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 |
|---|---|---|---|---|---|---|---|---|---|---|
tr_xrow m n i1 i2 (A : 'M_(m, n)) : (xrow i1 i2 A)^T = xcol i1 i2 A^T. | Proof. exact: tr_row_perm. Qed. | Lemma | tr_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",
... | [
"tr_row_perm",
"xcol",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_xcol m n j1 j2 (A : 'M_(m, n)) : (xcol j1 j2 A)^T = xrow j1 j2 A^T. | Proof. exact: tr_col_perm. Qed. | Lemma | tr_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",
... | [
"tr_col_perm",
"xcol",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_id n i (V : 'rV_n) : row i V = V. | Proof. by apply/rowP=> j; rewrite mxE [i]ord1. Qed. | Lemma | row_id | 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",
"ord1",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_id n j (V : 'cV_n) : col j V = V. | Proof. by apply/colP=> i; rewrite mxE [j]ord1. Qed. | Lemma | col_id | 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",
"ord1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_eq m1 m2 n i1 i2 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row i1 A1 = row i2 A2 -> A1 i1 =1 A2 i2. | Proof. by move/rowP=> eqA12 j; have /[!mxE] := eqA12 j. Qed. | Lemma | row_eq | 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",
... | [
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_eq m n1 n2 j1 j2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col j1 A1 = col j2 A2 -> A1^~ j1 =1 A2^~ j2. | Proof. by move/colP=> eqA12 i; have /[!mxE] := eqA12 i. Qed. | Lemma | col_eq | 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",
... | [
"col",
"colP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'_eq m n i0 (A B : 'M_(m, n)) :
row' i0 A = row' i0 B -> {in predC1 i0, A =2 B}. | Proof.
move=> /matrixP eqAB' i /[!inE]/[1!eq_sym]/unlift_some[i' -> _] j.
by have /[!mxE] := eqAB' i' j.
Qed. | Lemma | row'_eq | 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_sym",
"i0",
"inE",
"matrixP",
"mxE",
"predC1",
"row'",
"unlift_some"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col'_eq m n j0 (A B : 'M_(m, n)) :
col' j0 A = col' j0 B -> forall i, {in predC1 j0, A i =1 B i}. | Proof.
move=> /matrixP eqAB' i j /[!inE]/[1!eq_sym]/unlift_some[j' -> _].
by have /[!mxE] := eqAB' i j'.
Qed. | Lemma | col'_eq | 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",
... | [
"col'",
"eq_sym",
"inE",
"matrixP",
"mxE",
"predC1",
"unlift_some"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_row m n i0 (A : 'M_(m, n)) : (row i0 A)^T = col i0 A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | tr_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",
"col",
"i0",
"matrixP",
"mxE",
"row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_row' m n i0 (A : 'M_(m, n)) : (row' i0 A)^T = col' i0 A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | tr_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",
"col'",
"i0",
"matrixP",
"mxE",
"row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_col m n j0 (A : 'M_(m, n)) : (col j0 A)^T = row j0 A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | tr_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",
"row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_col' m n j0 (A : 'M_(m, n)) : (col' j0 A)^T = row' j0 A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | tr_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",
"row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_comp m1 m2 m3 n1 n2 n3
(f : 'I_m2 -> 'I_m1) (f' : 'I_m3 -> 'I_m2)
(g : 'I_n2 -> 'I_n1) (g' : 'I_n3 -> 'I_n2) (A : 'M_(m1, n1)) :
mxsub (f \o f') (g \o g') A = mxsub f' g' (mxsub f g A). | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | mxsub_comp | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowsub_comp m1 m2 m3 n
