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 |
|---|---|---|---|---|---|---|---|---|---|---|
unitmx_mul A B : (A *m B \in unitmx) = (A \in unitmx) && (B \in unitmx). | Proof. by rewrite -unitrM -det_mulmx. Qed. | Lemma | unitmx_mul | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_mulmx",
"unitmx",
"unitrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_inv (A : 'M_n) : (invmx A)^T = invmx (A^T). | Proof. by rewrite (fun_if trmx) linearZ /= trmx_adj -unitmx_tr -det_tr. Qed. | Lemma | trmx_inv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_tr",
"invmx",
"linearZ",
"trmx",
"trmx_adj",
"unitmx_tr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmxK : involutive invmx. | Proof.
move=> A; case uA : (A \in unitmx); last by rewrite /invmx !uA.
by apply: (can_inj (mulKVmx uA)); rewrite mulVmx // mulmxV ?unitmx_inv.
Qed. | Lemma | invmxK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"invmx",
"last",
"mulKVmx",
"mulVmx",
"mulmxV",
"unitmx",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx1_unit A B : A *m B = 1%:M -> A \in unitmx /\ B \in unitmx. | Proof. by move=> AB1; apply/andP; rewrite -unitmx_mul AB1 unitmx1. Qed. | Lemma | mulmx1_unit | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"unitmx",
"unitmx1",
"unitmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intro_unitmx A B : B *m A = 1%:M /\ A *m B = 1%:M -> unitmx A. | Proof. by case=> _ /mulmx1_unit[]. Qed. | Lemma | intro_unitmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmx1_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmx_out : {in [predC unitmx], invmx =1 id}. | Proof. by move=> A; rewrite inE /= /invmx -if_neg => ->. Qed. | Lemma | invmx_out | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"id",
"inE",
"invmx",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
detV (A : 'M_n) : \det A^-1 = (\det A)^-1. | Proof. exact: det_inv. Qed. | Lemma | detV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_inv"
] | Lemmas requiring that the coefficients are in a unit ring | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
unitr_trmx (A : 'M_n) : (A^T \is a GRing.unit) = (A \is a GRing.unit). | Proof. exact: unitmx_tr. Qed. | Lemma | unitr_trmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit",
"unitmx_tr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmxV (A : 'M_n) : A^-1^T = (A^T)^-1. | Proof. exact: trmx_inv. Qed. | Lemma | trmxV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"trmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mxV (s : 'S_n) : perm_mx s^-1 = (perm_mx s)^-1. | Proof.
rewrite -[_^-1]mul1r; apply: (canRL (mulmxK (unitmx_perm s))).
by rewrite -perm_mxM mulVg perm_mx1.
Qed. | Lemma | perm_mxV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mul1r",
"mulVg",
"mulmxK",
"perm_mx",
"perm_mx1",
"perm_mxM",
"unitmx_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mxV (A : 'M_n) : is_perm_mx A^-1 = is_perm_mx A. | Proof.
apply/is_perm_mxP/is_perm_mxP=> [] [s defA]; exists s^-1%g.
by rewrite -(invrK A) defA perm_mxV.
by rewrite defA perm_mxV.
Qed. | Lemma | is_perm_mxV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"invrK",
"is_perm_mx",
"is_perm_mxP",
"perm_mxV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_diag_mx_unit (R : comUnitRingType) n1 n2
(Aul : 'M[R]_n1) (Adr : 'M[R]_n2) :
(block_mx Aul 0 0 Adr \in unitmx) = (Aul \in unitmx) && (Adr \in unitmx). | Proof. by rewrite !unitmxE det_ublock unitrM. Qed. | Lemma | block_diag_mx_unit | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"det_ublock",
"unitmx",
"unitmxE",
"unitrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmx_block_diag (R : comUnitRingType) n1 n2
(Aul : 'M[R]_n1) (Adr : 'M[R]_n2) :
block_mx Aul 0 0 Adr \in unitmx ->
invmx (block_mx Aul 0 0 Adr) = block_mx (invmx Aul) 0 0 (invmx Adr). | Proof.
move=> /[dup] Aunit; rewrite block_diag_mx_unit => /andP[Aul_unit Adr_unit].
rewrite -[LHS]mul1mx; apply: (canLR (mulmxK _)) => //.
rewrite [RHS](mulmx_block (invmx Aul)) !(mulmx0, mul0mx, add0r, addr0).
by rewrite !mulVmx// -?scalar_mx_block.
