Library mathcomp.algebra.mxpoly
+ +
+(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
+ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+
+
++ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+
+ This file provides basic support for formal computation with matrices,
+ mainly results combining matrices and univariate polynomials, such as the
+ Cayley-Hamilton theorem; it also contains an extension of the first order
+ representation of algebra introduced in ssralg (GRing.term/formula).
+ rVpoly v == the little-endian decoding of the row vector v as a
+ polynomial p = \sum_i (v 0 i)%:P * 'X^i.
+ poly_rV p == the partial inverse to rVpoly, for polynomials of degree
+ less than d to 'rV_d (d is inferred from the context).
+ Sylvester_mx p q == the Sylvester matrix of p and q.
+ resultant p q == the resultant of p and q, i.e., \det (Sylvester_mx p q).
+ horner_mx A == the morphism from {poly R} to 'M_n (n of the form n'.+1)
+ mapping a (scalar) polynomial p to the value of its
+ scalar matrix interpretation at A (this is an instance of
+ the generic horner_morph construct defined in poly).
+ powers_mx A d == the d x (n ^ 2) matrix whose rows are the mxvec encodings
+ of the first d powers of A (n of the form n'.+1). Thus,
+ vec_mx (v *m powers_mx A d) = horner_mx A (rVpoly v).
+ char_poly A == the characteristic polynomial of A.
+ char_poly_mx A == a matrix whose detereminant is char_poly A.
+ mxminpoly A == the minimal polynomial of A, i.e., the smallest monic
+ polynomial that annihilates A (A must be nontrivial).
+ degree_mxminpoly A == the (positive) degree of mxminpoly A.
+ mx_inv_horner A == the inverse of horner_mx A for polynomials of degree
+ smaller than degree_mxminpoly A.
+ integralOver RtoK u <-> u is in the integral closure of the image of R
+ under RtoK : R -> K, i.e. u is a root of the image of a
+ monic polynomial in R.
+ algebraicOver FtoE u <-> u : E is algebraic over E; it is a root of the
+ image of a nonzero polynomial under FtoE; as F must be a
+ fieldType, this is equivalent to integralOver FtoE u.
+ integralRange RtoK <-> the integral closure of the image of R contains
+ all of K (:= forall u, integralOver RtoK u).
+ This toolkit for building formal matrix expressions is packaged in the
+ MatrixFormula submodule, and comprises the following:
+ eval_mx e == GRing.eval lifted to matrices (:= map_mx (GRing.eval e)).
+ mx_term A == GRing.Const lifted to matrices.
+ mulmx_term A B == the formal product of two matrices of terms.
+ mxrank_form m A == a GRing.formula asserting that the interpretation of
+ the term matrix A has rank m.
+ submx_form A B == a GRing.formula asserting that the row space of the
+ interpretation of the term matrix A is included in the
+ row space of the interpretation of B.
+ seq_of_rV v == the seq corresponding to a row vector.
+ row_env e == the flattening of a tensored environment e : seq 'rV_d.
+ row_var F d k == the term vector of width d such that for e : seq 'rV[F]_d
+ we have eval e 'X_k = eval_mx (row_env e) (row_var d k).
+
+
+
+
+Set Implicit Arguments.
+ +
+Import GRing.Theory.
+Import Monoid.Theory.
+ +
+Local Open Scope ring_scope.
+ +
+Import Pdiv.Idomain.
+
+
++Set Implicit Arguments.
+ +
+Import GRing.Theory.
+Import Monoid.Theory.
+ +
+Local Open Scope ring_scope.
+ +
+Import Pdiv.Idomain.
+
+ Row vector <-> bounded degree polynomial bijection
+
+
+Section RowPoly.
+ +
+Variables (R : ringType) (d : nat).
+Implicit Types u v : 'rV[R]_d.
+Implicit Types p q : {poly R}.
+ +
+Definition rVpoly v := \poly_(k < d) (if insub k is Some i then v 0 i else 0).
+Definition poly_rV p := \row_(i < d) p`_i.
+ +
+Lemma coef_rVpoly v k : (rVpoly v)`_k = if insub k is Some i then v 0 i else 0.
+ +
+Lemma coef_rVpoly_ord v (i : 'I_d) : (rVpoly v)`_i = v 0 i.
+ +
+Lemma rVpoly_delta i : rVpoly (delta_mx 0 i) = 'X^i.
+ +
+Lemma rVpolyK : cancel rVpoly poly_rV.
+ +
+Lemma poly_rV_K p : size p ≤ d → rVpoly (poly_rV p) = p.
+ +
+Lemma poly_rV_is_linear : linear poly_rV.
+ Canonical poly_rV_additive := Additive poly_rV_is_linear.
+Canonical poly_rV_linear := Linear poly_rV_is_linear.
+ +
+Lemma rVpoly_is_linear : linear rVpoly.
+Canonical rVpoly_additive := Additive rVpoly_is_linear.
+Canonical rVpoly_linear := Linear rVpoly_is_linear.
+ +
+End RowPoly.
+ +
+ +
+Section Resultant.
+ +
+Variables (R : ringType) (p q : {poly R}).
+ +
+Let dS := ((size q).-1 + (size p).-1)%N.
