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 |
|---|---|---|---|---|---|---|---|---|---|---|
ursubmxEsub : ursubmx = mxsub (lshift _) (@rshift _ _) A. | Proof. by rewrite /ursubmx rsubmxEsub usubmxEsub -mxsub_comp. Qed. | Lemma | ursubmxEsub | 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",
... | [
"lshift",
"mxsub",
"mxsub_comp",
"rshift",
"rsubmxEsub",
"ursubmx",
"usubmxEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drsubmxEsub : drsubmx = mxsub (@rshift _ _) (@rshift _ _) A. | Proof. by rewrite /drsubmx rsubmxEsub dsubmxEsub -mxsub_comp. Qed. | Lemma | drsubmxEsub | 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",
... | [
"drsubmx",
"dsubmxEsub",
"mxsub",
"mxsub_comp",
"rshift",
"rsubmxEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
A | := block_mx Aul Aur Adl Adr. | Let | A | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxEul i j : A (lshift m2 i) (lshift n2 j) = Aul i j. | Proof. by rewrite col_mxEu row_mxEl. Qed. | Lemma | block_mxEul | 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",
... | [
"col_mxEu",
"lshift",
"row_mxEl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxKul : ulsubmx A = Aul. | Proof. by rewrite /ulsubmx col_mxKu row_mxKl. Qed. | Lemma | block_mxKul | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mxKu",
"row_mxKl",
"ulsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxEur i j : A (lshift m2 i) (rshift n1 j) = Aur i j. | Proof. by rewrite col_mxEu row_mxEr. Qed. | Lemma | block_mxEur | 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",
... | [
"col_mxEu",
"lshift",
"row_mxEr",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxKur : ursubmx A = Aur. | Proof. by rewrite /ursubmx col_mxKu row_mxKr. Qed. | Lemma | block_mxKur | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mxKu",
"row_mxKr",
"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxEdl i j : A (rshift m1 i) (lshift n2 j) = Adl i j. | Proof. by rewrite col_mxEd row_mxEl. Qed. | Lemma | block_mxEdl | 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",
... | [
"col_mxEd",
"lshift",
"row_mxEl",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxKdl : dlsubmx A = Adl. | Proof. by rewrite /dlsubmx col_mxKd row_mxKl. Qed. | Lemma | block_mxKdl | 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",
... | [
"col_mxKd",
"dlsubmx",
"row_mxKl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxEdr i j : A (rshift m1 i) (rshift n1 j) = Adr i j. | Proof. by rewrite col_mxEd row_mxEr. Qed. | Lemma | block_mxEdr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mxEd",
"row_mxEr",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxKdr : drsubmx A = Adr. | Proof. by rewrite /drsubmx col_mxKd row_mxKr. Qed. | Lemma | block_mxKdr | 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",
... | [
"col_mxKd",
"drsubmx",
"row_mxKr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxEv : A = col_mx (row_mx Aul Aur) (row_mx Adl Adr). | Proof. by []. Qed. | Lemma | block_mxEv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mx",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_ulsub : (ulsubmx A)^T = ulsubmx A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | trmx_ulsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"ulsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_ursub : (ursubmx A)^T = dlsubmx A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | trmx_ursub | 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",
... | [
"apply",
"dlsubmx",
"matrixP",
"mxE",
"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_dlsub : (dlsubmx A)^T = ursubmx A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | trmx_dlsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finset",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"dlsubmx",
"matrixP",
"mxE",
"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_drsub : (drsubmx A)^T = drsubmx A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | trmx_drsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"drsubmx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_block_mx :
(block_mx Aul Aur Adl Adr)^T = block_mx Aul^T Adl^T Aur^T Adr^T. | Proof.
rewrite -[_^T]submxK -trmx_ulsub -trmx_ursub -trmx_dlsub -trmx_drsub.
by rewrite block_mxKul block_mxKur block_mxKdl block_mxKdr.
