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 |
|---|---|---|---|---|---|---|---|---|---|---|
col_mxcol n (B_ : forall i, 'M[T]_(p_ i, n)) j :
col j (\mxcol_i B_ i) = \mxcol_i (col j (B_ i)). | Proof. by apply/colP => l; rewrite !mxE. Qed. | Lemma | col_mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col",
"colP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxblock B_ i :
row i (\mxblock_(i, j) B_ i j) = \mxrow_j row (sig2 i) (B_ (sig1 i) j). | Proof. by apply/rowP => l; rewrite !mxE. Qed. | Lemma | row_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mxE",
"row",
"rowP",
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxblock B_ j :
col j (\mxblock_(i, j) B_ i j) = \mxcol_i col (sig2 j) (B_ i (sig1 j)). | Proof. by apply/colP => l; rewrite !mxE. Qed. | Lemma | col_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col",
"colP",
"mxE",
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxblock_ ( i < m , j < n ) E" | :=
(mxblock (fun (i : 'I_m) (j : 'I_ n) => E)) (only parsing) : ring_scope. | Notation | \mxblock_ ( i < m , j < n ) E | 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",
... | [
"mxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxblock_ ( i , j < n ) E" | :=
(\mxblock_(i < n, j < n) E) (only parsing) : ring_scope. | Notation | \mxblock_ ( i , j < n ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxblock_ ( i , j ) E" | := (\mxblock_(i < _, j < _) E) : ring_scope. | Notation | \mxblock_ ( i , j ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxrow_ ( j < m ) E" | := (mxrow (fun (j : 'I_m) => E))
(only parsing) : ring_scope. | Notation | \mxrow_ ( j < m ) E | 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",
... | [
"mxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxrow_ j E" | := (\mxrow_(j < _) E) : ring_scope. | Notation | \mxrow_ j E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxcol_ ( i < m ) E" | := (mxcol (fun (i : 'I_m) => E))
(only parsing) : ring_scope. | Notation | \mxcol_ ( i < m ) E | 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",
... | [
"mxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxcol_ i E" | := (\mxcol_(i < _) E) : ring_scope. | Notation | \mxcol_ i E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_mxblock {T : Type} {p q : nat} {p_ : 'I_p -> nat} {q_ : 'I_q -> nat}
(B_ : forall i j, 'M[T]_(p_ i, q_ j)) :
(\mxblock_(i, j) B_ i j)^T = \mxblock_(i, j) (B_ j i)^T. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | tr_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_mxrow n (B_ : forall j, 'M[T]_(n, p_ j)) :
(\mxrow_j B_ j)^T = \mxcol_i (B_ i)^T. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | tr_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_mxcol n (B_ : forall i, 'M[T]_(p_ i, n)) :
(\mxcol_i B_ i)^T = \mxrow_i (B_ i)^T. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | tr_mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_submxblock (A : 'M[T]_sp) i j :
(submxblock A i j)^T = (submxblock A^T j i). | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | tr_submxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_submxrow n (A : 'M[T]_(n, sp)) j :
(submxrow A j)^T = (submxcol A^T j). | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | tr_submxrow | 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",
"sp",
"submxcol",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_submxcol n (A : 'M[T]_(sp, n)) i :
(submxcol A i)^T = (submxrow A^T i). | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | tr_submxcol | 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",
"sp",
"submxcol",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsize_recl : (p_ ord0 + \sum_i p_ (lift ord0 i) = (\sum_i p_ i))%N. | Proof. by rewrite big_ord_recl. Qed. | Lemma | mxsize_recl | 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",
... | [
"big_ord_recl",
"lift",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrow_recl n (B_ : forall j, 'M[T]_(n, p_ j)) :
\mxrow_j B_ j = castmx (erefl, mxsize_recl)
(row_mx (B_ 0) (\mxrow_j B_ (lift ord0 j))). | Proof.
apply/mxrowP => i; rewrite mxrowK.
apply/matrixP => j k; rewrite !(castmxE, mxE)/=.
case: splitP => l /=; do [
rewrite [LHS]RankEsum big_mkcond big_ord_recl -big_mkcond/=;
rewrite /bump/= -addnA cast_ord_id;
under eq_bigl do rewrite add1n -ltn_predRL/=].
case: posnP => i0; last first.