+ +
+Definition Sylvester_mx : 'M[R]_dS := col_mx (band p) (band q).
+ +
+Lemma Sylvester_mxE (i j : 'I_dS) :
+ let S_ r k := r`_(j - k) *+ (k ≤ j) in
+ Sylvester_mx i j = match split i with inl k ⇒ S_ p k | inr k ⇒ S_ q k end.
+ +
+Definition resultant := \det Sylvester_mx.
+ +
+End Resultant.
+ +
+Lemma resultant_in_ideal (R : comRingType) (p q : {poly R}) :
+ size p > 1 → size q > 1 →
+ {uv : {poly R} × {poly R} | size uv.1 < size q ∧ size uv.2 < size p
+ & (resultant p q)%:P = uv.1 × p + uv.2 × q}.
+ +
+Lemma resultant_eq0 (R : idomainType) (p q : {poly R}) :
+ (resultant p q == 0) = (size (gcdp p q) > 1).
+ +
+Section HornerMx.
+ +
+Variables (R : comRingType) (n' : nat).
+Variable A : 'M[R]_n.
+Implicit Types p q : {poly R}.
+ +
+Definition horner_mx := horner_morph (fun a ⇒ scalar_mx_comm a A).
+Canonical horner_mx_additive := [additive of horner_mx].
+Canonical horner_mx_rmorphism := [rmorphism of horner_mx].
+ +
+Lemma horner_mx_C a : horner_mx a%:P = a%:M.
+ +
+Lemma horner_mx_X : horner_mx 'X = A.
+ +
+Lemma horner_mxZ : scalable horner_mx.
+ +
+Canonical horner_mx_linear := AddLinear horner_mxZ.
+Canonical horner_mx_lrmorphism := [lrmorphism of horner_mx].
+ +
+Definition powers_mx d := \matrix_(i < d) mxvec (A ^+ i).
+ +
+Lemma horner_rVpoly m (u : 'rV_m) :
+ horner_mx (rVpoly u) = vec_mx (u ×m powers_mx m).
+ +
+End HornerMx.
+ +
+Section CharPoly.
+ +
+Variables (R : ringType) (n : nat) (A : 'M[R]_n).
+Implicit Types p q : {poly R}.
+ +
+Definition char_poly_mx := 'X%:M - map_mx (@polyC R) A.
+Definition char_poly := \det char_poly_mx.
+ +
+Let diagA := [seq A i i | i : 'I_n].
+Let size_diagA : size diagA = n.
+ +
+Let split_diagA :
+ exists2 q, \prod_(x <- diagA) ('X - x%:P) + q = char_poly & size q ≤ n.-1.
+ +
+Lemma size_char_poly : size char_poly = n.+1.
+ +
+Lemma char_poly_monic : char_poly \is monic.
+ +
+Lemma char_poly_trace : n > 0 → char_poly`_n.-1 = - \tr A.
+ +
+Lemma char_poly_det : char_poly`_0 = (- 1) ^+ n × \det A.
+ +
+End CharPoly.
+ +
+Lemma mx_poly_ring_isom (R : ringType) n' (n := n'.+1) :
+ ∃ phi : {rmorphism 'M[{poly R}]_n → {poly 'M[R]_n}},
+ [/\ bijective phi,
+ ∀ p, phi p%:M = map_poly scalar_mx p,
+ ∀ A, phi (map_mx polyC A) = A%:P
+ & ∀ A i j k, (phi A)`_k i j = (A i j)`_k].
+ +
+Theorem Cayley_Hamilton (R : comRingType) n' (A : 'M[R]_n'.+1) :
+ horner_mx A (char_poly A) = 0.
+ +
+Lemma eigenvalue_root_char (F : fieldType) n (A : 'M[F]_n) a :
+ eigenvalue A a = root (char_poly A) a.
+ +
+Section MinPoly.
+ +
+Variables (F : fieldType) (n' : nat).
+Variable A : 'M[F]_n.
+Implicit Types p q : {poly F}.
+ +
+Fact degree_mxminpoly_proof : ∃ d, \rank (powers_mx A d.+1) ≤ d.
+ Definition degree_mxminpoly := ex_minn degree_mxminpoly_proof.
+ +
+Lemma mxminpoly_nonconstant : d > 0.
+ +
+Lemma minpoly_mx1 : (1%:M \in Ad)%MS.
+ +
+Lemma minpoly_mx_free : row_free Ad.
+ +
+Lemma horner_mx_mem p : (horner_mx A p \in Ad)%MS.
+ +
+Definition mx_inv_horner B := rVpoly (mxvec B ×m pinvmx Ad).
+ +
+Lemma mx_inv_horner0 : mx_inv_horner 0 = 0.
+ +
+Lemma mx_inv_hornerK B : (B \in Ad)%MS → horner_mx A (mx_inv_horner B) = B.
+ +
+Lemma minpoly_mxM B C : (B \in Ad → C \in Ad → B × C \in Ad)%MS.
+ +
+Lemma minpoly_mx_ring : mxring Ad.
+ +
+Definition mxminpoly := 'X^d - mx_inv_horner (A ^+ d).
