statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
integral_opp u : integralOver RtoK u -> integralOver RtoK (- u). | Proof. by rewrite -{1}[u]opprK => /intR_XsubC/integral_root_monic; apply. Qed. | Lemma | integral_opp | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"intR_XsubC",
"integralOver",
"integral_root_monic",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_horner (p : {poly K}) u :
{in p : seq K, integralRange RtoK} -> integralOver RtoK u ->
integralOver RtoK p.[u]. | Proof. by move=> ? /integral_opp/intR_XsubC/integral_horner_root; apply. Qed. | Lemma | integral_horner | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"intR_XsubC",
"integralOver",
"integralRange",
"integral_horner_root",
"integral_opp",
"poly",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_sub u v :
integralOver RtoK u -> integralOver RtoK v -> integralOver RtoK (u - v). | Proof.
move=> intRu /integral_opp/intR_XsubC/integral_horner/(_ intRu).
by rewrite !hornerE.
Qed. | Lemma | integral_sub | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"hornerE",
"intR_XsubC",
"integralOver",
"integral_horner",
"integral_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_add u v :
integralOver RtoK u -> integralOver RtoK v -> integralOver RtoK (u + v). | Proof. by rewrite -{2}[v]opprK => intRu /integral_opp; apply: integral_sub. Qed. | Lemma | integral_add | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"integralOver",
"integral_opp",
"integral_sub",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_mul u v :
integralOver RtoK u -> integralOver RtoK v -> integralOver RtoK (u * v). | Proof.
rewrite -{2}[v]hornerX -hornerZ => intRu; apply: integral_horner.
by apply/integral_poly=> i; rewrite coefZ coefX mulr_natr mulrb; case: ifP.
Qed. | Lemma | integral_mul | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coefX",
"coefZ",
"hornerX",
"hornerZ",
"integralOver",
"integral_horner",
"integral_poly",
"mulr_natr",
"mulrb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraicOver (fFtoE : F -> E) u | :=
exists2 p, p != 0 & root (map_poly fFtoE p) u. | Definition | algebraicOver | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"map_poly",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk_mon p | := ((lead_coef p)^-1 *: p). | Notation | mk_mon | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"lead_coef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_algebraic u : algebraicOver FtoE u <-> integralOver FtoE u. | Proof.
split=> [] [p p_nz pu0]; last by exists p; rewrite ?monic_neq0.
exists (mk_mon p); first by rewrite monicE lead_coefZ mulVf ?lead_coef_eq0.
by rewrite linearZ rootE hornerZ (rootP pu0) mulr0.
Qed. | Lemma | integral_algebraic | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"hornerZ",
"integralOver",
"last",
"lead_coefZ",
"lead_coef_eq0",
"linearZ",
"mk_mon",
"monicE",
"monic_neq0",
"mulVf",
"mulr0",
"rootE",
"rootP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_id a : algebraicOver FtoE (FtoE a). | Proof. exact/integral_algebraic/integral_id. Qed. | Lemma | algebraic_id | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"integral_algebraic",
"integral_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic0 : algebraicOver FtoE 0. | Proof. exact/integral_algebraic/integral0. Qed. | Lemma | algebraic0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"integral0",
"integral_algebraic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic1 : algebraicOver FtoE 1. | Proof. exact/integral_algebraic/integral1. Qed. | Lemma | algebraic1 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"integral1",
"integral_algebraic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_opp x : algebraicOver FtoE x -> algebraicOver FtoE (- x). | Proof. by move/integral_algebraic/integral_opp/integral_algebraic. Qed. | Lemma | algebraic_opp | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"integral_algebraic",
"integral_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_add x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x + y). | Proof.
move/integral_algebraic=> intFx /integral_algebraic intFy.
exact/integral_algebraic/integral_add.
Qed. | Lemma | algebraic_add | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"integral_add",
"integral_algebraic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_sub x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x - y). | Proof. by move=> algFx /algebraic_opp; apply: algebraic_add. Qed. | Lemma | algebraic_sub | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"algebraic_add",
"algebraic_opp",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_mul x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x * y). | Proof.
move/integral_algebraic=> intFx /integral_algebraic intFy.
exact/integral_algebraic/integral_mul.
