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<h1 class="libtitle">Library mathcomp.algebra.mxalgebra</h1>
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<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/>
Distributed under the terms of CeCILL-B. *)</span><br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/>
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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="9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">"</span></a>''A_' ( m , n )" := <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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="4b21ccac6bac793c6b443b82130bd598"><span class="id" title="notation">"</span></a>''A_' ( n )" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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="6a8cb243fd3c774d82487c7bb2ab694b"><span class="id" title="notation">"</span></a>''A_' n" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#4b21ccac6bac793c6b443b82130bd598"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#4b21ccac6bac793c6b443b82130bd598"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#4b21ccac6bac793c6b443b82130bd598"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#4b21ccac6bac793c6b443b82130bd598"><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="d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">"</span></a>''A' [ F ]_ ( m , n )" := <a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">[</span></a><span class="id" title="var">F</span><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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="f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">"</span></a>''A' [ F ]_ ( n )" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">[</span></a><span class="id" title="var">F</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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="dd112c0d02ab419975cf8b2396571f6f"><span class="id" title="notation">"</span></a>''A' [ F ]_ n" := <a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">[</span></a><span class="id" title="var">F</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#f8812adabd55a1f166fba128e2ab67df"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>, <span class="id" title="var">_</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a> ⇒<br/>
<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#705e11e709bb3e1492e885a674508f9a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#705e11e709bb3e1492e885a674508f9a"><span class="id" title="notation">pick</span></a> <span class="id" title="var">ij</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#705e11e709bb3e1492e885a674508f9a"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="mathcomp.ssreflect.fintype.html#705e11e709bb3e1492e885a674508f9a"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">j</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">L</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">U</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">r</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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#d70623330b2787db6b196e37db7d8f45"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">r</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6e9e3868d89c24ddc7d4f643f9ae0df"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6e9e3868d89c24ddc7d4f643f9ae0df"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6e9e3868d89c24ddc7d4f643f9ae0df"><span class="id" title="notation">]</span></a>%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><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#17d28d004d0863cb022d4ce832ddaaae"><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#17d28d004d0863cb022d4ce832ddaaae"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#17d28d004d0863cb022d4ce832ddaaae"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">]</span></a>.<br/>
<br/>
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<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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/>
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<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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#prod"><span class="id" title="inductive">prod</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#650edb4f13965b5befa677e688f789d4"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#650edb4f13965b5befa677e688f789d4"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><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#90d7b09f0c2effdccb449f69b6bb8499"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#17d28d004d0863cb022d4ce832ddaaae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#650edb4f13965b5befa677e688f789d4"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#17d28d004d0863cb022d4ce832ddaaae"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a> :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) ⇒<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a><br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) ⇒<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) ⇒<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#36107ffb18a8bbc5966401cfe543371b"><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#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9b077c369e19739ef880736ba34623ff"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9b077c369e19739ef880736ba34623ff"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</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#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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>.<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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">N</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#0efdd62d084b88acb1b2860eed745d54"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.matrix.html#0efdd62d084b88acb1b2860eed745d54"><span class="id" title="notation">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#81f8078534dcbb7e13a32d292f766525"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#81f8078534dcbb7e13a32d292f766525"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">D</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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>.<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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#90d7b09f0c2effdccb449f69b6bb8499"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/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/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#90d7b09f0c2effdccb449f69b6bb8499"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d7b09f0c2effdccb449f69b6bb8499"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><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#90d7b09f0c2effdccb449f69b6bb8499"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d7b09f0c2effdccb449f69b6bb8499"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#90d7b09f0c2effdccb449f69b6bb8499"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#650edb4f13965b5befa677e688f789d4"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#650edb4f13965b5befa677e688f789d4"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#650edb4f13965b5befa677e688f789d4"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> ((<a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><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#cbc2f2ab11c1c376b5c4511d28b14d74"><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#cbc2f2ab11c1c376b5c4511d28b14d74"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#81f8078534dcbb7e13a32d292f766525"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">rV_n</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#81f8078534dcbb7e13a32d292f766525"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">rV_n</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">rV_n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><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#b1eeadc2feabc7422252baa895418c7b"><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#b1eeadc2feabc7422252baa895418c7b"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#90d7b09f0c2effdccb449f69b6bb8499"><span class="id" title="notation"><</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#0cba8965977f5f50b779d6c4f6af1d2d"><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#650edb4f13965b5befa677e688f789d4"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">rV_n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#650edb4f13965b5befa677e688f789d4"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#650edb4f13965b5befa677e688f789d4"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>, ((<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#81f8078534dcbb7e13a32d292f766525"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#650edb4f13965b5befa677e688f789d4"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation"><<</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation"><<</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>>>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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>.