+ +
+Lemma size_mxminpoly : size p_A = d.+1.
+ +
+Lemma mxminpoly_monic : p_A \is monic.
+ +
+Lemma size_mod_mxminpoly p : size (p %% p_A) ≤ d.
+ +
+Lemma mx_root_minpoly : horner_mx A p_A = 0.
+ +
+Lemma horner_rVpolyK (u : 'rV_d) :
+ mx_inv_horner (horner_mx A (rVpoly u)) = rVpoly u.
+ +
+Lemma horner_mxK p : mx_inv_horner (horner_mx A p) = p %% p_A.
+ +
+Lemma mxminpoly_min p : horner_mx A p = 0 → p_A %| p.
+ +
+Lemma horner_rVpoly_inj : @injective 'M_n 'rV_d (horner_mx A \o rVpoly).
+ +
+Lemma mxminpoly_linear_is_scalar : (d ≤ 1) = is_scalar_mx A.
+ +
+Lemma mxminpoly_dvd_char : p_A %| char_poly A.
+ +
+Lemma eigenvalue_root_min a : eigenvalue A a = root p_A a.
+ +
+End MinPoly.
+ +
+
+
++ +
+Variables (R : ringType) (d : nat).
+Implicit Types u v : 'rV[R]_d.
+Implicit Types p q : {poly R}.
+ +
+Definition rVpoly v := \poly_(k < d) (if insub k is Some i then v 0 i else 0).
+Definition poly_rV p := \row_(i < d) p`_i.
+ +
+Lemma coef_rVpoly v k : (rVpoly v)`_k = if insub k is Some i then v 0 i else 0.
+ +
+Lemma coef_rVpoly_ord v (i : 'I_d) : (rVpoly v)`_i = v 0 i.
+ +
+Lemma rVpoly_delta i : rVpoly (delta_mx 0 i) = 'X^i.
+ +
+Lemma rVpolyK : cancel rVpoly poly_rV.
+ +
+Lemma poly_rV_K p : size p ≤ d → rVpoly (poly_rV p) = p.
+ +
+Lemma poly_rV_is_linear : linear poly_rV.
+ Canonical poly_rV_additive := Additive poly_rV_is_linear.
+Canonical poly_rV_linear := Linear poly_rV_is_linear.
+ +
+Lemma rVpoly_is_linear : linear rVpoly.
+Canonical rVpoly_additive := Additive rVpoly_is_linear.
+Canonical rVpoly_linear := Linear rVpoly_is_linear.
+ +
+End RowPoly.
+ +
+ +
+Section Resultant.
+ +
+Variables (R : ringType) (p q : {poly R}).
+ +
+Let dS := ((size q).-1 + (size p).-1)%N.
+ +
+Definition Sylvester_mx : 'M[R]_dS := col_mx (band p) (band q).
+ +
+Lemma Sylvester_mxE (i j : 'I_dS) :
+ let S_ r k := r`_(j - k) *+ (k ≤ j) in
+ Sylvester_mx i j = match split i with inl k ⇒ S_ p k | inr k ⇒ S_ q k end.
+ +
+Definition resultant := \det Sylvester_mx.
+ +
+End Resultant.
+ +
+Lemma resultant_in_ideal (R : comRingType) (p q : {poly R}) :
+ size p > 1 → size q > 1 →
+ {uv : {poly R} × {poly R} | size uv.1 < size q ∧ size uv.2 < size p
+ & (resultant p q)%:P = uv.1 × p + uv.2 × q}.
+ +
+Lemma resultant_eq0 (R : idomainType) (p q : {poly R}) :
+ (resultant p q == 0) = (size (gcdp p q) > 1).
+ +
+Section HornerMx.
+ +
+Variables (R : comRingType) (n' : nat).
+Variable A : 'M[R]_n.
+Implicit Types p q : {poly R}.
+ +
+Definition horner_mx := horner_morph (fun a ⇒ scalar_mx_comm a A).
+Canonical horner_mx_additive := [additive of horner_mx].
+Canonical horner_mx_rmorphism := [rmorphism of horner_mx].
+ +
+Lemma horner_mx_C a : horner_mx a%:P = a%:M.
+ +
+Lemma horner_mx_X : horner_mx 'X = A.
+ +
+Lemma horner_mxZ : scalable horner_mx.
+ +
+Canonical horner_mx_linear := AddLinear horner_mxZ.
+Canonical horner_mx_lrmorphism := [lrmorphism of horner_mx].
+ +
+Definition powers_mx d := \matrix_(i < d) mxvec (A ^+ i).
+ +
+Lemma horner_rVpoly m (u : 'rV_m) :
+ horner_mx (rVpoly u) = vec_mx (u ×m powers_mx m).
+ +
+End HornerMx.
+ +
+Section CharPoly.
+ +
+Variables (R : ringType) (n : nat) (A : 'M[R]_n).
+Implicit Types p q : {poly R}.
+ +
+Definition char_poly_mx := 'X%:M - map_mx (@polyC R) A.
+Definition char_poly := \det char_poly_mx.
+ +
+Let diagA := [seq A i i | i : 'I_n].
+Let size_diagA : size diagA = n.