Qed. | Lemma | tr_block_mx | 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",
... | [
"block_mx",
"block_mxKdl",
"block_mxKdr",
"block_mxKul",
"block_mxKur",
"submxK",
"trmx_dlsub",
"trmx_drsub",
"trmx_ulsub",
"trmx_ursub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxEh :
block_mx Aul Aur Adl Adr = row_mx (col_mx Aul Adl) (col_mx Aur Adr). | Proof. by apply: trmx_inj; rewrite tr_block_mx tr_row_mx 2!tr_col_mx. Qed. | Lemma | block_mxEh | 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",
"block_mx",
"col_mx",
"row_mx",
"tr_block_mx",
"tr_col_mx",
"tr_row_mx",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mxA m1 m2 m3 n1 n2 n3
(A11 : 'M_(m1, n1)) (A12 : 'M_(m1, n2)) (A13 : 'M_(m1, n3))
(A21 : 'M_(m2, n1)) (A22 : 'M_(m2, n2)) (A23 : 'M_(m2, n3))
(A31 : 'M_(m3, n1)) (A32 : 'M_(m3, n2)) (A33 : 'M_(m3, n3)) :
let cast := (esym (addnA m1 m2 m3), esym (addnA n1 n2 n3)) in
let row1 := row_mx A12 A13 in let c... | Proof.
rewrite /= block_mxEh !col_mxA -cast_row_mx -block_mxEv -block_mxEh.
rewrite block_mxEv block_mxEh !row_mxA -cast_col_mx -block_mxEh -block_mxEv.
by rewrite castmx_comp etrans_id.
Qed. | Lemma | block_mxA | 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",
... | [
"addnA",
"block_mx",
"block_mxEh",
"block_mxEv",
"cast_col_mx",
"cast_row_mx",
"castmx",
"castmx_comp",
"col1",
"col3",
"col_mx",
"col_mxA",
"row1",
"row_mx",
"row_mxA"
] | This lemma has Prenex Implicits to help RL rewrititng with castmx_sym. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
block_mxAx | := block_mxA. | Definition | block_mxAx | 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",
... | [
"block_mxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_ind m (P : forall n, 'M[R]_(m, n) -> Type) :
(forall A, P 0 A) ->
(forall n c A, P n A -> P (1 + n)%N (row_mx c A)) ->
forall n A, P n A. | Proof.
move=> P0 PS; elim=> [//|n IHn] A.
by rewrite -[n.+1]/(1 + n)%N in A *; rewrite -[A]hsubmxK; apply: PS.
Qed. | Lemma | row_ind | 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",
... | [
"P0",
"apply",
"hsubmxK",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_ind n (P : forall m, 'M[R]_(m, n) -> Type) :
(forall A, P 0 A) ->
(forall m r A, P m A -> P (1 + m)%N (col_mx r A)) ->
forall m A, P m A. | Proof.
move=> P0 PS; elim=> [//|m IHm] A.
by rewrite -[m.+1]/(1 + m)%N in A *; rewrite -[A]vsubmxK; apply: PS.
Qed. | Lemma | col_ind | 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",
... | [
"P0",
"apply",
"col_mx",
"vsubmxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_ind (P : forall m n, 'M[R]_(m, n) -> Type) :
(forall m A, P m 0 A) ->
(forall n A, P 0 n A) ->
(forall m n x r c A, P m n A -> P (1 + m)%N (1 + n)%N (block_mx x r c A)) ->
forall m n A, P m n A. | Proof.
move=> P0l P0r PS; elim=> [|m IHm] [|n] A; do ?by [apply: P0l|apply: P0r].
by rewrite -[A](@submxK 1 _ 1); apply: PS.
Qed. | Lemma | mx_ind | 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",
"block_mx",
"submxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_rect | := mx_ind. | Definition | matrix_rect | 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",
... | [
"mx_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_rec | := mx_ind. | Definition | matrix_rec | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mx_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_ind | := mx_ind. | Definition | matrix_ind | algebra | algebra/matrix.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mx_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqmx_ind (P : forall n, 'M[R]_n -> Type) :
(forall A, P 0 A) ->
(forall n x r c A, P n A -> P (1 + n)%N (block_mx x r c A)) ->
forall n A, P n A. | Proof.
by move=> P0 PS; elim=> [//|n IHn] A; rewrite -[A](@submxK 1 _ 1); apply: PS.