<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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><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#650edb4f13965b5befa677e688f789d4"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#90d7b09f0c2effdccb449f69b6bb8499"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) <span class="id" title="var">e</span> :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#46b28b862ebc4bda38b5188071e2f20c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/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#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/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#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#46b28b862ebc4bda38b5188071e2f20c"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><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#46b28b862ebc4bda38b5188071e2f20c"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#46b28b862ebc4bda38b5188071e2f20c"><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#46b28b862ebc4bda38b5188071e2f20c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#46b28b862ebc4bda38b5188071e2f20c"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#46b28b862ebc4bda38b5188071e2f20c"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#46b28b862ebc4bda38b5188071e2f20c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#46b28b862ebc4bda38b5188071e2f20c"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#46b28b862ebc4bda38b5188071e2f20c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#46b28b862ebc4bda38b5188071e2f20c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#46b28b862ebc4bda38b5188071e2f20c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#46b28b862ebc4bda38b5188071e2f20c"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#ae4d81913e6239182a9ac7467ffde8cd"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation"><<</span></a>(<a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
((<a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><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#622398b62523a74328f94700e42198d0"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u_</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><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#622398b62523a74328f94700e42198d0"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">)</span></a>) :<br/>
((<a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#839dc9de43c30e3fb6956e22c14c882a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">_n</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
((<a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#9b077c369e19739ef880736ba34623ff"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9b077c369e19739ef880736ba34623ff"><span class="id" title="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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9b077c369e19739ef880736ba34623ff"><span class="id" title="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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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#9482aae3d3b06e249765c1225dbb8cbb"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9b077c369e19739ef880736ba34623ff"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</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#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#46b28b862ebc4bda38b5188071e2f20c"><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#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#36107ffb18a8bbc5966401cfe543371b"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#650edb4f13965b5befa677e688f789d4"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) : (1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><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#21ad23dcb443b600284c1cd39a51fde1"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><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#21ad23dcb443b600284c1cd39a51fde1"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><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#21ad23dcb443b600284c1cd39a51fde1"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#21ad23dcb443b600284c1cd39a51fde1"><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#21ad23dcb443b600284c1cd39a51fde1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#839dc9de43c30e3fb6956e22c14c882a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#0cba8965977f5f50b779d6c4f6af1d2d"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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#46b28b862ebc4bda38b5188071e2f20c"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#46b28b862ebc4bda38b5188071e2f20c"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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#46b28b862ebc4bda38b5188071e2f20c"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">C</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#17d28d004d0863cb022d4ce832ddaaae"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#ef798e67335a7fc58f8072263210ea96"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#36107ffb18a8bbc5966401cfe543371b"><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#36107ffb18a8bbc5966401cfe543371b"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#36107ffb18a8bbc5966401cfe543371b"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#839dc9de43c30e3fb6956e22c14c882a"><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#ef798e67335a7fc58f8072263210ea96"><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#839dc9de43c30e3fb6956e22c14c882a"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#ef798e67335a7fc58f8072263210ea96"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#ef798e67335a7fc58f8072263210ea96"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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#ef798e67335a7fc58f8072263210ea96"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#ef798e67335a7fc58f8072263210ea96"><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#46b28b862ebc4bda38b5188071e2f20c"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#ef798e67335a7fc58f8072263210ea96"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#46b28b862ebc4bda38b5188071e2f20c"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#46b28b862ebc4bda38b5188071e2f20c"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><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#2caeadd15d44390ff4fc0106b8e96c32"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#d70623330b2787db6b196e37db7d8f45"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">V</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><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#0cba8965977f5f50b779d6c4f6af1d2d"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#46b28b862ebc4bda38b5188071e2f20c"><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#b3eea360671e1b32b18a26e15b3aace3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#wrapped"><span class="id" title="record">wrapped</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#wrapped"><span class="id" title="record">wrapped</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrfun.html#Wrap"><span