class="id" title="lemma">map_pinvmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#pinvmx"><span class="id" title="definition">pinvmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#pinvmx"><span class="id" title="definition">pinvmx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_kermx"><span class="id" title="lemma">map_kermx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#kermx"><span class="id" title="definition">kermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_cokermx"><span class="id" title="lemma">map_cokermx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#cokermx"><span class="id" title="definition">cokermx</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_submx"><span class="id" title="lemma">map_submx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09a21fbfc35503eeecaca8720742f7ab"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_ltmx"><span class="id" title="lemma">map_ltmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#74f1d33aea43cd94f177c35b7a221cde"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#74f1d33aea43cd94f177c35b7a221cde"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_eqmx"><span class="id" title="lemma">map_eqmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_genmx"><span class="id" title="lemma">map_genmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation">>></span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3962b76563fd8a8f45948950a775860e"><span class="id" title="notation">>></span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_addsmx"><span class="id" title="lemma">map_addsmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_capmx_gen"><span class="id" title="lemma">map_capmx_gen</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#capmx_gen"><span class="id" title="definition">capmx_gen</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_capmx"><span class="id" title="lemma">map_capmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92683a3ca3b0c0704351ce117beaffe3"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92683a3ca3b0c0704351ce117beaffe3"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_complmx"><span class="id" title="lemma">map_complmx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7772cb4a238f5fc3b7cf2f735c00df9d"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7772cb4a238f5fc3b7cf2f735c00df9d"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7772cb4a238f5fc3b7cf2f735c00df9d"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#7772cb4a238f5fc3b7cf2f735c00df9d"><span class="id" title="notation">C</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_diffmx"><span class="id" title="lemma">map_diffmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) :<br/> - (<a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f7be369075cd7f5f2c01766717911ece"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f7be369075cd7f5f2c01766717911ece"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_eigenspace"><span class="id" title="lemma">map_eigenspace</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_n</span></a>) <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenspace"><span class="id" title="definition">eigenspace</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenspace"><span class="id" title="definition">eigenspace</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#g"><span class="id" title="variable">g</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#MapMatrixSpaces.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eigenvalue_map"><span class="id" title="lemma">eigenvalue_map</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_n</span></a>) <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenvalue"><span class="id" title="definition">eigenvalue</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#g"><span class="id" title="variable">g</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#MapMatrixSpaces.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#eigenvalue"><span class="id" title="definition">eigenvalue</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a"><span class="id" title="variable">a</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="memmx_map"><span class="id" title="lemma">memmx_map</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#E"><span class="id" title="variable">E</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_mulsmx"><span class="id" title="lemma">map_mulsmx</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> (<span class="id" title="var">E1</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">E2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) :<br/> - ((<a class="idref" href="mathcomp.algebra.mxalgebra.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#17486d1fe34aeecf54f5debb0e4245b6"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#E2"><span class="id" title="variable">E2</span></a>)%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#E1"><span class="id" title="variable">E1</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#17486d1fe34aeecf54f5debb0e4245b6"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#E2"><span class="id" title="variable">E2</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_cent_mx"><span class="id" title="lemma">map_cent_mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#450fad4e10028541ec558897fa67947d"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_center_mx"><span class="id" title="lemma">map_center_mx</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#76a078f18670eba87f6da45223e154d2"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#64a7caee0a3cf4d848aa409baa5e70f5"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#c6c995a25415413a47df0a8d4a5b9d94"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#MapMatrixSpaces"><span class="id" title="section">MapMatrixSpaces</span></a>.<br/> -</div> -</div> - -<div id="footer"> -<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> -</div> - -</div> - -</body> -</html>
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