Qed. | Lemma | invmx_block_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"add0r",
"addr0",
"apply",
"block_diag_mx_unit",
"block_mx",
"invmx",
"mul0mx",
"mul1mx",
"mulVmx",
"mulmx0",
"mulmxK",
"mulmx_block",
"scalar_mx_block",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GLtype (R : finComUnitRingType) | := {unit 'M[R]_n.-1.+1}. | Definition | GLtype | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GLval R (u : GLtype R) : 'M[R]_n.-1.+1 | :=
let: FinRing.Unit A _ := u in A. | Coercion | GLval | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ ''GL_' n [ R ] }" | := (GLtype n R) : type_scope. | Notation | { ''GL_' n [ R ] } | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ ''GL_' n ( p ) }" | := {'GL_n['F_p]} : type_scope. | Notation | { ''GL_' n ( p ) } | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GLgroup | := [set: {'GL_n[R]}]. | Definition | GLgroup | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GLgroup_group | := Eval hnf in [group of GLgroup]. | Canonical | GLgroup_group | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLgroup",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_1E : GLval 1 = 1. | Proof. by []. Qed. | Lemma | GL_1E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_VE u : GLval u^-1 = (GLval u)^-1. | Proof. by []. Qed. | Lemma | GL_VE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_VxE u : GLval u^-1 = invmx u. | Proof. by []. Qed. | Lemma | GL_VxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLval",
"invmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_ME u v : GLval (u * v) = GLval u * GLval v. | Proof. by []. Qed. | Lemma | GL_ME | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_MxE u v : GLval (u * v) = u *m v. | Proof. by []. Qed. | Lemma | GL_MxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_unit u : GLval u \is a GRing.unit. | Proof. exact: valP. Qed. | Lemma | GL_unit | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLval",
"unit",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_unitmx u : val u \in unitmx. | Proof. exact: GL_unit. Qed. | Lemma | GL_unitmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GL_unit",
"unitmx",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_det u : \det u != 0. | Proof.
by apply: contraL (GL_unitmx u); rewrite unitmxE => /eqP->; rewrite unitr0.
Qed. | Lemma | GL_det | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GL_unitmx",
"apply",
"unitmxE",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''GL_' n [ R ]" | := (GLgroup n R)
(n at level 2, format "''GL_' n [ R ]") : group_scope. | Notation | ''GL_' n [ R ] | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLgroup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''GL_' n ( p )" | := 'GL_n['F_p]
(p at level 10, format "''GL_' n ( p )") : group_scope. | Notation | ''GL_' n ( p ) | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''GL_' n [ R ]" | := (GLgroup_group n R) : Group_scope. | Notation | ''GL_' n [ R ] | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLgroup_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''GL_' n ( p )" | := (GLgroup_group n 'F_p) : Group_scope. | Notation | ''GL_' n ( p ) | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"GLgroup_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemx_eq0 m n a (A : 'M[R]_(m, n)) :
(a *: A == 0) = (a == 0) || (A == 0). | Proof.
case nz_a: (a == 0) / eqP => [-> | _]; first by rewrite scale0r eqxx.
apply/eqP/eqP=> [aA0 | ->]; last exact: scaler0.
apply/matrixP=> i j; apply/eqP; move/matrixP/(_ i j)/eqP: aA0.
by rewrite !mxE mulf_eq0 nz_a.