by move=>... | Lemma | mxrow_recl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank2K",
"RankEsum",
"add1n",
"addnA",
"addnI",
"apply",
"big_mkcond",
"big_ord_recl",
"big_pred0_eq",
"bump",
"cast_ord_id",
"castmx",
"castmxE",
"eqRank",
"eq_bigl",
"i0",
"last",
"lift",
"ltn_ord",
"ltn_predRL",
"ltn_subLR",
"ltn_subRL",
"matrixP",
"mxE",
"mxrowK"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcol_recu {T : Type} {p : nat} {p_ : 'I_p.+1 -> nat} m
(B_ : forall j, 'M[T]_(p_ j, m)) :
\mxcol_j B_ j = castmx (mxsize_recl, erefl)
(col_mx (B_ 0) (\mxcol_j B_ (lift ord0 j))). | Proof.
by apply: trmx_inj; rewrite trmx_cast tr_col_mx !tr_mxcol mxrow_recl.
Qed. | Lemma | mxcol_recu | 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_mx",
"lift",
"mxrow_recl",
"mxsize_recl",
"nat",
"ord0",
"tr_col_mx",
"tr_mxcol",
"trmx_cast",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
e | := (mxsize_recl, mxsize_recl). | Notation | e | 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",
... | [
"mxsize_recl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
l0 | := (lift ord0). | Notation | l0 | 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",
... | [
"lift",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblock_recu {p q : nat} {p_ : 'I_p.+1 -> nat} {q_ : 'I_q -> nat}
(B_ : forall i j, 'M[T]_(p_ i, q_ j)) :
\mxblock_(i, j) B_ i j = castmx (mxsize_recl, erefl) (col_mx
(\mxrow_j B_ ord0 j)
(\mxblock_(i, j) B_ (l0 i) j)). | Proof. by rewrite !mxblockEv mxcol_recu. Qed. | Lemma | mxblock_recu | 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",
"l0",
"mxblockEv",
"mxcol_recu",
"mxsize_recl",
"nat",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblock_recl {p q : nat} {p_ : 'I_p -> nat} {q_ : 'I_q.+1 -> nat}
(B_ : forall i j, 'M[T]_(p_ i, q_ j)) :
\mxblock_(i, j) B_ i j = castmx (erefl, mxsize_recl)
(row_mx (\mxcol_i B_ i ord0) (\mxblock_(i, j) B_ i (l0 j))). | Proof. by rewrite !mxblockEh mxrow_recl. Qed. | Lemma | mxblock_recl | 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",
"l0",
"mxblockEh",
"mxrow_recl",
"mxsize_recl",
"nat",
"ord0",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblock_recul {p q : nat} {p_ : 'I_p.+1 -> nat} {q_ : 'I_q.+1 -> nat}
(B_ : forall i j, 'M[T]_(p_ i, q_ j)) :
\mxblock_(i, j) B_ i j = castmx e (block_mx
(B_ 0 0) (\mxrow_j B_ ord0 (l0 j))
(\mxcol_i B_ (l0 i) ord0) (\mxblock_(i, j) B_ (l0 i) (l0 j))). | Proof.
rewrite mxblock_recl mxcol_recu mxblock_recu -cast_row_mx -block_mxEh.
by rewrite castmx_comp; apply: eq_castmx.
Qed. | Lemma | mxblock_recul | 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",
"block_mxEh",
"cast_row_mx",
"castmx",
"castmx_comp",
"eq_castmx",
"l0",
"mxblock_recl",
"mxblock_recu",
"mxcol_recu",
"nat",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrowEblock {q : nat} {q_ : 'I_q -> nat} m
(R_ : forall j, 'M[T]_(m, q_ j)) :
(\mxrow_j R_ j) =
castmx (big_ord1 _ (fun=> m), erefl) (\mxblock_(i < 1, j < q) R_ j). | Proof.
rewrite mxblock_recu castmx_comp.
apply/matrixP => i j; rewrite !castmxE !mxE/=; case: splitP => //=.
by move=> k /val_inj->; rewrite ?cast_ord_id ?mxE//=.
by move=> [k klt]; suff: false by []; rewrite big_ord0 in klt.
Qed. | Lemma | mxrowEblock | 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",
"big_ord0",
"big_ord1",
"cast_ord_id",
"castmx",
"castmxE",
"castmx_comp",
"matrixP",
"mxE",
"mxblock_recu",
"nat",
"splitP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcolEblock {p : nat} {p_ : 'I_p -> nat} n
(C_ : forall i, 'M[T]_(p_ i, n)) :
(\mxcol_i C_ i) =
castmx (erefl, big_ord1 _ (fun=> n)) (\mxblock_(i < p, j < 1) C_ i). | Proof.
by apply: trmx_inj; rewrite tr_mxcol mxrowEblock trmx_cast tr_mxblock.