(f : 'I_m2 -> 'I_m1) (f' : 'I_m3 -> 'I_m2) (A : 'M_(m1, n)) :
rowsub (f \o f') A = rowsub f' (rowsub f A). | Proof. exact: mxsub_comp. Qed. | Lemma | rowsub_comp | 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",
... | [
"mxsub_comp",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colsub_comp m n n2 n3
(g : 'I_n2 -> 'I_n) (g' : 'I_n3 -> 'I_n2) (A : 'M_(m, n)) :
colsub (g \o g') A = colsub g' (colsub g A). | Proof. exact: mxsub_comp. Qed. | Lemma | colsub_comp | 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",
... | [
"colsub",
"mxsub_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsubrc m1 m2 n n2 f g (A : 'M_(m1, n)) :
mxsub f g A = rowsub f (colsub g A) :> 'M_(m2, n2). | Proof. exact: mxsub_comp. Qed. | Lemma | mxsubrc | 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",
... | [
"colsub",
"mxsub",
"mxsub_comp",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsubcr m1 m2 n n2 f g (A : 'M_(m1, n)) :
mxsub f g A = colsub g (rowsub f A) :> 'M_(m2, n2). | Proof. exact: mxsub_comp. Qed. | Lemma | mxsubcr | 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",
... | [
"colsub",
"mxsub",
"mxsub_comp",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowsub_cast m1 m2 n (eq_m : m1 = m2) (A : 'M_(m2, n)) :
rowsub (cast_ord eq_m) A = castmx (esym eq_m, erefl) A. | Proof. by case: _ / eq_m in A *; apply: (mxsub_eq_id (cast_ord_id _)). Qed. | Lemma | rowsub_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",
... | [
"apply",
"cast_ord",
"cast_ord_id",
"castmx",
"mxsub_eq_id",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colsub_cast m n1 n2 (eq_n : n1 = n2) (A : 'M_(m, n2)) :
colsub (cast_ord eq_n) A = castmx (erefl, esym eq_n) A. | Proof. by case: _ / eq_n in A *; apply: (mxsub_eq_id _ (cast_ord_id _)). Qed. | Lemma | colsub_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",
... | [
"apply",
"cast_ord",
"cast_ord_id",
"castmx",
"colsub",
"mxsub_eq_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_cast m1 m2 n1 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) A :
mxsub (cast_ord eq_m) (cast_ord eq_n) A = castmx (esym eq_m, esym eq_n) A. | Proof. by rewrite mxsubrc rowsub_cast colsub_cast castmx_comp/= etrans_id. Qed. | Lemma | mxsub_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",
... | [
"cast_ord",
"castmx",
"castmx_comp",
"colsub_cast",
"mxsub",
"mxsubrc",
"rowsub_cast"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmxEsub m1 m2 n1 n2 (eq_mn : (m1 = m2) * (n1 = n2)) A :
castmx eq_mn A = mxsub (cast_ord (esym eq_mn.1)) (cast_ord (esym eq_mn.2)) A. | Proof. by rewrite mxsub_cast !esymK; case: eq_mn. Qed. | Lemma | castmxEsub | 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",
"castmx",
"mxsub",
"mxsub_cast"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_mxsub m1 m2 n1 n2 f g (A : 'M_(m1, n1)) :
(mxsub f g A)^T = mxsub g f A^T :> 'M_(n2, m2). | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | trmx_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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxsub m1 m2 n1 n2
(f : 'I_m2 -> 'I_m1) (g : 'I_n2 -> 'I_n1) (A : 'M_(m1, n1)) i :
row i (mxsub f g A) = row (f i) (colsub g A). | Proof. by rewrite !rowEsub -!mxsub_comp. Qed. | Lemma | row_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",
... | [
"colsub",
"mxsub",
"mxsub_comp",
"row",
"rowEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxsub m1 m2 n1 n2
(f : 'I_m2 -> 'I_m1) (g : 'I_n2 -> 'I_n1) (A : 'M_(m1, n1)) i :
col i (mxsub f g A) = col (g i) (rowsub f A). | Proof. by rewrite !colEsub -!mxsub_comp. Qed. | Lemma | col_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",
... | [
"col",