Qed. | Lemma | algebraic_mul | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"integral_algebraic",
"integral_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_inv u : algebraicOver FtoE u -> algebraicOver FtoE u^-1. | Proof.
have [-> | /expf_neq0 nz_u_n] := eqVneq u 0; first by rewrite invr0.
case=> p nz_p pu0; exists (Poly (rev p)).
apply/eqP=> /polyP/(_ 0); rewrite coef_Poly coef0 nth_rev ?size_poly_gt0 //.
by apply/eqP; rewrite subn1 lead_coef_eq0.
apply/eqP/(mulfI (nz_u_n (size p).-1)); rewrite mulr0 -(rootP pu0).
rewrite (@... | Lemma | algebraic_inv | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Poly",
"addSn",
"addnC",
"algebraicOver",
"apply",
"coef0",
"coef_Poly",
"coef_map",
"eqVneq",
"eq_bigr",
"expf_neq0",
"exprD",
"exprVn",
"horner_coef",
"horner_coef_wide",
"invr0",
"lead_coef_eq0",
"mulfI",
"mulfK",
"mulr0",
"mulrCA",
"mulr_sumr",
"nth_rev",
"nz_p",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic_div x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x / y). | Proof. by move=> algFx /algebraic_inv; apply: algebraic_mul. Qed. | Lemma | algebraic_div | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraicOver",
"algebraic_inv",
"algebraic_mul",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_inv x : integralOver FtoE x -> integralOver FtoE x^-1. | Proof. by move/integral_algebraic/algebraic_inv/integral_algebraic. Qed. | Lemma | integral_inv | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"algebraic_inv",
"integralOver",
"integral_algebraic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_div x y :
integralOver FtoE x -> integralOver FtoE y -> integralOver FtoE (x / y). | Proof. by move=> algFx /integral_inv; apply: integral_mul. Qed. | Lemma | integral_div | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"integralOver",
"integral_inv",
"integral_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_root p u :
p != 0 -> root p u -> {in p : seq E, integralRange FtoE} ->
integralOver FtoE u. | Proof.
move=> nz_p pu0 algFp.
have mon_p1: mk_mon p \is monic.
by rewrite monicE lead_coefZ mulVf ?lead_coef_eq0.
have p1u0: root (mk_mon p) u by rewrite rootE hornerZ (rootP pu0) mulr0.
apply: integral_root_monic mon_p1 p1u0 _ => _ /(nthP 0)[i ltip <-].
rewrite coefZ mulrC; rewrite size_scale ?invr_eq0 ?lead_coef_eq... | Lemma | integral_root | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coefZ",
"hornerZ",
"integralOver",
"integralRange",
"integral_div",
"integral_root_monic",
"invr_eq0",
"lead_coefZ",
"lead_coef_eq0",
"mem_nth",
"mk_mon",
"monic",
"monicE",
"mulVf",
"mulr0",
"mulrC",
"nthP",
"nz_p",
"polySpred",
"root",
"rootE",
"rootP",
"seq... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
False | := GRing.False. | Notation | False | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
True | := GRing.True. | Notation | True | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
And | := GRing.And (only parsing). | Notation | And | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Add | := GRing.Add (only parsing). | Notation | Add | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bool b | := (GRing.Bool b%bool). | Notation | Bool | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
term | := (GRing.term F). | Notation | term | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form | := (GRing.formula F). | Notation | form | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"formula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval | := GRing.eval. | Notation | eval | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
holds | := GRing.holds. | Notation | holds | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_form | := GRing.qf_form. | Notation | qf_form | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_eval | := GRing.qf_eval. | Notation | qf_eval | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mx (e : seq F) | := @map_mx term F (eval e). | Definition | eval_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"eval",
"map_mx",
"seq",
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_term | := @map_mx F term GRing.Const. | Definition | mx_term | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"map_mx",
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mx_term e m n (A : 'M_(m, n)) : eval_mx e (mx_term A) = A. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | eval_mx_term | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"eval_mx",
"matrixP",
"mxE",
"mx_term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Definition | mulmx_term | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Add",
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Proof.
apply/matrixP=> i k; rewrite !mxE /= ((big_morph (eval e)) 0 +%R) //=.
by apply: eq_bigr => j _; rewrite /= !mxE.