Qed. | Lemma | mxcolEblock | 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",
"big_ord1",
"castmx",
"mxrowEblock",
"nat",
"tr_mxblock",
"tr_mxcol",
"trmx_cast",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxEmxrow m n (A : 'M[T]_(m, n)) :
A = castmx (erefl, big_ord1 _ (fun=> n)) (\mxrow__ A). | Proof.
apply/matrixP => i j; rewrite castmxE !mxE/= cast_ord_id.
congr (A i); set j' := cast_ord _ _.
suff -> : j' = (tagnat.Rank 0 j) by apply/val_inj; rewrite tagnat.Rank2K.
by apply/val_inj; rewrite [RHS]tagnat.RankEsum/= big_pred0_eq add0n.
Qed. | Lemma | mxEmxrow | 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",
... | [
"Rank",
"Rank2K",
"RankEsum",
"add0n",
"apply",
"big_ord1",
"big_pred0_eq",
"cast_ord",
"cast_ord_id",
"castmx",
"castmxE",
"matrixP",
"mxE",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxEmxcol m n (A : 'M[T]_(m, n)) :
A = castmx (big_ord1 _ (fun=> m), erefl) (\mxcol__ A). | Proof. by apply: trmx_inj; rewrite trmx_cast tr_mxcol [LHS]mxEmxrow. Qed. | Lemma | mxEmxcol | 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",
"big_ord1",
"castmx",
"mxEmxrow",
"tr_mxcol",
"trmx_cast",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxEmxblock m n (A : 'M[T]_(m, n)) :
A = castmx (big_ord1 _ (fun=> m), big_ord1 _ (fun=> n))
(\mxblock_(i < 1, j < 1) A). | Proof. by rewrite [LHS]mxEmxrow mxrowEblock castmx_comp; apply: eq_castmx. Qed. | Lemma | mxEmxblock | 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",
"big_ord1",
"castmx",
"castmx_comp",
"eq_castmx",
"mxEmxrow",
"mxrowEblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrowD m (R_ R'_ : forall j, 'M[V]_(m, q_ j)) :
\mxrow_j (R_ j + R'_ j) = \mxrow_j (R_ j) + \mxrow_j (R'_ j). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxrowD | 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 | |
mxrow0 m : \mxrow_j (0 : 'M[V]_(m, q_ j)) = 0. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxrow0 | 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 | |
mxrow_const m a : \mxrow_j (const_mx a : 'M[V]_(m, q_ j)) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxrow_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",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrow_sum (J : finType) m
(R_ : forall i j, 'M[V]_(m, q_ j)) (P : {pred J}) :
\mxrow_j (\sum_(i | P i) R_ i j) = \sum_(i | P i) \mxrow_j (R_ i j). | Proof.
apply/matrixP => i j; rewrite !(mxE, summxE).
by apply: eq_bigr => l; rewrite !mxE.
Qed. | Lemma | mxrow_sum | 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",
"eq_bigr",
"matrixP",
"mxE",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrowD m (B B' : 'M[V]_(m, sq)) j :
submxrow (B + B') j = submxrow B j + submxrow B' j. | Proof. by apply/matrixP => i i'; rewrite !mxE. Qed. | Lemma | submxrowD | 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",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrow0 m j : submxrow (0 : 'M[V]_(m, sq)) j = 0. | Proof. by apply/matrixP=> i i'; rewrite !mxE. Qed. | Lemma | submxrow0 | 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",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrow_sum (J : finType) m
(R_ : forall i, 'M[V]_(m, sq)) (P : {pred J}) j:
submxrow (\sum_(i | P i) R_ i) j = \sum_(i | P i) submxrow (R_ i) j. | Proof.
apply/matrixP => i i'; rewrite !(mxE, summxE).
by apply: eq_bigr => l; rewrite !mxE.