Qed. | Lemma | sqmx_ind | 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",
... | [
"P0",
"apply",
"block_mx",
"submxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ringmx_ind (P : forall n, 'M[R]_n.+1 -> Type) :
(forall x, P 0 x) ->
(forall n x (r : 'rV_n.+1) (c : 'cV_n.+1) A,
P n A -> P (1 + n)%N (block_mx x r c A)) ->
forall n A, P n A. | Proof.
by move=> P0 PS; elim=> [//|n IHn] A; rewrite -[A](@submxK 1 _ 1); apply: PS.
Qed. | Lemma | ringmx_ind | 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",
... | [
"P0",
"apply",
"block_mx",
"submxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_ind
(weight : forall m n, 'M[R]_(m, n) -> nat)
(sub : forall m n m' n', ('I_m' -> 'I_m) -> ('I_n' -> 'I_n) -> Prop)
(P : forall m n, 'M[R]_(m, n) -> Type) :
(forall m n (A : 'M[R]_(m, n)),
(forall m' n' f g, weight m' n' (mxsub f g A) < weight m n A ->
sub m n m' n' ... | Proof.
move=> Psub m n A; have [k] := ubnP (weight m n A).
elim: k => [//|k IHk] in m n A *.
rewrite ltnS => lt_A_k; apply: Psub => m' n' f g lt_A'_A ?.
by apply: IHk; apply: leq_trans lt_A_k.
Qed. | Lemma | mxsub_ind | 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",
... | [
"Psub",
"apply",
"leq_trans",
"ltnS",
"mxsub",
"n'",
"nat",
"sub",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_cast : #|{:'I_m * 'I_n}| = (m * n)%N. | Proof. by rewrite card_prod !card_ord. Qed. | Lemma | mxvec_cast | 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",
... | [
"card_ord",
"card_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_index (i : 'I_m) (j : 'I_n) | :=
cast_ord mxvec_cast (enum_rank (i, j)). | Definition | mxvec_index | 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",
... | [
"cast_ord",
"enum_rank",
"mxvec_cast"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_mxvec_index : 'I_(m * n) -> Type | :=
isMxvecIndex i j : is_mxvec_index (mxvec_index i j). | Variant | is_mxvec_index | 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",
... | [
"mxvec_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_indexP k : is_mxvec_index k. | Proof.
rewrite -[k](cast_ordK (esym mxvec_cast)) esymK.
by rewrite -[_ k]enum_valK; case: (enum_val _).
Qed. | Lemma | mxvec_indexP | 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",
... | [
"cast_ordK",
"enum_val",
"enum_valK",
"is_mxvec_index",
"mxvec_cast"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_of_mxvec_index k (i_k : is_mxvec_index k) | :=
let: isMxvecIndex i j := i_k in (i, j). | Coercion | pair_of_mxvec_index | 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",
... | [
"is_mxvec_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec (A : 'M[R]_(m, n)) | :=
castmx (erefl _, mxvec_cast) (\row_k A (enum_val k).1 (enum_val k).2). | Definition | mxvec | 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",
... | [
"castmx",
"enum_val",
"mxvec_cast"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vec_mx_key : unit. | Proof. by []. Qed. | Fact | vec_mx_key | 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",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vec_mx (u : 'rV[R]_(m * n)) | :=
\matrix[vec_mx_key]_(i, j) u 0 (mxvec_index i j). | Definition | vec_mx | 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",
... | [
"matrix",
"mxvec_index",
"vec_mx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvecE A i j : mxvec A 0 (mxvec_index i j) = A i j. | Proof. by rewrite castmxE mxE cast_ordK enum_rankK. Qed. | Lemma | mxvecE | 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",
... | [
"cast_ordK",
"castmxE",
"enum_rankK",
"mxE",
"mxvec",
"mxvec_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvecK : cancel mxvec vec_mx. | Proof. by move=> A; apply/matrixP=> i j; rewrite mxE mxvecE. Qed. | Lemma | mxvecK | 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",
"mxvec",
"mxvecE",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vec_mxK : cancel vec_mx mxvec. | Proof.
by move=> u; apply/rowP=> k; case/mxvec_indexP: k => i j; rewrite mxvecE mxE.