Qed. | Lemma | eval_mulmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"big_morph",
"eq_bigr",
"eval",
"eval_mx",
"matrixP",
"mulmx_term",
"mxE",
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphAnd f | := ((big_morph f) true andb). | Notation | morphAnd | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"big_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Let | Schur | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"dlsubmx",
"drsubmx",
"term",
"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 _ =>... | Fixpoint | mxrank_form | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Bool",
"Pick",
"Schur",
"form",
"n'",
"nat",
"xcol",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_form_qf r m n (A : 'M_(m, n)) : qf_form (mxrank_form r A). | Proof.
by elim: m r n A => [|m IHm] r [|n] A //=; rewrite GRing.Pick_form_qf /=.
Qed. | Lemma | mxrank_form_qf | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Pick_form_qf",
"mxrank_form",
"qf_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mxrank e r m n (A : 'M_(m, n)) :
qf_eval e (mxrank_form r A) = (\rank (eval_mx e A) == r). | Proof.
elim: m r n A => [|m IHm] r [|n] A /=; try by case r; rewrite unlock.
rewrite GRing.eval_Pick !unlock /=; set pf := fun _ => _.
rewrite -(@eq_pick _ pf) => [k|]; rewrite {}/pf ?mxE // eq_sym.
case: pick => [[i j]|] //=; set B := _ - _; have:= mxrankE B.
case: (Gaussian_elimination_ B) r => [[_ _] _] [|r] //= <-;... | Lemma | eval_mxrank | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Gaussian_elimination_",
"apply",
"big_ord1",
"eqSS",
"eq_pick",
"eq_sym",
"eval_Pick",
"eval_mx",
"matrixP",
"mxE",
"mxrankE",
"mxrank_form",
"pick",
"qf_eval",
"rank",
"tpermR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_vec_mx e m n (u : 'rV_(m * n)) :
eval_mx e (vec_mx u) = vec_mx (eval_mx e u). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | eval_vec_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"eval_mx",
"matrixP",
"mxE",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mxvec e m n (A : 'M_(m, n)) :
eval_mx e (mxvec A) = mxvec (eval_mx e A). | Proof. by rewrite -{2}[A]mxvecK eval_vec_mx vec_mxK. Qed. | Lemma | eval_mxvec | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"eval_mx",
"eval_vec_mx",
"mxvec",
"mxvecK",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_form | :=
\big[And/True]_(r < n.+1) (mxrank_form r (col_mx A B) ==> mxrank_form r B)%T. | Definition | submx_form | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"And",
"True",
"col_mx",
"mxrank_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_col_mx e :
eval_mx e (col_mx A B) = col_mx (eval_mx e A) (eval_mx e B). | Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. Qed. | Lemma | eval_col_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"col_mx",
"eval_mx",
"matrixP",
"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_form_qf : qf_form submx_form. | Proof.
by rewrite (morphAnd (@qf_form _)) ?big1 //= => r _; rewrite !mxrank_form_qf.
Qed. | Lemma | submx_form_qf | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"big1",
"morphAnd",
"mxrank_form_qf",
"qf_form",
"submx_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_submx e : qf_eval e submx_form = (eval_mx e A <= eval_mx e B)%MS. | Proof.
rewrite (morphAnd (qf_eval e)) //= big_andE /=.
apply/forallP/idP=> /= [|sAB d]; last first.
rewrite !eval_mxrank eval_col_mx -addsmxE; apply/implyP=> /eqP <-.
by rewrite mxrank_leqif_sup ?addsmxSr // addsmx_sub sAB /=.
move/(_ (inord (\rank (eval_mx e (col_mx A B))))).