Qed. | Lemma | scalemx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eqxx",
"last",
"matrixP",
"mulf_eq0",
"mxE",
"scale0r",
"scaler0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemx_inj m n a :
a != 0 -> injective ( *:%R a : 'M[R]_(m, n) -> 'M[R]_(m, n)). | Proof.
move=> nz_a A B eq_aAB; apply: contraNeq nz_a.
rewrite -[A == B]subr_eq0 -[a == 0]orbF => /negPf<-.
by rewrite -scalemx_eq0 linearB subr_eq0 /= eq_aAB.
Qed. | Lemma | scalemx_inj | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"contraNeq",
"linearB",
"scalemx_eq0",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det0P n (A : 'M[R]_n) :
reflect (exists2 v : 'rV[R]_n, v != 0 & v *m A = 0) (\det A == 0). | Proof.
apply: (iffP eqP) => [detA0 | [v n0v vA0]]; last first.
apply: contraNeq n0v => nz_detA; rewrite -(inj_eq (scalemx_inj nz_detA)).
by rewrite scaler0 -mul_mx_scalar -mul_mx_adj mulmxA vA0 mul0mx.
elim: n => [|n IHn] in A detA0 *.
by case/idP: (oner_eq0 R); rewrite -detA0 [A]thinmx0 -(thinmx0 1%:M) det1.
hav... | Lemma | det0P | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"A'",
"add0r",
"addr0",
"apply",
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"contraNneq",
"det1",
"eqVneq",
"eq_big",
"inj_eq",
"last",
"lift",
"liftK",
"linear0",
"matrixP",
"mul0mx",
"mul0r",
"mul_adj_mx",
"mul_mx_adj",
"mul_mx_row",
"mul_mx_scalar",
"mul_scalar_mx",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_inj {m n} : injective (map_mx f : 'M_(m, n) -> 'M_(m, n)). | Proof.
move=> A B eq_AB; apply/matrixP=> i j.
by move/matrixP/(_ i j): eq_AB => /[!mxE]; apply: fmorph_inj.
Qed. | Lemma | map_mx_inj | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"fmorph_inj",
"map_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_is_scalar n (A : 'M_n) : is_scalar_mx A^f = is_scalar_mx A. | Proof.
rewrite /is_scalar_mx; case: (insub _) => // i.
by rewrite mxE -map_scalar_mx inj_eq //; apply: map_mx_inj.
Qed. | Lemma | map_mx_is_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"inj_eq",
"insub",
"is_scalar_mx",
"map_mx_inj",
"map_scalar_mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_unitmx n (A : 'M_n) : (A^f \in unitmx) = (A \in unitmx). | Proof. by rewrite unitmxE det_map_mx // fmorph_unit // -unitfE. Qed. | Lemma | map_unitmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_map_mx",
"fmorph_unit",
"unitfE",
"unitmx",
"unitmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_unit n' (A : 'M_n'.+1) :
(A^f \is a GRing.unit) = (A \is a GRing.unit). | Proof. exact: map_unitmx. Qed. | Lemma | map_mx_unit | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"map_unitmx",
"n'",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_invmx n (A : 'M_n) : (invmx A)^f = invmx A^f. | Proof.
rewrite /invmx map_unitmx (fun_if (map_mx f)).
by rewrite map_mxZ map_mx_adj det_map_mx fmorphV.
Qed. | Lemma | map_invmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_map_mx",
"fmorphV",
"invmx",
"map_mx",
"map_mxZ",
"map_mx_adj",
"map_unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_inv n' (A : 'M_n'.+1) : A^-1^f = A^f^-1. | Proof. exact: map_invmx. Qed. | Lemma | map_mx_inv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"map_invmx",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_eq0 m n (A : 'M_(m, n)) : (A^f == 0) = (A == 0). | Proof. by rewrite -(inj_eq map_mx_inj) raddf0. Qed. | Lemma | map_mx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"inj_eq",
"map_mx_inj",
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cormen_lup {n} | :=
match n return let M := 'M[F]_n.+1 in M -> M * M * M with
| 0 => fun A => (1, 1, A)
| _.+1 => fun A =>
let k := odflt 0 [pick k | A k 0 != 0] in
let A1 : 'M_(1 + _) := xrow 0 k A in
let P1 : 'M_(1 + _) := tperm_mx 0 k in
let Schur := ((A k 0)^-1 *: dlsubmx A1) *m ursubmx A1 in
let: (P2, L2,... | Fixpoint | cormen_lup | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"P1",
"Schur",
"block_mx",
"dlsubmx",
"drsubmx",
"pick",
"tperm_mx",
"ulsubmx",
"ursubmx",
"xrow"
] | - U an upper triangular matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cormen_lup_perm n (A : 'M_n.+1) : is_perm_mx (cormen_lup A).1.1. | Proof.