"colEsub",
"mxsub",
"mxsub_comp",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_rowsub m1 m2 n (f : 'I_m2 -> 'I_m1) (A : 'M_(m1, n)) i :
row i (rowsub f A) = row (f i) A. | Proof. by rewrite row_mxsub mxsub_id. Qed. | Lemma | row_rowsub | 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",
... | [
"mxsub_id",
"row",
"row_mxsub",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_colsub m n1 n2 (g : 'I_n2 -> 'I_n1) (A : 'M_(m, n1)) i :
col i (colsub g A) = col (g i) A. | Proof. by rewrite col_mxsub mxsub_id. Qed. | Lemma | col_colsub | 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",
... | [
"col",
"col_mxsub",
"colsub",
"mxsub_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split_mxE | := apply/matrixP=> i j; do ![rewrite mxE | case: split => ?]. | Ltac | split_mxE | 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",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mx_key : unit. | Proof. by []. Qed. | Fact | row_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"
] | Concatenating two matrices, in either direction. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
row_mx (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) : 'M[R]_(m, n1 + n2) | :=
\matrix[row_mx_key]_(i, j)
match split j with inl j1 => A1 i j1 | inr j2 => A2 i j2 end. | Definition | 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",
... | [
"matrix",
"row_mx_key",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mx_key : unit. | Proof. by []. Qed. | Fact | col_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 | |
col_mx (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) : 'M[R]_(m1 + m2, n) | :=
\matrix[col_mx_key]_(i, j)
match split i with inl i1 => A1 i1 j | inr i2 => A2 i2 j end. | Definition | 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",
... | [
"col_mx_key",
"matrix",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lsubmx_key : unit. | Proof. by []. Qed. | Fact | lsubmx_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"
] | determines which submatrix is selected. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lsubmx (A : 'M[R]_(m, n1 + n2)) | :=
\matrix[lsubmx_key]_(i, j) A i (lshift n2 j). | Definition | 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",
... | [
"lshift",
"lsubmx_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsubmx_key : unit. | Proof. by []. Qed. | Fact | rsubmx_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 | |
rsubmx (A : 'M[R]_(m, n1 + n2)) | :=
\matrix[rsubmx_key]_(i, j) A i (rshift n1 j). | Definition | 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",
... | [
"matrix",
"rshift",
"rsubmx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
usubmx_key : unit. | Proof. by []. Qed. | Fact | usubmx_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 | |
usubmx (A : 'M[R]_(m1 + m2, n)) | :=
\matrix[usubmx_key]_(i, j) A (lshift m2 i) j. | Definition | 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",
... | [
"lshift",
"matrix",
"usubmx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dsubmx_key : unit. | Proof. by []. Qed. | Fact | dsubmx_key | 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",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dsubmx (A : 'M[R]_(m1 + m2, n)) | :=
\matrix[dsubmx_key]_(i, j) A (rshift m1 i) j. | Definition | 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",
... | [
"dsubmx_key",
"matrix",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxEl A1 A2 i j : row_mx A1 A2 i (lshift n2 j) = A1 i j. | Proof. by rewrite mxE (unsplitK (inl _ _)). Qed. | Lemma | row_mxEl | 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",
... | [
"lshift",
"mxE",