class="id" title="constructor">Wrap</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/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/8.8.0/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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/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#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#99ebe9d9a74694e8687b6f8adc097de4"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#99ebe9d9a74694e8687b6f8adc097de4"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#99ebe9d9a74694e8687b6f8adc097de4"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#99ebe9d9a74694e8687b6f8adc097de4"><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#99ebe9d9a74694e8687b6f8adc097de4"><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#99ebe9d9a74694e8687b6f8adc097de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#ea7e35bae15685d5cd3430a8e48be02b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ea7e35bae15685d5cd3430a8e48be02b"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ea7e35bae15685d5cd3430a8e48be02b"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#ea7e35bae15685d5cd3430a8e48be02b"><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#ea7e35bae15685d5cd3430a8e48be02b"><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#ea7e35bae15685d5cd3430a8e48be02b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#phantom"><span class="id" title="inductive">phantom</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phantom"><span class="id" title="constructor">Phantom</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><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/8.8.0/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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">rank</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#3cf4ddd1c78e2b0890c629753f41711f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#08fe8636f4b45ae6787c490d19de1366"><span class="id" title="notation">≥</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/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/8.8.0/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#46b28b862ebc4bda38b5188071e2f20c"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/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/8.8.0/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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#b1eeadc2feabc7422252baa895418c7b"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">M_n</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#b1eeadc2feabc7422252baa895418c7b"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#2caeadd15d44390ff4fc0106b8e96c32"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#unwrap"><span class="id" title="projection">unwrap</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">)</span></a>).<br/>
<br/>
<span class="id" title="keyword">CoInductive</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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#ae4d81913e6239182a9ac7467ffde8cd"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#ae4d81913e6239182a9ac7467ffde8cd"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#2caeadd15d44390ff4fc0106b8e96c32"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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#46b28b862ebc4bda38b5188071e2f20c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/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/8.8.0/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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#e36c289fb249221b43b9c978a67340fb"><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#e36c289fb249221b43b9c978a67340fb"><span class="id" title="notation">,</span></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#e36c289fb249221b43b9c978a67340fb"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">_n</span></a>).<br/>
<br/>
<span class="id" title="keyword">CoInductive</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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><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#622398b62523a74328f94700e42198d0"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><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#622398b62523a74328f94700e42198d0"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/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#93ce48d079482e776b2a4aab92a63444"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#93ce48d079482e776b2a4aab92a63444"><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#d70623330b2787db6b196e37db7d8f45"><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#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><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/8.8.0/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#b1eeadc2feabc7422252baa895418c7b"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><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#88dd4b2c2e6318afa7008df815f99a1a"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#81f8078534dcbb7e13a32d292f766525"><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#92e34d658f9a1b6a4d0f8de14c29c81c"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2bf09b7202225c789149165667752fab"><span class="id" title="notation">rV_n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#81f8078534dcbb7e13a32d292f766525"><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/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><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#b1eeadc2feabc7422252baa895418c7b"><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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#267fe5cb19a4f9c37deb7c452a25ccb6"><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#267fe5cb19a4f9c37deb7c452a25ccb6"><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>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="2841ad707bf668c5fe86250d8f31a3f6"><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="d5ec63f878af68490dd200946b5fc43e"><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="56c42908e63b585e8406ab6296f5d2e9"><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="a83de2bef5d483337931b658f4451b59"><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="9fb9809f0de6e5c70a07575d5458a53e"><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="a1b4ea64f146c6d055d065c894f8cf2a"><span class="id" title="notation">"</span></a>A <= B <= C" := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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="d9ea49cb89580bcf8a5574ea42be6467"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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="330a02895f982a3e7c04616cdee03771"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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="9d384dbebd7f19942980ee8ae3861f46"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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="5e36479739860cd244bd34c609f10109"><span class="id" title="notation">"</span></a>A == B" := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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="996fe23bb3b2a56fc494fe9a0a3c2cd1"><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="3aa1e041eb0c3f581bd44ed53c8f7182"><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="bce3bcafad88bdee58acbfcd89899a28"><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="09728f32fede5dee4dfccad9739422e8"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phantom"><span class="id" title="constructor">Phantom</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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="0fe18f7d3d06ab40e993f8a330b6b36a"><span class="id" title="notation">"</span></a>\sum_ ( i <- r | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><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#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><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#52c4d552b36d01307b4a33177122d4d1"><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="195521b4ce7300b84f06b410b6d69de0"><span class="id" title="notation">"</span></a>\sum_ ( i <- r ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><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#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><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="b0d2f22a1ea60cea1042a164a1a8fe11"><span