Qed. | Lemma | submxrow_sum | 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",
"eq_bigr",
"matrixP",
"mxE",
"sq",
"submxrow",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrowN m (R_ : forall j, 'M[V]_(m, q_ j)) :
\mxrow_j (- R_ j) = - \mxrow_j (R_ j). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxrowN | 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 | |
mxrowB m (R_ R'_ : forall j, 'M[V]_(m, q_ j)) :
\mxrow_j (R_ j - R'_ j) = \mxrow_j (R_ j) - \mxrow_j (R'_ j). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxrowB | 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 | |
submxrowN m (B : 'M[V]_(m, sq)) j :
submxrow (- B) j = - submxrow B j. | Proof. by apply/matrixP => i i'; rewrite !mxE. Qed. | Lemma | submxrowN | 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",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxrowB m (B B' : 'M[V]_(m, sq)) j :
submxrow (B - B') j = submxrow B j - submxrow B' j. | Proof. by apply/matrixP => i i'; rewrite !mxE. Qed. | Lemma | submxrowB | 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",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxrow m n' (A : 'M[R]_(m, n')) (R_ : forall j, 'M[R]_(n', q_ j)) :
A *m \mxrow_j R_ j= \mxrow_j (A *m R_ j). | Proof.
by apply/matrixP=> i s; rewrite !mxE; under [LHS]eq_bigr do rewrite !mxE.
Qed. | Lemma | mul_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"matrixP",
"mxE",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_submxrow m n' (A : 'M[R]_(m, n')) (B : 'M[R]_(n', sq)) j :
A *m submxrow B j= submxrow (A *m B) j. | Proof.
by apply/matrixP=> i s; rewrite !mxE; under [LHS]eq_bigr do rewrite !mxE.
Qed. | Lemma | mul_submxrow | 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",
"eq_bigr",
"matrixP",
"mxE",
"n'",
"sq",
"submxrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcolD m (C_ C'_ : forall i, 'M[V]_(p_ i, m)) :
\mxcol_i (C_ i + C'_ i) = \mxcol_i (C_ i) + \mxcol_i (C'_ i). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxcolD | 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 | |
mxcol0 m : \mxcol_i (0 : 'M[V]_(p_ i, m)) = 0. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxcol0 | 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 | |
mxcol_const m a : \mxcol_j (const_mx a : 'M[V]_(p_ j, m)) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxcol_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",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcol_sum
(I : finType) m (C_ : forall j i, 'M[V]_(p_ i, m)) (P : {pred I}):
\mxcol_i (\sum_(j | P j) C_ j i) = \sum_(j | P j) \mxcol_i (C_ j i). | Proof.
apply/matrixP => i j; rewrite !(mxE, summxE).
by apply: eq_bigr => l; rewrite !mxE.
Qed. | Lemma | mxcol_sum | 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",
"eq_bigr",
"matrixP",
"mxE",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcolD m (B B' : 'M[V]_(sp, m)) i :
submxcol (B + B') i = submxcol B i + submxcol B' i. | Proof. by apply/matrixP => j j'; rewrite !mxE. Qed. | Lemma | submxcolD | 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",
"sp",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcol0 m i : submxcol (0 : 'M[V]_(sp, m)) i = 0. | Proof. by apply/matrixP=> j j'; rewrite !mxE. Qed. | Lemma | submxcol0 | 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",
"sp",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcol_sum (I : finType) m
(C_ : forall j, 'M[V]_(sp, m)) (P : {pred I}) i :
submxcol (\sum_(j | P j) C_ j) i = \sum_(j | P j) submxcol (C_ j) i. | Proof.
apply/matrixP => j j'; rewrite !(mxE, summxE).
by apply: eq_bigr => l; rewrite !mxE.
Qed. | Lemma | submxcol_sum | 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",
"eq_bigr",
"matrixP",
"mxE",
"sp",
"submxcol",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcolN m (C_ : forall i, 'M[V]_(p_ i, m)) :
\mxcol_i (- C_ i) = - \mxcol_i (C_ i). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxcolN | 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 | |
mxcolB m (C_ C'_ : forall i, 'M[V]_(p_ i, m)) :
\mxcol_i (C_ i - C'_ i) = \mxcol_i (C_ i) - \mxcol_i (C'_ i). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxcolB | 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 | |
submxcolN m (B : 'M[V]_(sp, m)) i :
submxcol (- B) i = - submxcol B i. | Proof. by apply/matrixP => j j'; rewrite !mxE. Qed. | Lemma | submxcolN | 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",
"sp",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcolB m (B B' : 'M[V]_(sp, m)) i :
submxcol (B - B') i = submxcol B i - submxcol B' i. | Proof. by apply/matrixP => j j'; rewrite !mxE. Qed. | Lemma | submxcolB | 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",
"sp",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxcol_mul n' m (C_ : forall i, 'M[R]_(p_ i, n')) (A : 'M[R]_(n', m)) :
\mxcol_i C_ i *m A = \mxcol_i (C_ i *m A). | Proof.
by apply/matrixP=> i s; rewrite !mxE; under [LHS]eq_bigr do rewrite !mxE.