rewrite inordK ?ltnS ?rank_leq_col // ... | Lemma | eval_submx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addsmxE",
"addsmxSr",
"addsmx_sub",
"apply",
"big_andE",
"col_mx",
"eqxx",
"eval_col_mx",
"eval_mx",
"eval_mxrank",
"forallP",
"inord",
"inordK",
"last",
"ltnS",
"morphAnd",
"mxrank_leqif_sup",
"qf_eval",
"rank",
"rank_leq_col",
"submx_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_of_rV (v : 'rV_d) : seq F | := fgraph [ffun i => v 0 i]. | Definition | seq_of_rV | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"fgraph",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_seq_of_rV v : size (seq_of_rV v) = d. | Proof. by rewrite tuple.size_tuple card_ord. Qed. | Lemma | size_seq_of_rV | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"card_ord",
"seq_of_rV",
"size",
"size_tuple",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_seq_of_rV x0 v (i : 'I_d) : nth x0 (seq_of_rV v) i = v 0 i. | Proof. by rewrite nth_fgraph_ord ffunE. Qed. | Lemma | nth_seq_of_rV | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"ffunE",
"nth",
"nth_fgraph_ord",
"seq_of_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_var k : 'rV[term]_d | := \row_i ('X_(k * d + i))%T. | Definition | row_var | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_env (e : seq 'rV_d) | := flatten (map seq_of_rV e). | Definition | row_env | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"flatten",
"map",
"seq",
"seq_of_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_row_env e k (i : 'I_d) : (row_env e)`_(k * d + i) = e`_k 0 i. | Proof.
elim: e k => [|v e IHe] k; first by rewrite !nth_nil mxE.
rewrite /row_env /= nth_cat size_seq_of_rV.
case: k => [|k]; first by rewrite (valP i) nth_seq_of_rV.
by rewrite mulSn -addnA -if_neg -leqNgt leq_addr addKn IHe.
Qed. | Lemma | nth_row_env | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addKn",
"addnA",
"leqNgt",
"leq_addr",
"mulSn",
"mxE",
"nth_cat",
"nth_nil",
"nth_seq_of_rV",
"row_env",
"size_seq_of_rV",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_row_var e k : eval_mx (row_env e) (row_var k) = e`_k :> 'rV_d. | Proof. by apply/rowP=> i; rewrite !mxE /= nth_row_env. Qed. | Lemma | eval_row_var | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"eval_mx",
"mxE",
"nth_row_env",
"rowP",
"row_env",
"row_var"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Exists_row_form k (f : form) | :=
foldr GRing.Exists f (codom (fun i : 'I_d => k * d + i)%N). | Definition | Exists_row_form | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"codom",
"foldr",
"form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Exists_rowP e k f :
d > 0 ->
((exists v : 'rV[F]_d, holds (row_env (set_nth 0 e k v)) f)
<-> holds (row_env e) (Exists_row_form k f)). | Proof.
move=> d_gt0; pose i_ j := Ordinal (ltn_pmod j d_gt0).
have d_eq j: (j = j %/ d * d + i_ j)%N := divn_eq j d.
split=> [[v f_v] | ]; last case/GRing.foldExistsP=> e' ee' f_e'.
apply/GRing.foldExistsP; exists (row_env (set_nth 0 e k v)) => {f f_v}// j.
rewrite [j]d_eq !nth_row_env nth_set_nth /=; case: eqP => ... | Lemma | Exists_rowP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Exists_row_form",
"addn0",
"apply",
"d_gt0",
"divnMDl",
"divn_eq",
"divn_small",
"e'",
"ee'",
"eq_holds",
"foldExistsP",
"holds",
"imageP",
"last",
"ltn_pmod",
"mapP",
"mxE",
"nth_row_env",
"nth_set_nth",
"row_env",
"set_nth",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmx (m n : nat)
(V : 'M_(m, n)) (f : 'M[F]_n) : 'M_m | := V *m f *m pinvmx V. | Definition | conjmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"nat",
"pinvmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
restrictmx V | := (conjmx (row_base V)). | Notation | restrictmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"row_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx_comp (m n p : nat) (V : 'M[F]_(m, n)) (W : 'M_(n, p)) (f : 'M_p) :
stablemx W f -> stablemx V (conjmx W f) -> stablemx (V *m W) f. | Proof. by move=> Wf /(submxMr W); rewrite -mulmxA mulmxKpV// mulmxA. Qed. | Lemma | stablemx_comp | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"mulmxA",
"mulmxKpV",
"nat",
"stablemx",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx_restrict m n (A : 'M[F]_n) (V : 'M_n) (W : 'M_(m, \rank V)):
stablemx V A -> stablemx W (restrictmx V A) = stablemx (W *m row_base V) A. | Proof.