Qed. | Lemma | vec_mxK | 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",
"mxvec",
"mxvecE",
"mxvec_indexP",
"rowP",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
curry_mxvec_bij : {on 'I_(m * n), bijective (uncurry mxvec_index)}. | Proof.
exists (enum_val \o cast_ord (esym mxvec_cast)) => [[i j] _ | k _] /=.
by rewrite cast_ordK enum_rankK.
by case/mxvec_indexP: k => i j /=; rewrite cast_ordK enum_rankK.
Qed. | Lemma | curry_mxvec_bij | 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",
... | [
"cast_ord",
"cast_ordK",
"enum_rankK",
"enum_val",
"mxvec_cast",
"mxvec_index",
"mxvec_indexP",
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_key : unit. | Proof. by []. Qed. | Fact | map_mx_key | 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",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx m n (A : 'M_(m, n)) | := \matrix[map_mx_key]_(i, j) f (A i j). | Definition | map_mx | 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",
... | [
"map_mx_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ^f" | := (map_mx A) : ring_scope. | Notation | A ^f | 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",
... | [
"map_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_trmx : A^f^T = A^T^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_trmx | 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 | |
map_const_mx a : (const_mx a)^f = const_mx (f a) :> 'M_(m, n). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_const_mx | 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",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_row i : (row i A)^f = row i A^f. | Proof. by apply/rowP=> j; rewrite !mxE. Qed. | Lemma | map_row | 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 | |
map_col j : (col j A)^f = col j A^f. | Proof. by apply/colP=> i; rewrite !mxE. Qed. | Lemma | map_col | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_row' i0 : (row' i0 A)^f = row' i0 A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_row' | 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",
"i0",
"matrixP",
"mxE",
"row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_col' j0 : (col' j0 A)^f = col' j0 A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_col' | 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'",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxsub m' n' g h : (@mxsub _ _ _ m' n' g h A)^f = mxsub g h A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_mxsub | 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",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_row_perm s : (row_perm s A)^f = row_perm s A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_row_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",
... | [
"apply",
"matrixP",
"mxE",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_col_perm s : (col_perm s A)^f = col_perm s A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_col_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",
... | [
"apply",
"col_perm",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_xrow i1 i2 : (xrow i1 i2 A)^f = xrow i1 i2 A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_xrow | algebra | algebra/matrix.v | [
"HB",
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... | [
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"mxE",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_xcol j1 j2 : (xcol j1 j2 A)^f = xcol j1 j2 A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_xcol | algebra | algebra/matrix.v | [
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... | [
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"mxE",
"xcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_castmx m' n' c : (castmx c A)^f = castmx c A^f :> 'M_(m', n'). | Proof. by apply/matrixP=> i j; rewrite !(castmxE, mxE). Qed. | Lemma | map_castmx | algebra | algebra/matrix.v | [
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... | [
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"matrixP",
"mxE",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_conform_mx m' n' (B : 'M_(m', n')) :
(conform_mx B A)^f = conform_mx B^f A^f. | Proof.
move: B; have [[<- <-] B|] := eqVneq (m, n) (m', n').
by rewrite !conform_mx_id.
by rewrite negb_and => neq_mn B; rewrite !nonconform_mx.