elim: n => [|n IHn] /= in A *; first exact: is_perm_mx1.
set A' := _ - _; move/(_ A'): IHn; case: cormen_lup => [[P L U]] {A'}/=.
rewrite (is_perm_mxMr _ (perm_mx_is_perm _ _)).
by case/is_perm_mxP => s ->; apply: lift0_mx_is_perm.
Qed. | Lemma | cormen_lup_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"A'",
"apply",
"cormen_lup",
"is_perm_mx",
"is_perm_mx1",
"is_perm_mxMr",
"is_perm_mxP",
"lift0_mx_is_perm",
"perm_mx_is_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cormen_lup_correct n (A : 'M_n.+1) :
let: (P, L, U) := cormen_lup A in P * A = L * U. | Proof.
elim: n => [|n IHn] /= in A *; first by rewrite !mul1r.
set k := odflt _ _; set A1 : 'M_(1 + _) := xrow _ _ _.
set A' := _ - _; move/(_ A'): IHn; case: cormen_lup => [[P' L' U']] /= IHn.
rewrite -mulrA -!mulmxE -xrowE -/A1 /= -[n.+2]/(1 + n.+1)%N -{1}(submxK A1).
rewrite !mulmx_block !mul0mx !mulmx0 !add0r !addr... | Lemma | cormen_lup_correct | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
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"L'",
"add0r",
"addr0",
"addrC",
"apply",
"block_mx",
"colP",
"cormen_lup",
"dlsubmx",
"lshift0",
"mul0mx",
"mul1mx",
"mul1r",
"mulVf",
"mulmx0",
"mulmx1",
"mulmxA",
"mulmxE",
"mulmx_block",
"mulrA",
"mulrDr",
"mx11_scalar",
"mxE",
"pickP",
"scalemxAl",
"sca... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cormen_lup_detL n (A : 'M_n.+1) : \det (cormen_lup A).1.2 = 1. | Proof.
elim: n => [|n IHn] /= in A *; first by rewrite det1.
set A' := _ - _; move/(_ A'): IHn; case: cormen_lup => [[P L U]] {A'}/= detL.
by rewrite (@det_lblock _ 1) det1 mul1r.
Qed. | Lemma | cormen_lup_detL | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"A'",
"cormen_lup",
"det1",
"det_lblock",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cormen_lup_lower n A (i j : 'I_n.+1) :
i <= j -> (cormen_lup A).1.2 i j = (i == j)%:R. | Proof.
elim: n => [|n IHn] /= in A i j *; first by rewrite [i]ord1 [j]ord1 mxE.
set A' := _ - _; move/(_ A'): IHn; case: cormen_lup => [[P L U]] {A'}/= Ll.
rewrite !mxE split1; case: unliftP => [i'|] -> /=; rewrite !mxE split1.
by case: unliftP => [j'|] -> //; apply: Ll.
by case: unliftP => [j'|] ->; rewrite /= mxE.