move=> A_stabV; rewrite mulmxA -[in RHS]mulmxA.
rewrite -(submxMfree _ W (row_base_free V)) mulmxKpV //.
by rewrite mulmx_sub ?stablemx_row_base.
Qed. | Lemma | stablemx_restrict | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"mulmxA",
"mulmxKpV",
"mulmx_sub",
"rank",
"restrictmx",
"row_base",
"row_base_free",
"stablemx",
"stablemx_row_base",
"submxMfree"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmxM (m n : nat) (V : 'M[F]_(m, n)) :
{in [pred f | stablemx V f] &, {morph conjmx V : f g / f *m g}}. | Proof.
move=> f g; rewrite !inE => Vf Vg /=.
by rewrite /conjmx 2!mulmxA mulmxA mulmxKpV ?stablemx_row_base.
Qed. | Lemma | conjmxM | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"inE",
"mulmxA",
"mulmxKpV",
"nat",
"stablemx",
"stablemx_row_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjMmx (m n p : nat) (V : 'M[F]_(m, n)) (W : 'M_(n, p)) (f : 'M_p) :
row_free (V *m W) -> stablemx W f -> stablemx V (conjmx W f) ->
conjmx (V *m W) f = conjmx V (conjmx W f). | Proof.
move=> rfVW Wf VWf; apply: (row_free_inj rfVW); rewrite mulmxKpV ?stablemx_comp//.
by rewrite mulmxA mulmxKpV// -[RHS]mulmxA mulmxKpV ?mulmxA.
Qed. | Lemma | conjMmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"mulmxA",
"mulmxKpV",
"nat",
"row_free",
"row_free_inj",
"stablemx",
"stablemx_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjuMmx (m n : nat) (V : 'M[F]_m) (W : 'M_(m, n)) (f : 'M_n) :
V \in unitmx -> row_free W -> stablemx W f ->
conjmx (V *m W) f = conjmx V (conjmx W f). | Proof.
move=> Vu rfW Wf; rewrite conjMmx ?stablemx_unit//.
by rewrite /row_free mxrankMfree// -/(row_free V) row_free_unit.
Qed. | Lemma | conjuMmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjMmx",
"conjmx",
"mxrankMfree",
"nat",
"row_free",
"row_free_unit",
"stablemx",
"stablemx_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjMumx (m n : nat) (V : 'M[F]_(m, n)) (W f : 'M_n) :
W \in unitmx -> row_free V -> stablemx V (conjmx W f) ->
conjmx (V *m W) f = conjmx V (conjmx W f). | Proof.
move=> Wu rfW Wf; rewrite conjMmx ?stablemx_unit//.
by rewrite /row_free mxrankMfree ?row_free_unit.