Qed. | Lemma | map_conform_mx | algebra | algebra/matrix.v | [
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... | [
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"eqVneq",
"n'",
"nonconform_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxvec : (mxvec A)^f = mxvec A^f. | Proof. by apply/rowP=> i; rewrite !(castmxE, mxE). Qed. | Lemma | map_mxvec | algebra | algebra/matrix.v | [
"HB",
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... | [
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"mxE",
"mxvec",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_vec_mx (v : 'rV_(m * n)) : (vec_mx v)^f = vec_mx v^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_vec_mx | algebra | algebra/matrix.v | [
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"mathcomp",
"ssreflect",
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... | [
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"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_row_mx : (row_mx Aul Aur)^f = row_mx Aul^f Aur^f. | Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. Qed. | Lemma | map_row_mx | algebra | algebra/matrix.v | [
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... | [
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"row_mx",
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_col_mx : (col_mx Aul Adl)^f = col_mx Aul^f Adl^f. | Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. Qed. | Lemma | map_col_mx | algebra | algebra/matrix.v | [
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... | [
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"matrixP",
"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_block_mx :
(block_mx Aul Aur Adl Adr)^f = block_mx Aul^f Aur^f Adl^f Adr^f. | Proof. by apply/matrixP=> i j; do 3![rewrite !mxE //; case: split => ?]. Qed. | Lemma | map_block_mx | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
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"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_lsubmx : (lsubmx Bh)^f = lsubmx Bh^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_lsubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rsubmx : (rsubmx Bh)^f = rsubmx Bh^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_rsubmx | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
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"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_usubmx : (usubmx Bv)^f = usubmx Bv^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_usubmx | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
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"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_dsubmx : (dsubmx Bv)^f = dsubmx Bv^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_dsubmx | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
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"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_ulsubmx : (ulsubmx B)^f = ulsubmx B^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_ulsubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
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"ulsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_ursubmx : (ursubmx B)^f = ursubmx B^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_ursubmx | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
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"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_dlsubmx : (dlsubmx B)^f = dlsubmx B^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_dlsubmx | algebra | algebra/matrix.v | [
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... | [
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"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_drsubmx : (drsubmx B)^f = drsubmx B^f. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map_drsubmx | algebra | algebra/matrix.v | [
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... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"M ^ phi" | := (map_mx phi M). | Notation | M ^ phi | algebra | algebra/matrix.v | [
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... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_comp (f : R -> S) (g : S -> T)
(M : 'M_(m, n)) : M ^ (g \o f) = (M ^ f) ^ g. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | map_mx_comp | algebra | algebra/matrix.v | [
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... | [
"apply",
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"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_map_mx (g f : R -> S) (M : 'M_(m, n)) :
(forall i j, f (M i j) = g (M i j)) -> M ^ f = M ^ g. | Proof. by move=> fg; apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | eq_in_map_mx | algebra | algebra/matrix.v | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_map_mx (g f : R -> S) : f =1 g ->
forall (M : 'M_(m, n)), M ^ f = M ^ g. | Proof. by move=> eq_fg M; apply/eq_in_map_mx. Qed. | Lemma | eq_map_mx | algebra | algebra/matrix.v | [
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... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_id_in (f : R -> R) (M : 'M_(m, n)) :
(forall i j, f (M i j) = M i j) -> M ^ f = M. | Proof. by move=> fM; apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | map_mx_id_in | algebra | algebra/matrix.v | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_id (f : R -> R) : f =1 id -> forall M : 'M_(m, n), M ^ f = M. | Proof. by move=> fid M; rewrite map_mx_id_in. Qed. | Lemma | map_mx_id | algebra | algebra/matrix.v | [
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... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx_key : unit. | Proof. by []. Qed. | Fact | map2_mx_key | algebra | algebra/matrix.v | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx m n (A : 'M_(m, n)) (B : 'M_(m, n)) | :=
\matrix[map2_mx_key]_(i, j) f (A i j) (B i j). | Definition | map2_mx | algebra | algebra/matrix.v | [
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... | [
"map2_mx_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_trmx : (map2_mx A B)^T = map2_mx A^T B^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_trmx | algebra | algebra/matrix.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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... | [