"row_mx",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxKl A1 A2 : lsubmx (row_mx A1 A2) = A1. | Proof. by apply/matrixP=> i j; rewrite mxE row_mxEl. Qed. | Lemma | row_mxKl | 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",
"matrixP",
"mxE",
"row_mx",
"row_mxEl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxEr A1 A2 i j : row_mx A1 A2 i (rshift n1 j) = A2 i j. | Proof. by rewrite mxE (unsplitK (inr _ _)). Qed. | Lemma | row_mxEr | 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",
... | [
"mxE",
"row_mx",
"rshift",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxKr A1 A2 : rsubmx (row_mx A1 A2) = A2. | Proof. by apply/matrixP=> i j; rewrite mxE row_mxEr. Qed. | Lemma | row_mxKr | 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_mx",
"row_mxEr",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hsubmxK A : row_mx (lsubmx A) (rsubmx A) = A. | Proof. by apply/matrixP=> i j /[!mxE]; case: split_ordP => k -> /[!mxE]. Qed. | Lemma | hsubmxK | 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",
"matrixP",
"mxE",
"row_mx",
"rsubmx",
"split_ordP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxEu A1 A2 i j : col_mx A1 A2 (lshift m2 i) j = A1 i j. | Proof. by rewrite mxE (unsplitK (inl _ _)). Qed. | Lemma | col_mxEu | 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",
... | [
"col_mx",
"lshift",
"mxE",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxKu A1 A2 : usubmx (col_mx A1 A2) = A1. | Proof. by apply/matrixP=> i j; rewrite mxE col_mxEu. Qed. | Lemma | col_mxKu | 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",
"col_mxEu",
"matrixP",
"mxE",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxEd A1 A2 i j : col_mx A1 A2 (rshift m1 i) j = A2 i j. | Proof. by rewrite mxE (unsplitK (inr _ _)). Qed. | Lemma | col_mxEd | 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",
... | [
"col_mx",
"mxE",
"rshift",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxKd A1 A2 : dsubmx (col_mx A1 A2) = A2. | Proof. by apply/matrixP=> i j; rewrite mxE col_mxEd. Qed. | Lemma | col_mxKd | 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",
"col_mxEd",
"dsubmx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lsubmxEsub : lsubmx = colsub (lshift _). | Proof. by rewrite /lsubmx /mxsub !unlock. Qed. | Lemma | lsubmxEsub | 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",
... | [
"colsub",
"lshift",
"lsubmx",
"mxsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsubmxEsub : rsubmx = colsub (@rshift _ _). | Proof. by rewrite /rsubmx /mxsub !unlock. Qed. | Lemma | rsubmxEsub | 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",
... | [
"colsub",
"mxsub",
"rshift",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
usubmxEsub : usubmx = rowsub (lshift _). | Proof. by rewrite /usubmx /mxsub !unlock. Qed. | Lemma | usubmxEsub | 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",
... | [
"lshift",
"mxsub",
"rowsub",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dsubmxEsub : dsubmx = rowsub (@rshift _ _). | Proof. by rewrite /dsubmx /mxsub !unlock. Qed. | Lemma | dsubmxEsub | 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",
... | [
"dsubmx",
"mxsub",
"rowsub",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_row_mx A1 A2 B1 B2 : row_mx A1 A2 = row_mx B1 B2 -> A1 = B1 /\ A2 = B2. | Proof.
move=> eqAB; move: (congr1 lsubmx eqAB) (congr1 rsubmx eqAB).
by rewrite !(row_mxKl, row_mxKr).
Qed. | Lemma | eq_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",
... | [
"lsubmx",
"row_mx",
"row_mxKl",
"row_mxKr",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_col_mx A1 A2 B1 B2 : col_mx A1 A2 = col_mx B1 B2 -> A1 = B1 /\ A2 = B2. | Proof.
move=> eqAB; move: (congr1 usubmx eqAB) (congr1 dsubmx eqAB).
by rewrite !(col_mxKu, col_mxKd).