+ +
+Let split_diagA :
+ exists2 q, \prod_(x <- diagA) ('X - x%:P) + q = char_poly & size q ≤ n.-1.
+ +
+Lemma size_char_poly : size char_poly = n.+1.
+ +
+Lemma char_poly_monic : char_poly \is monic.
+ +
+Lemma char_poly_trace : n > 0 → char_poly`_n.-1 = - \tr A.
+ +
+Lemma char_poly_det : char_poly`_0 = (- 1) ^+ n × \det A.
+ +
+End CharPoly.
+ +
+Lemma mx_poly_ring_isom (R : ringType) n' (n := n'.+1) :
+ ∃ phi : {rmorphism 'M[{poly R}]_n → {poly 'M[R]_n}},
+ [/\ bijective phi,
+ ∀ p, phi p%:M = map_poly scalar_mx p,
+ ∀ A, phi (map_mx polyC A) = A%:P
+ & ∀ A i j k, (phi A)`_k i j = (A i j)`_k].
+ +
+Theorem Cayley_Hamilton (R : comRingType) n' (A : 'M[R]_n'.+1) :
+ horner_mx A (char_poly A) = 0.
+ +
+Lemma eigenvalue_root_char (F : fieldType) n (A : 'M[F]_n) a :
+ eigenvalue A a = root (char_poly A) a.
+ +
+Section MinPoly.
+ +
+Variables (F : fieldType) (n' : nat).
+Variable A : 'M[F]_n.
+Implicit Types p q : {poly F}.
+ +
+Fact degree_mxminpoly_proof : ∃ d, \rank (powers_mx A d.+1) ≤ d.
+ Definition degree_mxminpoly := ex_minn degree_mxminpoly_proof.
+ +
+Lemma mxminpoly_nonconstant : d > 0.
+ +
+Lemma minpoly_mx1 : (1%:M \in Ad)%MS.
+ +
+Lemma minpoly_mx_free : row_free Ad.
+ +
+Lemma horner_mx_mem p : (horner_mx A p \in Ad)%MS.
+ +
+Definition mx_inv_horner B := rVpoly (mxvec B ×m pinvmx Ad).
+ +
+Lemma mx_inv_horner0 : mx_inv_horner 0 = 0.
+ +
+Lemma mx_inv_hornerK B : (B \in Ad)%MS → horner_mx A (mx_inv_horner B) = B.
+ +
+Lemma minpoly_mxM B C : (B \in Ad → C \in Ad → B × C \in Ad)%MS.
+ +
+Lemma minpoly_mx_ring : mxring Ad.
+ +
+Definition mxminpoly := 'X^d - mx_inv_horner (A ^+ d).
+ +
+Lemma size_mxminpoly : size p_A = d.+1.
+ +
+Lemma mxminpoly_monic : p_A \is monic.
+ +
+Lemma size_mod_mxminpoly p : size (p %% p_A) ≤ d.
+ +
+Lemma mx_root_minpoly : horner_mx A p_A = 0.
+ +
+Lemma horner_rVpolyK (u : 'rV_d) :
+ mx_inv_horner (horner_mx A (rVpoly u)) = rVpoly u.
+ +
+Lemma horner_mxK p : mx_inv_horner (horner_mx A p) = p %% p_A.
+ +
+Lemma mxminpoly_min p : horner_mx A p = 0 → p_A %| p.
+ +
+Lemma horner_rVpoly_inj : @injective 'M_n 'rV_d (horner_mx A \o rVpoly).
+ +
+Lemma mxminpoly_linear_is_scalar : (d ≤ 1) = is_scalar_mx A.
+ +
+Lemma mxminpoly_dvd_char : p_A %| char_poly A.
+ +
+Lemma eigenvalue_root_min a : eigenvalue A a = root p_A a.
+ +
+End MinPoly.
+ +
+
+ Parametricity.
+
+
+Section MapRingMatrix.
+ +
+Variables (aR rR : ringType) (f : {rmorphism aR → rR}).
+Variables (d n : nat) (A : 'M[aR]_n).
+ +
+Lemma map_rVpoly (u : 'rV_d) : fp (rVpoly u) = rVpoly u^f.
+ +
+Lemma map_poly_rV p : (poly_rV p)^f = poly_rV (fp p) :> 'rV_d.
+ +
+Lemma map_char_poly_mx : map_mx fp (char_poly_mx A) = char_poly_mx A^f.
+ +
+Lemma map_char_poly : fp (char_poly A) = char_poly A^f.
+ +
+End MapRingMatrix.
+ +
+Section MapResultant.
+ +
+Lemma map_resultant (aR rR : ringType) (f : {rmorphism {poly aR} → rR}) p q :
+ f (lead_coef p) != 0 → f (lead_coef q) != 0 →
+ f (resultant p q)= resultant (map_poly f p) (map_poly f q).
+ +
+End MapResultant.
+ +
+Section MapComRing.
+ +
+Variables (aR rR : comRingType) (f : {rmorphism aR → rR}).
+Variables (n' : nat) (A : 'M[aR]_n'.+1).
+ +
+Lemma map_powers_mx e : (powers_mx A e)^f = powers_mx A^f e.