Qed. | Lemma | conjMumx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjMmx",
"conjmx",
"mxrankMfree",
"nat",
"row_free",
"row_free_unit",
"stablemx",
"stablemx_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjuMumx (n : nat) (V W f : 'M[F]_n) :
V \in unitmx -> W \in unitmx ->
conjmx (V *m W) f = conjmx V (conjmx W f). | Proof. by move=> Vu Wu; rewrite conjuMmx ?stablemx_unit ?row_free_unit. Qed. | Lemma | conjuMumx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"conjuMmx",
"nat",
"row_free_unit",
"stablemx_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmx_scalar (m n : nat) (V : 'M[F]_(m, n)) (a : F) :
row_free V -> conjmx V a%:M = a%:M. | Proof. by move=> rfV; rewrite /conjmx scalar_mxC mulmxKp. Qed. | Lemma | conjmx_scalar | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"mulmxKp",
"nat",
"row_free",
"scalar_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj0mx (m n : nat) f : conjmx (0 : 'M[F]_(m, n)) f = 0. | Proof. by rewrite /conjmx !mul0mx. Qed. | Lemma | conj0mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"mul0mx",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmx0 (m n : nat) (V : 'M[F]_(m, n)) : conjmx V 0 = 0. | Proof. by rewrite /conjmx mulmx0 mul0mx. Qed. | Lemma | conjmx0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"mul0mx",
"mulmx0",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjumx (n : nat) (V : 'M_n) (f : 'M[F]_n) : V \in unitmx ->
conjmx V f = V *m f *m invmx V. | Proof. by move=> uV; rewrite /conjmx pinvmxE. Qed. | Lemma | conjumx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"invmx",
"nat",
"pinvmxE",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj1mx (n : nat) (f : 'M[F]_n) : conjmx 1%:M f = f. | Proof. by rewrite conjumx ?unitmx1// invmx1 mulmx1 mul1mx. Qed. | Lemma | conj1mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"conjumx",
"invmx1",
"mul1mx",
"mulmx1",
"nat",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjVmx (n : nat) (V : 'M_n) (f : 'M[F]_n) : V \in unitmx ->
conjmx (invmx V) f = invmx V *m f *m V. | Proof. by move=> Vunit; rewrite conjumx ?invmxK ?unitmx_inv. Qed. | Lemma | conjVmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"conjumx",
"invmx",
"invmxK",
"nat",
"unitmx",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmxK (n : nat) (V f : 'M[F]_n) :
V \in unitmx -> conjmx (invmx V) (conjmx V f) = f. | Proof. by move=> Vu; rewrite -conjuMumx ?unitmx_inv// mulVmx ?conj1mx. Qed. | Lemma | conjmxK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conj1mx",
"conjmx",
"conjuMumx",
"invmx",
"mulVmx",
"nat",
"unitmx",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmxVK (n : nat) (V f : 'M[F]_n) :
V \in unitmx -> conjmx V (conjmx (invmx V) f) = f. | Proof. by move=> Vu; rewrite -conjuMumx ?unitmx_inv// mulmxV ?conj1mx. Qed. | Lemma | conjmxVK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conj1mx",
"conjmx",
"conjuMumx",
"invmx",
"mulmxV",
"nat",
"unitmx",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_conj m n p (V : 'M[F]_(n.+1, m.+1)) (f : 'M_m.+1) :
row_free V -> stablemx V f ->
horner_mx (conjmx V f) p = conjmx V (horner_mx f p). | Proof.
move=> V_free V_stab; rewrite/conjmx; elim/poly_ind: p => [|p c].
by rewrite !rmorph0 mulmx0 mul0mx.
rewrite !(rmorphD, rmorphM)/= !(horner_mx_X, horner_mx_C) => ->.
rewrite [_ * _]mulmxA [_ *m (V *m _)]mulmxA mulmxKpV ?horner_mx_stable//.
apply: (row_free_inj V_free); rewrite [_ *m V]mulmxDl.
pose stablemxE :... | Lemma | horner_mx_conj | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"horner_mx_stable",
"mul0mx",
"mulmx0",
"mulmxA",
"mulmxDl",
"mulmxDr",
"mulmxKpV",
"poly_ind",
"rmorph0",
"rmorphD",
"rmorphM",
"row_free",
"row_free_inj",
"scalar_mxC",
"stablemx",
"stablemxC",
"stablemxD",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_uconj n p (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \is a GRing.unit ->
horner_mx (V *m f *m invmx V) p = V *m horner_mx f p *m invmx V. | Proof.
move=> V_unit; rewrite -!conjumx//.
by rewrite horner_mx_conj ?row_free_unit ?stablemx_unit.