Qed. | Lemma | eq_col_mx | 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",
... | [
"col_mx",
"col_mxKd",
"col_mxKu",
"dsubmx",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lsubmx_const (r : R) : lsubmx (const_mx r : 'M_(m, n1 + n2)) = const_mx r. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | lsubmx_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",
"lsubmx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsubmx_const (r : R) : rsubmx (const_mx r : 'M_(m, n1 + n2)) = const_mx r. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | rsubmx_const | 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",
"const_mx",
"matrixP",
"mxE",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mx_const a : row_mx (const_mx a) (const_mx a) = const_mx a. | Proof. by split_mxE. Qed. | Lemma | row_mx_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"const_mx",
"row_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mx_const a : col_mx (const_mx a) (const_mx a) = const_mx a. | Proof. by split_mxE. Qed. | Lemma | col_mx_const | 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_mx",
"const_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_usubmx A i : row i (usubmx A) = row (lshift m2 i) A. | Proof. by apply/rowP=> j; rewrite !mxE; congr (A _ _); apply/val_inj. Qed. | Lemma | row_usubmx | 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",
"lshift",
"mxE",
"row",
"rowP",
"usubmx",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_dsubmx A i : row i (dsubmx A) = row (rshift m1 i) A. | Proof. by apply/rowP=> j; rewrite !mxE; congr (A _ _); apply/val_inj. Qed. | Lemma | row_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",
"mxE",
"row",
"rowP",
"rshift",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_lsubmx A i : col i (lsubmx A) = col (lshift n2 i) A. | Proof. by apply/colP=> j; rewrite !mxE; congr (A _ _); apply/val_inj. Qed. | Lemma | col_lsubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col",
"colP",
"lshift",
"lsubmx",
"mxE",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_rsubmx A i : col i (rsubmx A) = col (rshift n1 i) A. | Proof. by apply/colP=> j; rewrite !mxE; congr (A _ _); apply/val_inj. Qed. | Lemma | col_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",
"col",
"colP",
"mxE",
"rshift",
"rsubmx",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_thin_mx m n (A : 'M_(m,0)) (B : 'M_(m,n)) : row_mx A B = B. | Proof.
apply/matrixP=> i j; rewrite mxE; case: splitP=> [|k H]; first by case.
by congr fun_of_matrix; exact: val_inj.
Qed. | Lemma | row_thin_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"fun_of_matrix",
"matrixP",
"mxE",
"row_mx",
"splitP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_flat_mx m n (A : 'M_(0,n)) (B : 'M_(m,n)) : col_mx A B = B. | Proof.
apply/matrixP=> i j; rewrite mxE; case: splitP => [|k H]; first by case.
by congr fun_of_matrix; exact: val_inj.
Qed. | Lemma | col_flat_mx | 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",
"col_mx",
"fun_of_matrix",
"matrixP",
"mxE",
"splitP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_lsub m n1 n2 (A : 'M_(m, n1 + n2)) : (lsubmx A)^T = usubmx A^T. | Proof. by split_mxE. Qed. | Lemma | trmx_lsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lsubmx",
"split_mxE",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_rsub m n1 n2 (A : 'M_(m, n1 + n2)) : (rsubmx A)^T = dsubmx A^T. | Proof. by split_mxE. Qed. | Lemma | trmx_rsub | 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",
... | [
"dsubmx",
"rsubmx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_row_mx m n1 n2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
(row_mx A1 A2)^T = col_mx A1^T A2^T. | Proof. by split_mxE. Qed. | Lemma | tr_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",
... | [
"col_mx",
"row_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_col_mx m1 m2 n (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
(col_mx A1 A2)^T = row_mx A1^T A2^T. | Proof. by split_mxE. Qed. | Lemma | tr_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",
... | [
"col_mx",
"row_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_usub m1 m2 n (A : 'M_(m1 + m2, n)) : (usubmx A)^T = lsubmx A^T. | Proof. by split_mxE. Qed. | Lemma | trmx_usub | 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",