+ +
+Lemma map_horner_mx p : (horner_mx A p)^f = horner_mx A^f (fp p).
+ +
+End MapComRing.
+ +
+Section MapField.
+ +
+Variables (aF rF : fieldType) (f : {rmorphism aF → rF}).
+Variables (n' : nat) (A : 'M[aF]_n'.+1).
+ +
+Lemma degree_mxminpoly_map : degree_mxminpoly A^f = degree_mxminpoly A.
+ +
+Lemma mxminpoly_map : mxminpoly A^f = fp (mxminpoly A).
+ +
+Lemma map_mx_inv_horner u : fp (mx_inv_horner A u) = mx_inv_horner A^f u^f.
+ +
+End MapField.
+ +
+Section IntegralOverRing.
+ +
+Definition integralOver (R K : ringType) (RtoK : R → K) (z : K) :=
+ exists2 p, p \is monic & root (map_poly RtoK p) z.
+ +
+Definition integralRange R K RtoK := ∀ z, @integralOver R K RtoK z.
+ +
+Variables (B R K : ringType) (BtoR : B → R) (RtoK : {rmorphism R → K}).
+ +
+Lemma integral_rmorph x :
+ integralOver BtoR x → integralOver (RtoK \o BtoR) (RtoK x).
+ +
+Lemma integral_id x : integralOver RtoK (RtoK x).
+ +
+Lemma integral_nat n : integralOver RtoK n%:R.
+ +
+Lemma integral0 : integralOver RtoK 0.
+ +
+Lemma integral1 : integralOver RtoK 1.
+ +
+Lemma integral_poly (p : {poly K}) :
+ (∀ i, integralOver RtoK p`_i) ↔ {in p : seq K, integralRange RtoK}.
+ +
+End IntegralOverRing.
+ +
+Section IntegralOverComRing.
+ +
+Variables (R K : comRingType) (RtoK : {rmorphism R → K}).
+ +
+Lemma integral_horner_root w (p q : {poly K}) :
+ p \is monic → root p w →
+ {in p : seq K, integralRange RtoK} → {in q : seq K, integralRange RtoK} →
+ integralOver RtoK q.[w].
+ +
+Lemma integral_root_monic u p :
+ p \is monic → root p u → {in p : seq K, integralRange RtoK} →
+ integralOver RtoK u.
+ +
+Hint Resolve (integral0 RtoK) (integral1 RtoK) (@monicXsubC K).
+ +
+Let XsubC0 (u : K) : root ('X - u%:P) u.
+Let intR_XsubC u :
+ integralOver RtoK (- u) → {in 'X - u%:P : seq K, integralRange RtoK}.
+ +
+Lemma integral_opp u : integralOver RtoK u → integralOver RtoK (- u).
+ +
+Lemma integral_horner (p : {poly K}) u :
+ {in p : seq K, integralRange RtoK} → integralOver RtoK u →
+ integralOver RtoK p.[u].
+ +
+Lemma integral_sub u v :
+ integralOver RtoK u → integralOver RtoK v → integralOver RtoK (u - v).
+ +
+Lemma integral_add u v :
+ integralOver RtoK u → integralOver RtoK v → integralOver RtoK (u + v).
+ +
+Lemma integral_mul u v :
+ integralOver RtoK u → integralOver RtoK v → integralOver RtoK (u × v).
+ +
+End IntegralOverComRing.
+ +
+Section IntegralOverField.
+ +
+Variables (F E : fieldType) (FtoE : {rmorphism F → E}).
+ +
+Definition algebraicOver (fFtoE : F → E) u :=
+ exists2 p, p != 0 & root (map_poly fFtoE p) u.
+ +
+Notation mk_mon p := ((lead_coef p)^-1 *: p).
+ +
+Lemma integral_algebraic u : algebraicOver FtoE u ↔ integralOver FtoE u.
+ +
+Lemma algebraic_id a : algebraicOver FtoE (FtoE a).
+ +
+Lemma algebraic0 : algebraicOver FtoE 0.
+ +
+Lemma algebraic1 : algebraicOver FtoE 1.
+ +
+Lemma algebraic_opp x : algebraicOver FtoE x → algebraicOver FtoE (- x).
+ +
+Lemma algebraic_add x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x + y).
+ +
+Lemma algebraic_sub x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x - y).
+ +
+Lemma algebraic_mul x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x × y).
+ +
+Lemma algebraic_inv u : algebraicOver FtoE u → algebraicOver FtoE u^-1.
+ +
+Lemma algebraic_div x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x / y).
+ +
+Lemma integral_inv x : integralOver FtoE x → integralOver FtoE x^-1.
+ +
+Lemma integral_div x y :
+ integralOver FtoE x → integralOver FtoE y → integralOver FtoE (x / y).
+ +
+Lemma integral_root p u :
+ p != 0 → root p u → {in p : seq E, integralRange FtoE} →
+ integralOver FtoE u.
+ +
+End IntegralOverField.
+ +
+
+
++ +
+Variables (aR rR : ringType) (f : {rmorphism aR → rR}).
+Variables (d n : nat) (A : 'M[aR]_n).
+ +
+Lemma map_rVpoly (u : 'rV_d) : fp (rVpoly u) = rVpoly u^f.