Qed. | Lemma | horner_mx_uconj | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjumx",
"horner_mx",
"horner_mx_conj",
"invmx",
"row_free_unit",
"stablemx_unit",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_uconjC n p (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \is a GRing.unit ->
horner_mx (invmx V *m f *m V) p = invmx V *m horner_mx f p *m V. | Proof.
move=> V_unit; rewrite -[X in _ *m X](invmxK V).
by rewrite horner_mx_uconj ?invmxK ?unitmx_inv.
Qed. | Lemma | horner_mx_uconjC | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx",
"horner_mx_uconj",
"invmx",
"invmxK",
"unit",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_conj m n (V : 'M[F]_(m.+1, n.+1)) (f : 'M_n.+1) :
row_free V -> stablemx V f -> mxminpoly (conjmx V f) %| mxminpoly f. | Proof.
by move=> *; rewrite mxminpoly_min// horner_mx_conj// mx_root_minpoly conjmx0.
Qed. | Lemma | mxminpoly_conj | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"conjmx0",
"horner_mx_conj",
"mx_root_minpoly",
"mxminpoly",
"mxminpoly_min",
"row_free",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_uconj n (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \in unitmx -> mxminpoly (conjmx V f) = mxminpoly f. | Proof.
have simp := (row_free_unit, stablemx_unit, unitmx_inv, unitmx1).
move=> Vu; apply/eqP; rewrite -eqp_monic ?mxminpoly_monic// /eqp.
apply/andP; split; first by rewrite mxminpoly_conj ?simp.
by rewrite -[f in X in X %| _](conjmxK _ Vu) mxminpoly_conj ?simp.
Qed. | Lemma | mxminpoly_uconj | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"conjmxK",
"eqp",
"eqp_monic",
"mxminpoly",
"mxminpoly_conj",
"mxminpoly_monic",
"row_free_unit",
"simp",
"split",
"stablemx_unit",
"unitmx",
"unitmx1",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_kermxpoly_conjmx V f p W : stablemx V f -> row_free V ->
(W <= kermxpoly (conjmx V f) p)%MS = (W *m V <= kermxpoly f p)%MS. | Proof.
case: n m => [|n'] [|m'] in V f W * => fV rfV; rewrite ?thinmx0//.
by rewrite /row_free mxrank.unlock in rfV.
by rewrite mul0mx !sub0mx.
apply/sub_kermxP/sub_kermxP; rewrite horner_mx_conj//; last first.
by move=> /(congr1 (mulmxr (pinvmx V)))/=; rewrite mul0mx !mulmxA.
move=> /(congr1 (mulmxr V))/=; rewri... | Lemma | sub_kermxpoly_conjmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"horner_mx_conj",
"horner_mx_stable",
"kermxpoly",
"last",
"mul0mx",
"mulmxA",
"mulmxKpV",
"mulmx_sub",
"mulmxr",
"mxrank",
"n'",
"pinvmx",
"row_free",
"stablemx",
"sub0mx",
"sub_kermxP",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_eigenspace_conjmx V f a W : stablemx V f -> row_free V ->
(W <= eigenspace (conjmx V f) a)%MS = (W *m V <= eigenspace f a)%MS. | Proof. by move=> fV rfV; rewrite !eigenspace_poly sub_kermxpoly_conjmx. Qed. | Lemma | sub_eigenspace_conjmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"eigenspace",
"eigenspace_poly",
"row_free",
"stablemx",
"sub_kermxpoly_conjmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenpoly_conjmx V f : stablemx V f -> row_free V ->
{subset eigenpoly (conjmx V f) <= eigenpoly f}. | Proof.
move=> fV rfV a /eigenpolyP [x]; rewrite sub_kermxpoly_conjmx//.
move=> xV_le_fa x_neq0; apply/eigenpolyP.
by exists (x *m V); rewrite ?mulmx_free_eq0.