... | [
"lsubmx",
"split_mxE",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_dsub m1 m2 n (A : 'M_(m1 + m2, n)) : (dsubmx A)^T = rsubmx A^T. | Proof. by split_mxE. Qed. | Lemma | trmx_dsub | 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",
... | [
"dsubmx",
"rsubmx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vsubmxK m1 m2 n (A : 'M_(m1 + m2, n)) : col_mx (usubmx A) (dsubmx A) = A. | Proof. by apply: trmx_inj; rewrite tr_col_mx trmx_usub trmx_dsub hsubmxK. Qed. | Lemma | vsubmxK | 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",
"dsubmx",
"hsubmxK",
"tr_col_mx",
"trmx_dsub",
"trmx_inj",
"trmx_usub",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_row_mx m m' n1 n2 (eq_m : m = m') A1 A2 :
castmx (eq_m, erefl _) (row_mx A1 A2)
= row_mx (castmx (eq_m, erefl n1) A1) (castmx (eq_m, erefl n2) A2). | Proof. by case: m' / eq_m. Qed. | Lemma | cast_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",
... | [
"castmx",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_col_mx m1 m2 n n' (eq_n : n = n') A1 A2 :
castmx (erefl _, eq_n) (col_mx A1 A2)
= col_mx (castmx (erefl m1, eq_n) A1) (castmx (erefl m2, eq_n) A2). | Proof. by case: n' / eq_n. Qed. | Lemma | cast_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",
... | [
"castmx",
"col_mx",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxA m n1 n2 n3 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) (A3 : 'M_(m, n3)) :
let cast := (erefl m, esym (addnA n1 n2 n3)) in
row_mx A1 (row_mx A2 A3) = castmx cast (row_mx (row_mx A1 A2) A3). | Proof.
apply: (canRL (castmxKV _ _)); apply/matrixP=> i j.
rewrite castmxE !mxE cast_ord_id; case: splitP => j1 /= def_j.
have: (j < n1 + n2) && (j < n1) by rewrite def_j lshift_subproof /=.
by move: def_j; do 2![case: splitP => // ? ->; rewrite ?mxE] => /ord_inj->.
case: splitP def_j => j2 ->{j} def_j /[!mxE].
h... | Lemma | row_mxA | 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",
... | [
"addnA",
"addnI",
"apply",
"cast_ord_id",
"castmx",
"castmxE",
"castmxKV",
"leqNgt",
"leq_add2l",
"leq_addr",
"lshift_subproof",
"ltn_add2l",
"matrixP",
"mxE",
"ord_inj",
"row_mx",
"splitP",
"val_inj"
] | This lemma has Prenex Implicits to help RL rewriting with castmx_sym. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
row_mxAx | := row_mxA. | Definition | row_mxAx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"row_mxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxA m1 m2 m3 n (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) (A3 : 'M_(m3, n)) :
let cast := (esym (addnA m1 m2 m3), erefl n) in
col_mx A1 (col_mx A2 A3) = castmx cast (col_mx (col_mx A1 A2) A3). | Proof. by apply: trmx_inj; rewrite trmx_cast !tr_col_mx -row_mxA. Qed. | Lemma | col_mxA | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addnA",
"apply",
"castmx",
"col_mx",
"row_mxA",
"tr_col_mx",
"trmx_cast",
"trmx_inj"
] | This lemma has Prenex Implicits to help RL rewrititng with castmx_sym. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
col_mxAx | := col_mxA. | Definition | col_mxAx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_row_mx m n1 n2 i0 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
row i0 (row_mx A1 A2) = row_mx (row i0 A1) (row i0 A2). | Proof.
by apply/matrixP=> i j /[!mxE]; case: (split j) => j' /[1!mxE].
Qed. | Lemma | row_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",
"i0",
"matrixP",
"mxE",
"row",
"row_mx",
"split"
] | bypass Prenex Implicits. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
col_col_mx m1 m2 n j0 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
col j0 (col_mx A1 A2) = col_mx (col j0 A1) (col j0 A2). | Proof. by apply: trmx_inj; rewrite !(tr_col, tr_col_mx, row_row_mx). Qed. | Lemma | col_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",
"col_mx",
"row_row_mx",
"tr_col",
"tr_col_mx",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'_row_mx m n1 n2 i0 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
row' i0 (row_mx A1 A2) = row_mx (row' i0 A1) (row' i0 A2). | Proof.
by apply/matrixP=> i j /[!mxE]; case: (split j) => j' /[1!mxE].