... | Lemma | cormen_lup_lower | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"A'",
"apply",
"cormen_lup",
"mxE",
"ord1",
"split1",
"unliftP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cormen_lup_upper n A (i j : 'I_n.+1) :
j < i -> (cormen_lup A).2 i j = 0 :> F. | Proof.
elim: n => [|n IHn] /= in A i j *; first by rewrite [i]ord1.
set A' := _ - _; move/(_ A'): IHn; case: cormen_lup => [[P L U]] {A'}/= Uu.
rewrite !mxE split1; case: unliftP => [i'|] -> //=; rewrite !mxE split1.
by case: unliftP => [j'|] ->; [apply: Uu | rewrite /= mxE].
Qed. | Lemma | cormen_lup_upper | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
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... | [
"A'",
"Uu",
"apply",
"cormen_lup",
"mxE",
"ord1",
"split1",
"unliftP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_pred (S : {pred T}) | :=
fun M : 'M[T]_(m, n) => [forall i, [forall j, M i j \in S]]. | Definition | mxOver_pred | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver (S : {pred T}) | := [qualify a M | mxOver_pred S M]. | Definition | mxOver | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxOver_pred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOverP {S : {pred T}} {M : 'M[T]__} :
reflect (forall i j, M i j \in S) (M \is a mxOver S). | Proof. exact/'forall_forallP. Qed. | Lemma | mxOverP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOverS (S1 S2 : {pred T}) :
{subset S1 <= S2} -> {subset mxOver S1 <= mxOver S2}. | Proof. by move=> sS12 M /mxOverP S1M; apply/mxOverP=> i j; apply/sS12/S1M. Qed. | Lemma | mxOverS | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"S1",
"S2",
"apply",
"mxOver",
"mxOverP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_const c S : c \in S -> const_mx c \is a mxOver S. | Proof. by move=> cS; apply/mxOverP => i j; rewrite !mxE. Qed. | Lemma | mxOver_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"mxE",
"mxOver",
"mxOverP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_constE c S : (m > 0)%N -> (n > 0)%N ->
(const_mx c \is a mxOver S) = (c \in S). | Proof.
move=> m_gt0 n_gt0; apply/idP/idP; last exact: mxOver_const.
by move=> /mxOverP /(_ (Ordinal m_gt0) (Ordinal n_gt0)); rewrite mxE.
Qed. | Lemma | mxOver_constE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"last",
"mxE",
"mxOver",
"mxOverP",
"mxOver_const",
"n_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
thinmxOver {n : nat} {T : Type} (M : 'M[T]_(n, 0)) S : M \is a mxOver S. | Proof. by apply/mxOverP => ? []. Qed. | Lemma | thinmxOver | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxOver",
"mxOverP",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
flatmxOver {n : nat} {T : Type} (M : 'M[T]_(0, n)) S : M \is a mxOver S. | Proof. by apply/mxOverP => - []. Qed. | Lemma | flatmxOver | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxOver",
"mxOverP",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver0 S : 0 \in S -> 0 \is a @mxOver m n _ S. | Proof. exact: mxOver_const. Qed. | Lemma | mxOver0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxOver",
"mxOver_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_nmod_closed : nmod_closed (@mxOver m n _ addS). | Proof.
split=> [|p q Sp Sq]; first by rewrite mxOver0 // ?rpred0.
by apply/mxOverP=> i j; rewrite mxE rpredD // !(mxOverP _).
Qed. | Fact | mxOver_nmod_closed | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"mxOver",
"mxOver0",
"mxOverP",
"nmod_closed",
"rpred0",
"rpredD",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_opp_subproof : oppr_closed (@mxOver m n _ oppS). | Proof. by move=> A /mxOverP SA; apply/mxOverP=> i j; rewrite mxE rpredN. Qed. | Fact | mxOver_opp_subproof | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"mxOver",
"mxOverP",
"oppr_closed",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_scalar S c : 0 \in S -> c \in S -> c%:M \is a @mxOver n n R S. | Proof. by move=> S0 cS; apply/mxOverP => i j; rewrite !mxE; case: eqP. Qed. | Lemma | mxOver_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"S0",
"apply",
"mxE",
"mxOver",
"mxOverP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_scalarE S c : (n > 0)%N ->
(c%:M \is a @mxOver n n R S) = ((n > 1) ==> (0 \in S)) && (c \in S). | Proof.
case: n => [|[|k]]//= _.
by apply/mxOverP/idP => [/(_ ord0 ord0)|cij i j]; rewrite ?mxE ?ord1.
apply/mxOverP/andP => [cij|[S0 cij] i j]; last by rewrite !mxE; case: eqP.
by split; [have := cij 0 1|have := cij 0 0]; rewrite !mxE.