Qed. | Lemma | mxcol_mul | 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",
"eq_bigr",
"matrixP",
"mxE",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxcol_mul n' m (B : 'M[R]_(sp, n')) (A : 'M[R]_(n', m)) i :
submxcol B i *m A = submxcol (B *m A) i. | Proof.
by apply/matrixP=> j s; rewrite !mxE; under [LHS]eq_bigr do rewrite !mxE.
Qed. | Lemma | submxcol_mul | 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",
"eq_bigr",
"matrixP",
"mxE",
"n'",
"sp",
"submxcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblockD (B_ B'_ : forall i j, 'M[V]_(p_ i, q_ j)) :
\mxblock_(i, j) (B_ i j + B'_ i j) =
\mxblock_(i, j) (B_ i j) + \mxblock_(i, j) (B'_ i j). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxblockD | 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 | |
mxblock0 : \mxblock_(i, j) (0 : 'M[V]_(p_ i, q_ j)) = 0. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxblock0 | 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 | |
mxblock_const a :
\mxblock_(i, j) (const_mx a : 'M[V]_(p_ i, q_ j)) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxblock_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",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblock_sum (I : finType)
(B_ : forall k i j, 'M[V]_(p_ i, q_ j)) (P : {pred I}):
\mxblock_(i, j) (\sum_(k | P k) B_ k i j) =
\sum_(k | P k) \mxblock_(i, j) (B_ k i j). | Proof.
apply/matrixP => i j; rewrite !(mxE, summxE).
by apply: eq_bigr => l; rewrite !mxE.
Qed. | Lemma | mxblock_sum | 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",
"eq_bigr",
"matrixP",
"mxE",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblockD (B B' : 'M[V]_(sp, sq)) i j :
submxblock (B + B') i j = submxblock B i j + submxblock B' i j. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | submxblockD | 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",
"sp",
"sq",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblock0 i j : submxblock (0 : 'M[V]_(sp, sq)) i j = 0. | Proof. by apply/matrixP=> k l; rewrite !mxE. Qed. | Lemma | submxblock0 | 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",
"sp",
"sq",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblock_sum (I : finType)
(B_ : forall k, 'M[V]_(sp, sq)) (P : {pred I}) i j :
submxblock (\sum_(k | P k) B_ k) i j = \sum_(k | P k) submxblock (B_ k) i j. | Proof.
apply/matrixP => k l; rewrite !(mxE, summxE).
by apply: eq_bigr => p; rewrite !mxE.
Qed. | Lemma | submxblock_sum | 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",
"eq_bigr",
"matrixP",
"mxE",
"sp",
"sq",
"submxblock",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxblockN (B_ : forall i j, 'M[V]_(p_ i, q_ j)) :
\mxblock_(i, j) (- B_ i j) = - \mxblock_(i, j) (B_ i j). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxblockN | 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 | |
mxblockB (B_ B'_ : forall i j, 'M[V]_(p_ i, q_ j)) :
\mxblock_(i, j) (B_ i j - B'_ i j) =
\mxblock_(i, j) (B_ i j) - \mxblock_(i, j) (B'_ i j). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | mxblockB | 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 | |
submxblockN (B : 'M[V]_(sp, sq)) i j :
submxblock (- B) i j = - submxblock B i j. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | submxblockN | 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",
"sp",
"sq",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblockB (B B' : 'M[V]_(sp, sq)) i j :
submxblock (B - B') i j = submxblock B i j - submxblock B' i j. | Proof. by apply/matrixP => k l; rewrite !mxE. Qed. | Lemma | submxblockB | 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",
"sp",
"sq",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxrow_mxcol m n
(R_ : forall j, 'M[R]_(m, p_ j)) (C_ : forall i, 'M[R]_(p_ i, n)) :
\mxrow_j R_ j *m \mxcol_i C_ i = \sum_i (R_ i *m C_ i). | Proof.
apply/matrixP => i j; rewrite !mxE summxE; under [RHS]eq_bigr do rewrite !mxE.
rewrite sig_big_dep/= (reindex _ tagnat.sig_bij_on)/=.
by apply: eq_bigr=> l _; rewrite !mxE.
Qed. | Lemma | mul_mxrow_mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"matrixP",
"mxE",
"reindex",
"sig_big_dep",
"sig_bij_on",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxcol_mxrow m
(C_ : forall i, 'M[R]_(p_ i, m)) (R_ : forall j, 'M[R]_(m, q_ j)) :
\mxcol_i C_ i*m \mxrow_j R_ j = \mxblock_(i, j) (C_ i *m R_ j). | Proof.
apply/mxblockP => i j; rewrite mxblockK.
by rewrite submxblockEh -mul_submxrow -submxcol_mul mxcolK mxrowK.