Qed. | Lemma | row'_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",
"i0",
"matrixP",
"mxE",
"row'",
"row_mx",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col'_col_mx m1 m2 n j0 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
col' j0 (col_mx A1 A2) = col_mx (col' j0 A1) (col' j0 A2). | Proof. by apply: trmx_inj; rewrite !(tr_col', tr_col_mx, row'_row_mx). Qed. | Lemma | col'_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'",
"col_mx",
"row'_row_mx",
"tr_col'",
"tr_col_mx",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colKl m n1 n2 j1 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col (lshift n2 j1) (row_mx A1 A2) = col j1 A1. | Proof. by apply/matrixP=> i j; rewrite !(row_mxEl, mxE). Qed. | Lemma | colKl | 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",
"lshift",
"matrixP",
"mxE",
"row_mx",
"row_mxEl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colKr m n1 n2 j2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col (rshift n1 j2) (row_mx A1 A2) = col j2 A2. | Proof. by apply/matrixP=> i j; rewrite !(row_mxEr, mxE). Qed. | Lemma | colKr | 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",
"row_mx",
"row_mxEr",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowKu m1 m2 n i1 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row (lshift m2 i1) (col_mx A1 A2) = row i1 A1. | Proof. by apply/matrixP=> i j; rewrite !(col_mxEu, mxE). Qed. | Lemma | rowKu | 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",
"col_mxEu",
"lshift",
"matrixP",
"mxE",
"row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowKd m1 m2 n i2 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row (rshift m1 i2) (col_mx A1 A2) = row i2 A2. | Proof. by apply/matrixP=> i j; rewrite !(col_mxEd, mxE). Qed. | Lemma | rowKd | 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",
"col_mxEd",
"matrixP",
"mxE",
"row",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col'Kl m n1 n2 j1 (A1 : 'M_(m, n1.+1)) (A2 : 'M_(m, n2)) :
col' (lshift n2 j1) (row_mx A1 A2) = row_mx (col' j1 A1) A2. | Proof.
apply/matrixP=> i /= j; symmetry; rewrite 2!mxE; case: split_ordP => j' ->.
by rewrite mxE -(row_mxEl _ A2); congr (row_mx _ _ _); apply: ord_inj.
rewrite -(row_mxEr A1); congr (row_mx _ _ _); apply: ord_inj => /=.
by rewrite /bump -ltnS -addSn ltn_addr.
Qed. | Lemma | col'Kl | 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",
... | [
"addSn",
"apply",
"bump",
"col'",
"lshift",
"ltnS",
"ltn_addr",
"matrixP",
"mxE",
"ord_inj",
"row_mx",
"row_mxEl",
"row_mxEr",
"split_ordP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'Ku m1 m2 n i1 (A1 : 'M_(m1.+1, n)) (A2 : 'M_(m2, n)) :
row' (lshift m2 i1) (@col_mx m1.+1 m2 n A1 A2) = col_mx (row' i1 A1) A2. | Proof.
by apply: trmx_inj; rewrite tr_col_mx !(@tr_row' _.+1) (@tr_col_mx _.+1) col'Kl.
Qed. | Lemma | row'Ku | 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'Kl",
"col_mx",
"lshift",
"row'",
"tr_col_mx",
"tr_row'",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx'_cast m n : 'I_n -> (m + n.-1)%N = (m + n).-1. | Proof. by case=> j /ltn_predK <-; rewrite addnS. Qed. | Lemma | mx'_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",
... | [
"addnS",
"ltn_predK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col'Kr m n1 n2 j2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col' (rshift n1 j2) (@row_mx m n1 n2 A1 A2)
= castmx (erefl m, mx'_cast n1 j2) (row_mx A1 (col' j2 A2)). | Proof.
apply/matrixP=> i j; symmetry; rewrite castmxE mxE cast_ord_id.
case: splitP => j' /= def_j.