Qed. | Lemma | eigenpoly_conjmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"eigenpoly",
"eigenpolyP",
"mulmx_free_eq0",
"row_free",
"stablemx",
"sub_kermxpoly_conjmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue_conjmx V f : stablemx V f -> row_free V ->
{subset eigenvalue (conjmx V f) <= eigenvalue f}. | Proof.
by move=> fV rfV a; rewrite ![_ \in _]eigenvalue_poly; apply: eigenpoly_conjmx.
Qed. | Lemma | eigenvalue_conjmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"eigenpoly_conjmx",
"eigenvalue",
"eigenvalue_poly",
"row_free",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmx_eigenvalue a V f : (V <= eigenspace f a)%MS -> row_free V ->
conjmx V f = a%:M. | Proof.
by move=> /eigenspaceP Vfa rfV; rewrite /conjmx Vfa -mul_scalar_mx mulmxKp.
Qed. | Lemma | conjmx_eigenvalue | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"eigenspace",
"eigenspaceP",
"mul_scalar_mx",
"mulmxKp",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmx_to_for {F : fieldType} {m n}
(P : 'M_(m, n)) A (S : {pred 'M[F]_m}) | := S (conjmx P A). | Definition | simmx_to_for | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ~_ P '{' 'in' S '}'" | := (simmx_to_for P A S)
(at level 70, P at level 0, format "A ~_ P '{' 'in' S '}'") :
ring_scope. | Notation | A ~_ P '{' 'in' S '}' | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"simmx_to_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmx_for P A B | := (A ~_P {in PredOfSimpl.coerce (pred1 B)}). | Notation | simmx_for | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ~_ P B" | := (simmx_for P A B) (format "A ~_ P B"). | Notation | A ~_ P B | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"simmx_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmx_in S A B | := (exists2 P, P \in S & A ~_P B). | Notation | simmx_in | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A '~_{in' S '}' B" | := (simmx_in S A B)
(at level 70, format "A '~_{in' S '}' B"). | Notation | A '~_{in' S '}' B | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"simmx_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmx_in_to S A S' | := (exists2 P, P \in S & A ~_P {in S'}). | Notation | simmx_in_to | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A '~_{in' S '}' '{' 'in' S' '}'" | := (simmx_in_to S A S')
(format "A '~_{in' S '}' '{' 'in' S' '}'"). | Notation | A '~_{in' S '}' '{' 'in' S' '}' | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"simmx_in_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_simmx_in S As S' | :=
(exists2 P, P \in S & all [pred A | A ~_P {in S'}] As). | Notation | all_simmx_in | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_for P A | := (A ~_P {in is_diag_mx}). | Notation | diagonalizable_for | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"is_diag_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_in S A | := (A ~_{in S} {in is_diag_mx}). | Notation | diagonalizable_in | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"is_diag_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable A | := (diagonalizable_in unitmx A). | Notation | diagonalizable | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"diagonalizable_in",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizable_in S As | := (all_simmx_in S As is_diag_mx). | Notation | codiagonalizable_in | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"all_simmx_in",
"is_diag_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizable As | := (codiagonalizable_in unitmx As). | Notation | codiagonalizable | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"codiagonalizable_in",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmxPp m n {P : 'M[F]_(m, n)} {A B} :
stablemx P A -> A ~_P B -> P *m A = B *m P. | Proof. by move=> stablemxPA /eqP <-; rewrite mulmxKpV. Qed. | Lemma | simmxPp | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"mulmxKpV",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmxW m n {P : 'M[F]_(m, n)} {A B} : row_free P ->
P *m A = B *m P -> A ~_P B. | Proof. by rewrite /(_ ~__ _)/= /conjmx => fP ->; rewrite mulmxKp. Qed. | Lemma | simmxW | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"fP",
"mulmxKp",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmxP {P A B} : P \in unitmx ->
reflect (P *m A = B *m P) (A ~_P B). | Proof.
move=> p_unit; apply: (iffP idP); first exact/simmxPp/stablemx_unit.
by apply: simmxW; rewrite row_free_unit.
Qed. | Lemma | simmxP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"row_free_unit",
"simmxPp",
"simmxW",
"stablemx_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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