Library mathcomp.algebra.mxpoly
(* (c) Copyright 2006-2015 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.
Open Local 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.
Implicit Arguments poly_rV [R d].
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.
Implicit Arguments poly_rV [R d].
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.