+ +
+Lemma map_poly_rV p : (poly_rV p)^f = poly_rV (fp p) :> 'rV_d.
+ +
+Lemma map_char_poly_mx : map_mx fp (char_poly_mx A) = char_poly_mx A^f.
+ +
+Lemma map_char_poly : fp (char_poly A) = char_poly A^f.
+ +
+End MapRingMatrix.
+ +
+Section MapResultant.
+ +
+Lemma map_resultant (aR rR : ringType) (f : {rmorphism {poly aR} → rR}) p q :
+ f (lead_coef p) != 0 → f (lead_coef q) != 0 →
+ f (resultant p q)= resultant (map_poly f p) (map_poly f q).
+ +
+End MapResultant.
+ +
+Section MapComRing.
+ +
+Variables (aR rR : comRingType) (f : {rmorphism aR → rR}).
+Variables (n' : nat) (A : 'M[aR]_n'.+1).
+ +
+Lemma map_powers_mx e : (powers_mx A e)^f = powers_mx A^f e.
+ +
+Lemma map_horner_mx p : (horner_mx A p)^f = horner_mx A^f (fp p).
+ +
+End MapComRing.
+ +
+Section MapField.
+ +
+Variables (aF rF : fieldType) (f : {rmorphism aF → rF}).
+Variables (n' : nat) (A : 'M[aF]_n'.+1).
+ +
+Lemma degree_mxminpoly_map : degree_mxminpoly A^f = degree_mxminpoly A.
+ +
+Lemma mxminpoly_map : mxminpoly A^f = fp (mxminpoly A).
+ +
+Lemma map_mx_inv_horner u : fp (mx_inv_horner A u) = mx_inv_horner A^f u^f.
+ +
+End MapField.
+ +
+Section IntegralOverRing.
+ +
+Definition integralOver (R K : ringType) (RtoK : R → K) (z : K) :=
+ exists2 p, p \is monic & root (map_poly RtoK p) z.
+ +
+Definition integralRange R K RtoK := ∀ z, @integralOver R K RtoK z.
+ +
+Variables (B R K : ringType) (BtoR : B → R) (RtoK : {rmorphism R → K}).
+ +
+Lemma integral_rmorph x :
+ integralOver BtoR x → integralOver (RtoK \o BtoR) (RtoK x).
+ +
+Lemma integral_id x : integralOver RtoK (RtoK x).
+ +
+Lemma integral_nat n : integralOver RtoK n%:R.
+ +
+Lemma integral0 : integralOver RtoK 0.
+ +
+Lemma integral1 : integralOver RtoK 1.
+ +
+Lemma integral_poly (p : {poly K}) :
+ (∀ i, integralOver RtoK p`_i) ↔ {in p : seq K, integralRange RtoK}.
+ +
+End IntegralOverRing.
+ +
+Section IntegralOverComRing.
+ +
+Variables (R K : comRingType) (RtoK : {rmorphism R → K}).
+ +
+Lemma integral_horner_root w (p q : {poly K}) :
+ p \is monic → root p w →
+ {in p : seq K, integralRange RtoK} → {in q : seq K, integralRange RtoK} →
+ integralOver RtoK q.[w].
+ +
+Lemma integral_root_monic u p :
+ p \is monic → root p u → {in p : seq K, integralRange RtoK} →
+ integralOver RtoK u.
+ +
+Hint Resolve (integral0 RtoK) (integral1 RtoK) (@monicXsubC K).
+ +
+Let XsubC0 (u : K) : root ('X - u%:P) u.
+Let intR_XsubC u :
+ integralOver RtoK (- u) → {in 'X - u%:P : seq K, integralRange RtoK}.
+ +
+Lemma integral_opp u : integralOver RtoK u → integralOver RtoK (- u).
+ +
+Lemma integral_horner (p : {poly K}) u :
+ {in p : seq K, integralRange RtoK} → integralOver RtoK u →
+ integralOver RtoK p.[u].
+ +
+Lemma integral_sub u v :
+ integralOver RtoK u → integralOver RtoK v → integralOver RtoK (u - v).
+ +
+Lemma integral_add u v :
+ integralOver RtoK u → integralOver RtoK v → integralOver RtoK (u + v).
+ +
+Lemma integral_mul u v :
+ integralOver RtoK u → integralOver RtoK v → integralOver RtoK (u × v).
+ +
+End IntegralOverComRing.
+ +
+Section IntegralOverField.
+ +
+Variables (F E : fieldType) (FtoE : {rmorphism F → E}).
+ +
+Definition algebraicOver (fFtoE : F → E) u :=
+ exists2 p, p != 0 & root (map_poly fFtoE p) u.
+ +
+Notation mk_mon p := ((lead_coef p)^-1 *: p).
+ +
+Lemma integral_algebraic u : algebraicOver FtoE u ↔ integralOver FtoE u.
+ +
+Lemma algebraic_id a : algebraicOver FtoE (FtoE a).
+ +
+Lemma algebraic0 : algebraicOver FtoE 0.
+ +
+Lemma algebraic1 : algebraicOver FtoE 1.