class="id" title="notation">"</span></a>\sum_ ( m <= i < n | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><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#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><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#f420cd67a470642ef8830577affa92e5"><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="fbd97e292e5e4635d769277a0ae94b8b"><span class="id" title="notation">"</span></a>\sum_ ( m <= i < n ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><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#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><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="5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">"</span></a>\sum_ ( i | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><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#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><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#8850ee6edf9a388b1213678f3d3ee856"><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="8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">"</span></a>\sum_ i B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><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#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><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="eb45384230c3de55a7664b9c512bf78a"><span class="id" title="notation">"</span></a>\sum_ ( i : t | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><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#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><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#ec673a52d55e56af63579baa68d352ee"><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="d616ea0def93d66606af266470d875c4"><span class="id" title="notation">"</span></a>\sum_ ( i : t ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><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#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><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="6304699ff6314894dadbdf939c913ca3"><span class="id" title="notation">"</span></a>\sum_ ( i < n | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><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#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><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#dc42c7ad0ea9096c0f795649807315df"><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="aeaf7456b5f63e1187e10faf83214324"><span class="id" title="notation">"</span></a>\sum_ ( i < n ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><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#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><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="39629a528edaa7245c83047fdb4e6f4e"><span class="id" title="notation">"</span></a>\sum_ ( i 'in' A | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><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#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><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#a9a46078b76c2e36303d504b8fb5bbb3"><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="1aeb4e1e3c663187981498cfc7be766c"><span class="id" title="notation">"</span></a>\sum_ ( i 'in' A ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><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#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">/</span></a>0%<span class="id" title="var">R</span><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><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="100e927ba2b04df0505700ca9d0edd64"><span class="id" title="notation">"</span></a>\bigcap_ ( i <- r | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><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#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><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#52c4d552b36d01307b4a33177122d4d1"><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="845a32e2b6ac0c8cb7ff31d46b5effbe"><span class="id" title="notation">"</span></a>\bigcap_ ( i <- r ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><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#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation"><-</span></a> <span class="id" title="var">r</span><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><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="4c61e2e86aaaed7199d7969c55dde83a"><span class="id" title="notation">"</span></a>\bigcap_ ( m <= i < n | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><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#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#f420cd67a470642ef8830577affa92e5"><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#f420cd67a470642ef8830577affa92e5"><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="50bd67d6e1114dc77e277f97fc8c5e87"><span class="id" title="notation">"</span></a>\bigcap_ ( m <= i < n ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><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#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#db346c83cc8192751cf56eb8b0029d40"><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="832ade9269dc0679b2f47948785795cb"><span class="id" title="notation">"</span></a>\bigcap_ ( i | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><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#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><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#8850ee6edf9a388b1213678f3d3ee856"><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="54e7d9f4b977a8c6a97cf3c43491cf52"><span class="id" title="notation">"</span></a>\bigcap_ i B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><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#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a0ddbff8fbef0617dd5dab072904e591"><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="a0706a1c863da06a9ac8711872d1a241"><span class="id" title="notation">"</span></a>\bigcap_ ( i : t | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><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#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#ec673a52d55e56af63579baa68d352ee"><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#ec673a52d55e56af63579baa68d352ee"><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="14ab555d1cb58909e936ca429ff4e681"><span class="id" title="notation">"</span></a>\bigcap_ ( i : t ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><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#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><span class="id" title="notation">:</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.bigop.html#7c24ccda1da6510c0183e6d456463b39"><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="6d00948f0afb96dc8f2c0b3c2b4f5963"><span class="id" title="notation">"</span></a>\bigcap_ ( i < n | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><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#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><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#dc42c7ad0ea9096c0f795649807315df"><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="cd0b2dfadab058fb8fcd2a17029fa3d9"><span class="id" title="notation">"</span></a>\bigcap_ ( i < n ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><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#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation"><</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><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="e774fc61187b067979331225b8061991"><span class="id" title="notation">"</span></a>\bigcap_ ( i 'in' A | P ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><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#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a9a46078b76c2e36303d504b8fb5bbb3"><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#a9a46078b76c2e36303d504b8fb5bbb3"><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="33d67d3525c53acfe6774eee5cf035af"><span class="id" title="notation">"</span></a>\bigcap_ ( i 'in' A ) B" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><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#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">in</span></a> <span class="id" title="var">A</span><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><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/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><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#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#d5eb23b08bc98c3329b2748a3ba944ae"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d5eb23b08bc98c3329b2748a3ba944ae"><span class="id" title="notation">rV</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d5eb23b08bc98c3329b2748a3ba944ae"><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#d5eb23b08bc98c3329b2748a3ba944ae"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d5eb23b08bc98c3329b2748a3ba944ae"><span class="id" title="notation">_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">GL_n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><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#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">(</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#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">,</span></a> 2<a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation"><</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#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><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#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> 1<a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><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#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">GL_n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><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#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">(</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#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">,</span></a> 2<a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation"><</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#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><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#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> 1<a class="idref" href="mathcomp.ssreflect.bigop.html#a394fade46b35354f996bd50f29a270c"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">GL_1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><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#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">GL_2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><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#9d15bf4f7c2b7d6b94fee6bdd940e5b4"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> 2 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6fd9f66265063abfc4b6d9b6bff4ad18"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6fd9f66265063abfc4b6d9b6bff4ad18"><span class="id" title="notation">GL_n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6fd9f66265063abfc4b6d9b6bff4ad18"><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#6fd9f66265063abfc4b6d9b6bff4ad18"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">(</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#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">,</span></a> 2<a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) : 0 <a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#6a8cb243fd3c774d82487c7bb2ab694b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6a8cb243fd3c774d82487c7bb2ab694b"><span class="id" title="notation">A_n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#1ce49b162eb757fc4a2e0ce4df0ee5cd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1ce49b162eb757fc4a2e0ce4df0ee5cd"><span class="id" title="notation">M_n</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#a83de2bef5d483337931b658f4451b59"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#5e36479739860cd244bd34c609f10109"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">D</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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/8.8.0/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#1ce49b162eb757fc4a2e0ce4df0ee5cd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1ce49b162eb757fc4a2e0ce4df0ee5cd"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A_</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><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#622398b62523a74328f94700e42198d0"><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/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#5bdeaec12a667f4fb2d5ea436c1979c7"><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#5bdeaec12a667f4fb2d5ea436c1979c7"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) : <br/>
(1<a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :=<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#d5ec63f878af68490dd200946b5fc43e"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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#fd554c34965aa6852b3a09f211aef8b0"><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#d5ec63f878af68490dd200946b5fc43e"><span class="id" title="notation">>></span></a>%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#fd554c34965aa6852b3a09f211aef8b0"><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#a83de2bef5d483337931b658f4451b59"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R4</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#a83de2bef5d483337931b658f4451b59"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#a83de2bef5d483337931b658f4451b59"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#fd554c34965aa6852b3a09f211aef8b0"><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#a83de2bef5d483337931b658f4451b59"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>)<br/>
(<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R4</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#fd554c34965aa6852b3a09f211aef8b0"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A1</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">A2</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation"><</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#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><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#fd554c34965aa6852b3a09f211aef8b0"><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#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#fd554c34965aa6852b3a09f211aef8b0"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#fd554c34965aa6852b3a09f211aef8b0"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R3</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#fd554c34965aa6852b3a09f211aef8b0"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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#fd554c34965aa6852b3a09f211aef8b0"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#fd554c34965aa6852b3a09f211aef8b0"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#fd554c34965aa6852b3a09f211aef8b0"><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#a83de2bef5d483337931b658f4451b59"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#fd554c34965aa6852b3a09f211aef8b0"><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#a83de2bef5d483337931b658f4451b59"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) <span class="id" title="var">e</span> :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><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#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">)</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#a83de2bef5d483337931b658f4451b59"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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/8.8.0/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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">A</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d28ed5889dd5bd7699eb66e93fcf8a1c"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d28ed5889dd5bd7699eb66e93fcf8a1c"><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#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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#d837c1a28d718b1ce93b8aa0ad2f20fe"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#d837c1a28d718b1ce93b8aa0ad2f20fe"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#35189e9513aa6c3f11385f2c0e19be6d"><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#bce3bcafad88bdee58acbfcd89899a28"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><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#6f6a817e8af480225304556c4b1a37bc"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><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#6f6a817e8af480225304556c4b1a37bc"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6bc5aad53caab585f4bb088e10501342"><span