Qed. | Lemma | mxOver_scalarE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"bigop",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"S0",
"apply",
"last",
"mxE",
"mxOver",
"mxOverP",
"ord0",
"ord1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOverZ (S : mulrClosed R) :
{in S & mxOver S, forall a : R, forall v : 'M[R]_(m, n),
a *: v \is a mxOver S}. | Proof.
by move=> a v aS /mxOverP vS; apply/mxOverP => i j; rewrite !mxE rpredM.
Qed. | Lemma | mxOverZ | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"mxOver",
"mxOverP",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_diag (S : {pred R}) k (D : 'rV[R]_k) :
0 \in S -> D \is a mxOver S -> diag_mx D \is a mxOver S. | Proof.
move=> S0 DS; apply/mxOverP => i j; rewrite !mxE.
by case: eqP => //; rewrite (mxOverP DS).
Qed. | Lemma | mxOver_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"S0",
"apply",
"diag_mx",
"mxE",
"mxOver",
"mxOverP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_diagE (S : {pred R}) k (D : 'rV[R]_k) : k > 0 ->
(diag_mx D \is a mxOver S) = ((k > 1) ==> (0 \in S)) && (D \is a mxOver S). | Proof.
case: k => [|[|k]]//= in D * => _.
by rewrite [diag_mx _]mx11_scalar [D in RHS]mx11_scalar !mxE.
apply/idP/andP => [/mxOverP DS|[S0 DS]]; last exact: mxOver_diag.
split; first by have /[!mxE] := DS 0 1.
by apply/mxOverP => i j; have := DS j j; rewrite ord1 !mxE eqxx.
Qed. | Lemma | mxOver_diagE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"S0",
"apply",
"diag_mx",
"eqxx",
"last",
"mx11_scalar",
"mxE",
"mxOver",
"mxOverP",
"mxOver_diag",
"ord1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOverM (S : semiringClosed R) p q r : {in mxOver S & mxOver S,
forall u : 'M[R]_(p, q), forall v : 'M[R]_(q, r), u *m v \is a mxOver S}. | Proof.
move=> M N /mxOverP MS /mxOverP NS; apply/mxOverP => i j.
by rewrite !mxE rpred_sum // => k _; rewrite rpredM.
Qed. | Lemma | mxOverM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"mxOver",
"mxOverP",
"rpredM",
"rpred_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxOver_mul_subproof : mulr_closed (@mxOver n n _ S). | Proof. by split; rewrite ?mxOver_scalar ?rpred0 ?rpred1//; apply: mxOverM. Qed. | Fact | mxOver_mul_subproof | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mulr_closed",
"mxOver",
"mxOverM",
"mxOver_scalar",
"rpred0",
"rpred1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sp | := (\sum_i p_ i)%N. | Notation | sp | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sq | := (\sum_i q_ i)%N. | Notation | sq | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblock (B_ : forall i j, 'M[T]_(p_ i, q_ j)) | :=
\matrix_(j, k) B_ (sig1 j) (sig1 k) (sig2 j) (sig2 k). | Definition | mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxblock_ ( i , j ) E" | := (mxblock (fun i j => E)) : ring_scope. | Notation | \mxblock_ ( i , j ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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... | [
"mxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrow m (B_ : forall j, 'M[T]_(m, q_ j)) | :=
\matrix_(j, k) B_ (sig1 k) j (sig2 k). | Definition | mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxrow_ i E" | := (mxrow (fun i => E)) : ring_scope. | Notation | \mxrow_ i E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
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"GRing.Theory",
... | [
"mxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcol n (B_ : forall i, 'M[T]_(p_ i, n)) | :=
\matrix_(j, k) B_ (sig1 j) (sig2 j) k. | Definition | mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxcol_ i E" | := (mxcol (fun i => E)) : ring_scope. | Notation | \mxcol_ i E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblock (A : 'M[T]_(sp, sq)) i j | := mxsub (Rank i) (Rank j) A. | Definition | submxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
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"sp",