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"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_const_mx a b :
map2_mx (const_mx a) (const_mx b) = const_mx (f a b) :> 'M_(m, n). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_const_mx | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
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"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_row i : map2_mx (row i A) (row i B) = row i (map2_mx A B). | Proof. by apply/rowP=> j; rewrite !mxE. Qed. | Lemma | map2_row | 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",
"map2_mx",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_col j : map2_mx (col j A) (col j B) = col j (map2_mx A B). | Proof. by apply/colP=> i; rewrite !mxE. Qed. | Lemma | map2_col | 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",
"map2_mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_row' i0 : map2_mx (row' i0 A) (row' i0 B) = row' i0 (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_row' | 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",
"i0",
"map2_mx",
"matrixP",
"mxE",
"row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_col' j0 : map2_mx (col' j0 A) (col' j0 B) = col' j0 (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_col' | 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'",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mxsub m' n' g h :
map2_mx (@mxsub _ _ _ m' n' g h A) (@mxsub _ _ _ m' n' g h B) =
mxsub g h (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_mxsub | 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",
"map2_mx",
"matrixP",
"mxE",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_row_perm s :
map2_mx (row_perm s A) (row_perm s B) = row_perm s (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_row_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",
... | [
"apply",
"map2_mx",
"matrixP",
"mxE",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_col_perm s :
map2_mx (col_perm s A) (col_perm s B) = col_perm s (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_col_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",
... | [
"apply",
"col_perm",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_xrow i1 i2 :
map2_mx (xrow i1 i2 A) (xrow i1 i2 B) = xrow i1 i2 (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_xrow | 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",
"map2_mx",
"matrixP",
"mxE",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_xcol j1 j2 :
map2_mx (xcol j1 j2 A) (xcol j1 j2 B) = xcol j1 j2 (map2_mx A B). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_xcol | 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",
"map2_mx",
"matrixP",
"mxE",
"xcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_castmx m' n' c :
map2_mx (castmx c A) (castmx c B) = castmx c (map2_mx A B) :> 'M_(m', n'). | Proof. by apply/matrixP=> i j; rewrite !(castmxE, mxE). Qed. | Lemma | map2_castmx | 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",
"castmx",
"castmxE",
"map2_mx",
"matrixP",
"mxE",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_conform_mx m' n' (A' : 'M_(m', n')) (B' : 'M_(m', n')) :
map2_mx (conform_mx A' A) (conform_mx B' B) =
conform_mx (map2_mx A' B') (map2_mx A B). | Proof.
move: A' B'; have [[<- <-] A' B'|] := eqVneq (m, n) (m', n').
by rewrite !conform_mx_id.
by rewrite negb_and => neq_mn A' B'; rewrite !nonconform_mx.
Qed. | Lemma | map2_conform_mx | 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'",
"conform_mx",
"conform_mx_id",
"eqVneq",
"map2_mx",
"n'",
"nonconform_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mxvec : map2_mx (mxvec A) (mxvec B) = mxvec (map2_mx A B). | Proof. by apply/rowP=> i; rewrite !(castmxE, mxE). Qed. | Lemma | map2_mxvec | 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",
"castmxE",
"map2_mx",
"mxE",
"mxvec",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_vec_mx (v : 'rV_(m * n)) (w : 'rV_(m * n)) :
map2_mx (vec_mx v) (vec_mx w) = vec_mx (map2_mx v w). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_vec_mx | 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",
"map2_mx",
"matrixP",
"mxE",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_row_mx :
map2_mx (row_mx Aul Aur) (row_mx A'ul A'ur) =
row_mx (map2_mx Aul A'ul) (map2_mx Aur A'ur). | Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. Qed. | Lemma | map2_row_mx | 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",
"map2_mx",
"matrixP",
"mxE",
"row_mx",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_col_mx :
map2_mx (col_mx Aul Adl) (col_mx A'ul A'dl) =
col_mx (map2_mx Aul A'ul) (map2_mx Adl A'dl). | Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. Qed. | Lemma | map2_col_mx | 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_mx",
"map2_mx",
"matrixP",
"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_block_mx :
map2_mx (block_mx Aul Aur Adl Adr) (block_mx A'ul A'ur A'dl A'dr) =
block_mx
(map2_mx Aul A'ul) (map2_mx Aur A'ur) (map2_mx Adl A'dl) (map2_mx Adr A'dr). | Proof. by apply/matrixP=> i j; do 3![rewrite !mxE //; case: split => ?]. Qed. | Lemma | map2_block_mx | 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",
"block_mx",
"map2_mx",
"matrixP",
"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_lsubmx : map2_mx (lsubmx Bh) (lsubmx B'h) = lsubmx (map2_mx Bh B'h). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_lsubmx | 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",
"lsubmx",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_rsubmx : map2_mx (rsubmx Bh) (rsubmx B'h) = rsubmx (map2_mx Bh B'h). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_rsubmx | 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",
"map2_mx",
"matrixP",
"mxE",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_usubmx : map2_mx (usubmx Bv) (usubmx B'v) = usubmx (map2_mx Bv B'v). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_usubmx | 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",
"map2_mx",
"matrixP",
"mxE",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_dsubmx : map2_mx (dsubmx Bv) (dsubmx B'v) = dsubmx (map2_mx Bv B'v). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_dsubmx | 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",
"dsubmx",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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