+ +
+Lemma algebraic_opp x : algebraicOver FtoE x → algebraicOver FtoE (- x).
+ +
+Lemma algebraic_add x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x + y).
+ +
+Lemma algebraic_sub x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x - y).
+ +
+Lemma algebraic_mul x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x × y).
+ +
+Lemma algebraic_inv u : algebraicOver FtoE u → algebraicOver FtoE u^-1.
+ +
+Lemma algebraic_div x y :
+ algebraicOver FtoE x → algebraicOver FtoE y → algebraicOver FtoE (x / y).
+ +
+Lemma integral_inv x : integralOver FtoE x → integralOver FtoE x^-1.
+ +
+Lemma integral_div x y :
+ integralOver FtoE x → integralOver FtoE y → integralOver FtoE (x / y).
+ +
+Lemma integral_root p u :
+ p != 0 → root p u → {in p : seq E, integralRange FtoE} →
+ integralOver FtoE u.
+ +
+End IntegralOverField.
+ +
+
+ Lifting term, formula, envs and eval to matrices. Wlog, and for the sake
+ of simplicity, we only lift (tensor) envs to row vectors; we can always
+ use mxvec/vec_mx to store and retrieve matrices.
+ We don't provide definitions for addition, subtraction, scaling, etc,
+ because they have simple matrix expressions.
+
+
+Module MatrixFormula.
+ +
+Section MatrixFormula.
+ +
+Variable F : fieldType.
+ +
+ +
+Definition eval_mx (e : seq F) := map_mx (eval e).
+ +
+Definition mx_term := map_mx (@GRing.Const F).
+ +
+Lemma eval_mx_term e m n (A : 'M_(m, n)) : eval_mx e (mx_term A) = A.
+ +
+Definition mulmx_term m n p (A : 'M[term]_(m, n)) (B : 'M_(n, p)) :=
+ \matrix_(i, k) (\big[Add/0]_j (A i j × B j k))%T.
+ +
+Lemma eval_mulmx e m n p (A : 'M[term]_(m, n)) (B : 'M_(n, p)) :
+ eval_mx e (mulmx_term A B) = eval_mx e A ×m eval_mx e B.
+ +
+ +
+Let Schur m n (A : 'M[term]_(1 + m, 1 + n)) (a := A 0 0) :=
+ \matrix_(i, j) (drsubmx A i j - a^-1 × dlsubmx A i 0%R × ursubmx A 0%R j)%T.
+ +
+Fixpoint mxrank_form (r m n : nat) : 'M_(m, n) → form :=
+ match m, n return 'M_(m, n) → form with
+ | m'.+1, n'.+1 ⇒ fun A : 'M_(1 + m', 1 + n') ⇒
+ let nzA k := A k.1 k.2 != 0 in
+ let xSchur k := Schur (xrow k.1 0%R (xcol k.2 0%R A)) in
+ let recf k := Bool (r > 0) ∧ mxrank_form r.-1 (xSchur k) in
+ GRing.Pick nzA recf (Bool (r == 0%N))
+ | _, _ ⇒ fun _ ⇒ Bool (r == 0%N)
+ end%T.
+ +
+Lemma mxrank_form_qf r m n (A : 'M_(m, n)) : qf_form (mxrank_form r A).
+ +
+Lemma eval_mxrank e r m n (A : 'M_(m, n)) :
+ qf_eval e (mxrank_form r A) = (\rank (eval_mx e A) == r).
+ +
+Lemma eval_vec_mx e m n (u : 'rV_(m × n)) :
+ eval_mx e (vec_mx u) = vec_mx (eval_mx e u).
+ +
+Lemma eval_mxvec e m n (A : 'M_(m, n)) :
+ eval_mx e (mxvec A) = mxvec (eval_mx e A).
+ +
+Section Subsetmx.
+ +
+Variables (m1 m2 n : nat) (A : 'M[term]_(m1, n)) (B : 'M[term]_(m2, n)).
+ +
+Definition submx_form :=
+ \big[And/True]_(r < n.+1) (mxrank_form r (col_mx A B) ==> mxrank_form r B)%T.
+ +
+Lemma eval_col_mx e :
+ eval_mx e (col_mx A B) = col_mx (eval_mx e A) (eval_mx e B).
+ +
+Lemma submx_form_qf : qf_form submx_form.
+ +
+Lemma eval_submx e : qf_eval e submx_form = (eval_mx e A ≤ eval_mx e B)%MS.
+ +
+End Subsetmx.
+ +
+Section Env.
+ +
+Variable d : nat.
+ +
+Definition seq_of_rV (v : 'rV_d) : seq F := fgraph [ffun i ⇒ v 0 i].
+ +
+Lemma size_seq_of_rV v : size (seq_of_rV v) = d.
+ +
+Lemma nth_seq_of_rV x0 v (i : 'I_d) : nth x0 (seq_of_rV v) i = v 0 i.
+ +
+Definition row_var k : 'rV[term]_d := \row_i ('X_(k × d + i))%T.
+ +
+Definition row_env (e : seq 'rV_d) := flatten (map seq_of_rV e).
+ +
+Lemma nth_row_env e k (i : 'I_d) : (row_env e)`_(k × d + i) = e`_k 0 i.