"sq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrow m (A : 'M[T]_(m, sq)) j | := colsub (Rank j) A. | Definition | submxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
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... | [
"Rank",
"colsub",
"sq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcol n (A : 'M[T]_(sp, n)) i | := rowsub (Rank i) A. | Definition | submxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank",
"rowsub",
"sp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblockEh B_ : \mxblock_(i, j) B_ i j = \mxrow_j \mxcol_i B_ i j. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | mxblockEh | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblockEv B_ : \mxblock_(i, j) B_ i j = \mxcol_i \mxrow_j B_ i j. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | mxblockEv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblockEh A i j : submxblock A i j = submxcol (submxrow A j) i. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | submxblockEh | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"submxblock",
"submxcol",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblockEv A i j : submxblock A i j = submxrow (submxcol A i) j. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | submxblockEv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"submxblock",
"submxcol",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblockK B_ i j : submxblock (\mxblock_(i, j) B_ i j) i j = B_ i j. | Proof.
apply/matrixP => k l; rewrite !mxE !Rank2K.
by do !case: _ / esym; rewrite !cast_ord_id.
Qed. | Lemma | mxblockK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank2K",
"apply",
"cast_ord_id",
"matrixP",
"mxE",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrowK m B_ j : @submxrow m (\mxrow_j B_ j) j = B_ j. | Proof.
apply/matrixP => k l; rewrite !mxE !Rank2K.
by do !case: _ / esym; rewrite !cast_ord_id.
Qed. | Lemma | mxrowK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank2K",
"apply",
"cast_ord_id",
"matrixP",
"mxE",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcolK n B_ i : @submxcol n (\mxcol_i B_ i) i = B_ i. | Proof.
apply/matrixP => k l; rewrite !mxE !Rank2K.
by do !case: _ / esym; rewrite !cast_ord_id.
Qed. | Lemma | mxcolK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank2K",
"apply",
"cast_ord_id",
"matrixP",
"mxE",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrow_matrix B_ j :
submxrow (\mxblock_(i, j) B_ i j) j = \mxcol_i B_ i j. | Proof. by rewrite mxblockEh mxrowK. Qed. | Lemma | submxrow_matrix | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxblockEh",
"mxrowK",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcol_matrix B_ i :
submxcol (\mxblock_(i, j) B_ i j) i = \mxrow_j B_ i j. | Proof. by rewrite mxblockEv mxcolK. Qed. | Lemma | submxcol_matrix | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxblockEv",
"mxcolK",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblockK A : \mxblock_(i, j) (submxblock A i j) = A. | Proof. by apply/matrixP => k l; rewrite !mxE !sig2K. Qed. | Lemma | submxblockK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"sig2K",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrowK m (A : 'M[T]_(m, sq)) : \mxrow_j (submxrow A j) = A. | Proof. by apply/matrixP => k l; rewrite !mxE !sig2K. Qed. | Lemma | submxrowK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"sig2K",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcolK n (A : 'M[T]_(sp, n)) : \mxcol_i (submxcol A i) = A. | Proof. by apply/matrixP => k l; rewrite !mxE !sig2K. Qed. | Lemma | submxcolK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"sig2K",
"sp",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblockP A B :
(forall i j, submxblock A i j = submxblock B i j) <-> A = B. | Proof.
split=> [eqAB|->//]; apply/matrixP=> s t;
have /matrixP := eqAB (sig1 s) (sig1 t).
by move=> /(_ (sig2 s) (sig2 t)); rewrite !mxE !sig2K.