class="id" title="notation">M</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><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#6f6a817e8af480225304556c4b1a37bc"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a83de2bef5d483337931b658f4451b59"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9c6b777e699b0b93592b907e7450465e"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c6b777e699b0b93592b907e7450465e"><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#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#002b20fea7ea6146f072b539b8de9620"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><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#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><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#6f6a817e8af480225304556c4b1a37bc"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</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#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#6f6a817e8af480225304556c4b1a37bc"><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#6f6a817e8af480225304556c4b1a37bc"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3aa1e041eb0c3f581bd44ed53c8f7182"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) <span class="id" title="var">R_</span> :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><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#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#8aff942cd5cd388036490acbb1397b96"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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#9fdeed72b47a12f54ffaf8faf3f4e8a4"><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/>
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<span class="id" title="keyword">Notation</span> <a name="bfc118b745d1a8ee504472dad1db645c"><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="e1933dd1dc5a439f13af26926cf1eb7b"><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="e5b97b8191a3f5cf8ea44049784b107f"><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="03fcd60414435adb7df518261801fd6f"><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#bce3bcafad88bdee58acbfcd89899a28"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><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="4c1ecc3d7e0e67d99a4d5a27544cc715"><span class="id" title="notation">"</span></a>''C_' ( R ) ( S )" := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#03fcd60414435adb7df518261801fd6f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#03fcd60414435adb7df518261801fd6f"><span class="id" title="notation">C_R</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#03fcd60414435adb7df518261801fd6f"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#03fcd60414435adb7df518261801fd6f"><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="972f5ef28830d59d0a4b5dfdfda2843a"><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/>
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<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#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><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/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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#0c709ebe43ddbd7719f75250a7b916d9"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#2841ad707bf668c5fe86250d8f31a3f6"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#a83de2bef5d483337931b658f4451b59"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#a83de2bef5d483337931b658f4451b59"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#9fb9809f0de6e5c70a07575d5458a53e"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9fb9809f0de6e5c70a07575d5458a53e"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><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#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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#d5ec63f878af68490dd200946b5fc43e"><span class="id" title="notation">>></span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#d5ec63f878af68490dd200946b5fc43e"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#3aa1e041eb0c3f581bd44ed53c8f7182"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#3aa1e041eb0c3f581bd44ed53c8f7182"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) :<br/>
<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#bce3bcafad88bdee58acbfcd89899a28"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#bce3bcafad88bdee58acbfcd89899a28"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><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#56c42908e63b585e8406ab6296f5d2e9"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#56c42908e63b585e8406ab6296f5d2e9"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#56c42908e63b585e8406ab6296f5d2e9"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#56c42908e63b585e8406ab6296f5d2e9"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#cb37620352ad6b90a047a361359e2f04"><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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">,</span></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#cb37620352ad6b90a047a361359e2f04"><span class="id" title="notation">)</span></a>) :<br/>
(<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#09728f32fede5dee4dfccad9739422e8"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#09728f32fede5dee4dfccad9739422e8"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#1ce49b162eb757fc4a2e0ce4df0ee5cd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1ce49b162eb757fc4a2e0ce4df0ee5cd"><span class="id" title="notation">M_n</span></a>) <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#1ce49b162eb757fc4a2e0ce4df0ee5cd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1ce49b162eb757fc4a2e0ce4df0ee5cd"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#bfc118b745d1a8ee504472dad1db645c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#bfc118b745d1a8ee504472dad1db645c"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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#bfc118b745d1a8ee504472dad1db645c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#bfc118b745d1a8ee504472dad1db645c"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">E2</span> : <a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#e1933dd1dc5a439f13af26926cf1eb7b"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e1933dd1dc5a439f13af26926cf1eb7b"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><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#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">)</span></a>%<span class="id" title="var">MS</span><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#e5b97b8191a3f5cf8ea44049784b107f"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">A_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><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#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#9e73b6d2baafc158f1b2b2f6e40fc548"><span class="id" title="notation">)</span></a>) : (<a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><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#972f5ef28830d59d0a4b5dfdfda2843a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#996fe23bb3b2a56fc494fe9a0a3c2cd1"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><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#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b131265909f3a801cb4367eec910d732"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#972f5ef28830d59d0a4b5dfdfda2843a"><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/>
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