+ +
+Lemma eval_row_var e k : eval_mx (row_env e) (row_var k) = e`_k :> 'rV_d.
+ +
+Definition Exists_row_form k (f : form) :=
+ foldr GRing.Exists f (codom (fun i : 'I_d ⇒ k × d + i)%N).
+ +
+Lemma Exists_rowP e k f :
+ d > 0 →
+ ((∃ v : 'rV[F]_d, holds (row_env (set_nth 0 e k v)) f)
+ ↔ holds (row_env e) (Exists_row_form k f)).
+ +
+End Env.
+ +
+End MatrixFormula.
+ +
+End MatrixFormula.
+
++ +
+Section MatrixFormula.
+ +
+Variable F : fieldType.
+ +
+ +
+Definition eval_mx (e : seq F) := map_mx (eval e).
+ +
+Definition mx_term := map_mx (@GRing.Const F).
+ +
+Lemma eval_mx_term e m n (A : 'M_(m, n)) : eval_mx e (mx_term A) = A.
+ +
+Definition mulmx_term m n p (A : 'M[term]_(m, n)) (B : 'M_(n, p)) :=
+ \matrix_(i, k) (\big[Add/0]_j (A i j × B j k))%T.
+ +
+Lemma eval_mulmx e m n p (A : 'M[term]_(m, n)) (B : 'M_(n, p)) :
+ eval_mx e (mulmx_term A B) = eval_mx e A ×m eval_mx e B.
+ +
+ +
+Let Schur m n (A : 'M[term]_(1 + m, 1 + n)) (a := A 0 0) :=
+ \matrix_(i, j) (drsubmx A i j - a^-1 × dlsubmx A i 0%R × ursubmx A 0%R j)%T.
+ +
+Fixpoint mxrank_form (r m n : nat) : 'M_(m, n) → form :=
+ match m, n return 'M_(m, n) → form with
+ | m'.+1, n'.+1 ⇒ fun A : 'M_(1 + m', 1 + n') ⇒
+ let nzA k := A k.1 k.2 != 0 in
+ let xSchur k := Schur (xrow k.1 0%R (xcol k.2 0%R A)) in
+ let recf k := Bool (r > 0) ∧ mxrank_form r.-1 (xSchur k) in
+ GRing.Pick nzA recf (Bool (r == 0%N))
+ | _, _ ⇒ fun _ ⇒ Bool (r == 0%N)
+ end%T.
+ +
+Lemma mxrank_form_qf r m n (A : 'M_(m, n)) : qf_form (mxrank_form r A).
+ +
+Lemma eval_mxrank e r m n (A : 'M_(m, n)) :
+ qf_eval e (mxrank_form r A) = (\rank (eval_mx e A) == r).
+ +
+Lemma eval_vec_mx e m n (u : 'rV_(m × n)) :
+ eval_mx e (vec_mx u) = vec_mx (eval_mx e u).
+ +
+Lemma eval_mxvec e m n (A : 'M_(m, n)) :
+ eval_mx e (mxvec A) = mxvec (eval_mx e A).
+ +
+Section Subsetmx.
+ +
+Variables (m1 m2 n : nat) (A : 'M[term]_(m1, n)) (B : 'M[term]_(m2, n)).
+ +
+Definition submx_form :=
+ \big[And/True]_(r < n.+1) (mxrank_form r (col_mx A B) ==> mxrank_form r B)%T.
+ +
+Lemma eval_col_mx e :
+ eval_mx e (col_mx A B) = col_mx (eval_mx e A) (eval_mx e B).
+ +
+Lemma submx_form_qf : qf_form submx_form.
+ +
+Lemma eval_submx e : qf_eval e submx_form = (eval_mx e A ≤ eval_mx e B)%MS.
+ +
+End Subsetmx.
+ +
+Section Env.
+ +
+Variable d : nat.
+ +
+Definition seq_of_rV (v : 'rV_d) : seq F := fgraph [ffun i ⇒ v 0 i].
+ +
+Lemma size_seq_of_rV v : size (seq_of_rV v) = d.
+ +
+Lemma nth_seq_of_rV x0 v (i : 'I_d) : nth x0 (seq_of_rV v) i = v 0 i.
+ +
+Definition row_var k : 'rV[term]_d := \row_i ('X_(k × d + i))%T.
+ +
+Definition row_env (e : seq 'rV_d) := flatten (map seq_of_rV e).
+ +
+Lemma nth_row_env e k (i : 'I_d) : (row_env e)`_(k × d + i) = e`_k 0 i.
+ +
+Lemma eval_row_var e k : eval_mx (row_env e) (row_var k) = e`_k :> 'rV_d.
+ +
+Definition Exists_row_form k (f : form) :=
+ foldr GRing.Exists f (codom (fun i : 'I_d ⇒ k × d + i)%N).
+ +
+Lemma Exists_rowP e k f :
+ d > 0 →
+ ((∃ v : 'rV[F]_d, holds (row_env (set_nth 0 e k v)) f)
+ ↔ holds (row_env e) (Exists_row_form k f)).
+ +
+End Env.
+ +
+End MatrixFormula.
+ +
+End MatrixFormula.
+