rewrite mxE -(row_mxEl _ A2); congr (row_mx _ _ _); apply: ord_inj.
by rewrite /= def_j /bump leqNgt ltn_addr.
rewrite 2!mxE -(row_mxEr A1); congr (row_mx _ _ _ _); apply: ord_inj.
by rewrite /= def_j /bump leq_add2l... | Lemma | col'Kr | 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",
... | [
"addnCA",
"apply",
"bump",
"cast_ord_id",
"castmx",
"castmxE",
"col'",
"leqNgt",
"leq_add2l",
"ltn_addr",
"matrixP",
"mx'_cast",
"mxE",
"ord_inj",
"row_mx",
"row_mxEl",
"row_mxEr",
"rshift",
"splitP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'Kd m1 m2 n i2 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row' (rshift m1 i2) (col_mx A1 A2)
= castmx (mx'_cast m1 i2, erefl n) (col_mx A1 (row' i2 A2)). | Proof. by apply: trmx_inj; rewrite trmx_cast !(tr_row', tr_col_mx) col'Kr. Qed. | Lemma | row'Kd | 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",
"col'Kr",
"col_mx",
"mx'_cast",
"row'",
"rshift",
"tr_col_mx",
"tr_row'",
"trmx_cast",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mx Aul Aur Adl Adr : 'M_(m1 + m2, n1 + n2) | :=
col_mx (row_mx Aul Aur) (row_mx Adl Adr). | Definition | 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",
... | [
"col_mx",
"row_mx"
] | up left, up right, down left and down right components | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_block_mx Aul Aur Adl Adr Bul Bur Bdl Bdr :
block_mx Aul Aur Adl Adr = block_mx Bul Bur Bdl Bdr ->
[/\ Aul = Bul, Aur = Bur, Adl = Bdl & Adr = Bdr]. | Proof. by case/eq_col_mx; do 2!case/eq_row_mx=> -> ->. Qed. | Lemma | eq_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",
... | [
"block_mx",
"eq_col_mx",
"eq_row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mx_const a :
block_mx (const_mx a) (const_mx a) (const_mx a) (const_mx a) = const_mx a. | Proof. by split_mxE. Qed. | Lemma | block_mx_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",
... | [
"block_mx",
"const_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ulsubmx | := lsubmx (usubmx A). | Definition | ulsubmx | 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",
... | [
"lsubmx",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ursubmx | := rsubmx (usubmx A). | Definition | ursubmx | 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",
... | [
"rsubmx",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dlsubmx | := lsubmx (dsubmx A). | Definition | dlsubmx | 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",
... | [
"dsubmx",
"lsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drsubmx | := rsubmx (dsubmx A). | Definition | drsubmx | 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",
... | [
"dsubmx",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxK : block_mx ulsubmx ursubmx dlsubmx drsubmx = A. | Proof. by rewrite /block_mx !hsubmxK vsubmxK. Qed. | Lemma | submxK | 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",
... | [
"block_mx",
"dlsubmx",
"drsubmx",
"hsubmxK",
"ulsubmx",
"ursubmx",
"vsubmxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ulsubmxEsub : ulsubmx = mxsub (lshift _) (lshift _) A. | Proof. by rewrite /ulsubmx lsubmxEsub usubmxEsub -mxsub_comp. Qed. | Lemma | ulsubmxEsub | 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",
... | [
"lshift",
"lsubmxEsub",
"mxsub",
"mxsub_comp",
"ulsubmx",
"usubmxEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dlsubmxEsub : dlsubmx = mxsub (@rshift _ _) (lshift _) A. | Proof. by rewrite /dlsubmx lsubmxEsub dsubmxEsub -mxsub_comp. Qed. | Lemma | dlsubmxEsub | 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",
... | [
"dlsubmx",
"dsubmxEsub",
"lshift",
"lsubmxEsub",
"mxsub",
"mxsub_comp",
"rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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