Qed. | Lemma | mxblockP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"sig1",
"sig2",
"sig2K",
"split",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrowP m (A B : 'M_(m, sq)) :
(forall j, submxrow A j = submxrow B j) <-> A = B. | Proof.
split=> [eqAB|->//]; apply/matrixP=> i t; have /matrixP := eqAB (sig1 t).
by move=> /(_ i (sig2 t)); rewrite !mxE !sig2K.
Qed. | Lemma | mxrowP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"sig1",
"sig2",
"sig2K",
"split",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcolP n (A B : 'M_(sp, n)) :
(forall i, submxcol A i = submxcol B i) <-> A = B. | Proof.
split=> [eqAB|->//]; apply/matrixP=> s j; have /matrixP := eqAB (sig1 s).
by move=> /(_ (sig2 s) j); rewrite !mxE !sig2K.
Qed. | Lemma | mxcolP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"sig1",
"sig2",
"sig2K",
"sp",
"split",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxblockP A_ B_ :
(forall i j, A_ i j = B_ i j) <->
(\mxblock_(i, j) A_ i j = \mxblock_(i, j) B_ i j). | Proof.
split; first by move=> e; apply/mxblockP => i j; rewrite !mxblockK.
by move=> + i j => /mxblockP/(_ i j); rewrite !mxblockK.
Qed. | Lemma | eq_mxblockP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxblockK",
"mxblockP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxblock A_ B_ :
(forall i j, A_ i j = B_ i j) ->
(\mxblock_(i, j) A_ i j = \mxblock_(i, j) B_ i j). | Proof. by move=> /eq_mxblockP. Qed. | Lemma | eq_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"eq_mxblockP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxrowP m (A_ B_ : forall j, 'M[T]_(m, q_ j)) :
(forall j, A_ j = B_ j) <-> (\mxrow_j A_ j = \mxrow_j B_ j). | Proof.
split; first by move=> e; apply/mxrowP => j; rewrite !mxrowK.
by move=> + j => /mxrowP/(_ j); rewrite !mxrowK.
Qed. | Lemma | eq_mxrowP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxrowK",
"mxrowP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxrow m (A_ B_ : forall j, 'M[T]_(m, q_ j)) :
(forall j, A_ j = B_ j) -> (\mxrow_j A_ j = \mxrow_j B_ j). | Proof. by move=> /eq_mxrowP. Qed. | Lemma | eq_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"eq_mxrowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxcolP n (A_ B_ : forall i, 'M[T]_(p_ i, n)) :
(forall i, A_ i = B_ i) <-> (\mxcol_i A_ i = \mxcol_i B_ i). | Proof.
split; first by move=> e; apply/mxcolP => i; rewrite !mxcolK.
by move=> + i => /mxcolP/(_ i); rewrite !mxcolK.
Qed. | Lemma | eq_mxcolP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxcolK",
"mxcolP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxcol n (A_ B_ : forall i, 'M[T]_(p_ i, n)) :
(forall i, A_ i = B_ i) -> (\mxcol_i A_ i = \mxcol_i B_ i). | Proof. by move=> /eq_mxcolP. Qed. | Lemma | eq_mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"eq_mxcolP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxrow m (B_ : forall j, 'M[T]_(m, q_ j)) i :
row i (\mxrow_j B_ j) = \mxrow_j (row i (B_ j)). | Proof. by apply/rowP => l; rewrite !mxE. Qed. | Lemma | row_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxrow m (B_ : forall j, 'M[T]_(m, q_ j)) j :
col j (\mxrow_j B_ j) = col (sig2 j) (B_ (sig1 j)). | Proof. by apply/colP => l; rewrite !mxE. Qed. | Lemma | col_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col",
"colP",
"mxE",
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxcol n (B_ : forall i, 'M[T]_(p_ i, n)) i :
row i (\mxcol_i B_ i) = row (sig2 i) (B_ (sig1 i)). | Proof. by apply/rowP => l; rewrite !mxE. Qed. | Lemma | row_mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"row",
"rowP",
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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