Qed. | Lemma | mul_mxcol_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mul_submxrow",
"mxblockK",
"mxblockP",
"mxcolK",
"mxrowK",
"submxblockEh",
"submxcol_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxrow_mxblock m
(R_ : forall i, 'M[R]_(m, p_ i)) (B_ : forall i j, 'M[R]_(p_ i, q_ j)) :
\mxrow_i R_ i *m \mxblock_(i, j) B_ i j = \mxrow_j (\sum_i (R_ i *m B_ i j)). | Proof.
rewrite mxblockEv mul_mxrow_mxcol mxrow_sum.
by apply: eq_bigr => i _; rewrite mul_mxrow.
Qed. | Lemma | mul_mxrow_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"mul_mxrow",
"mul_mxrow_mxcol",
"mxblockEv",
"mxrow_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxblock_mxrow m
(B_ : forall i j, 'M[R]_(q_ i, p_ j)) (C_ : forall i, 'M[R]_(p_ i, m)) :
\mxblock_(i, j) B_ i j *m \mxcol_j C_ j = \mxcol_i (\sum_j (B_ i j *m C_ j)). | Proof.
rewrite mxblockEh mul_mxrow_mxcol mxcol_sum.
by apply: eq_bigr => i _; rewrite mxcol_mul.
Qed. | Lemma | mul_mxblock_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"mul_mxrow_mxcol",
"mxblockEh",
"mxcol_mul",
"mxcol_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxblock {R : pzSemiRingType} {p q r : nat}
{p_ : 'I_p -> nat} {q_ : 'I_q -> nat} {r_ : 'I_r -> nat}
(A_ : forall i j, 'M[R]_(p_ i, q_ j)) (B_ : forall j k, 'M_(q_ j, r_ k)) :
\mxblock_(i, j) A_ i j *m \mxblock_(j, k) B_ j k =
\mxblock_(i, k) \sum_j (A_ i j *m B_ j k). | Proof.
rewrite mxblockEh mul_mxrow_mxblock mxblockEh; apply: eq_mxrow => i.
by under [LHS]eq_bigr do rewrite mxcol_mul; rewrite -mxcol_sum.
Qed. | Lemma | mul_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"eq_mxrow",
"mul_mxrow_mxblock",
"mxblockEh",
"mxcol_mul",
"mxcol_sum",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_trig_mxblockP (B_ : forall i j, 'M[V]_(p_ i, p_ j)) :
reflect [/\ forall (i j : 'I_p), (i < j)%N -> B_ i j = 0 &
forall i, is_trig_mx (B_ i i)]
(is_trig_mx (\mxblock_(i, j) B_ i j)). | Proof.
apply: (iffP is_trig_mxP); last first.
move=> [Blt1 /(_ _)/is_trig_mxP Blt2]/= s s'; rewrite !mxE.
rewrite -[_ < _]lt_sig ltEsig/= /sig1 /sig2 leEord.
case: ltngtP => //= ii'; first by rewrite (Blt1 _ _ ii') mxE.
move: (sig s) (sig s') ii' => -[/= i j] [/= i' +] /val_inj ii'.
by case: _ / ii' => j'; re... | Lemma | is_trig_mxblockP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank",
"Rank2K",
"apply",
"cast_ord_id",
"is_trig_mx",
"is_trig_mxP",
"last",
"leEord",
"leqnn",
"ltEsig",
"lt_rank",
"lt_sig",
"ltnW",
"ltn_geF",
"ltngtP",
"matrixP",
"mxE",
"sig",
"sig1",
"sig2",
"split",
"tagged_asE",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_trig_mxblock (B_ : forall i j, 'M[V]_(p_ i, p_ j)) :
is_trig_mx (\mxblock_(i, j) B_ i j) =
([forall i : 'I_p, forall j : 'I_p, (i < j)%N ==> (B_ i j == 0)] &&
[forall i, is_trig_mx (B_ i i)]). | Proof.
by apply/is_trig_mxblockP/andP => -[] => [/(_ _ _ _)/eqP|]
=> /'forall_'forall_implyP => [|/(_ _ _ _)/eqP] Blt /forallP.
Qed. | Lemma | is_trig_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"forallP",
"is_trig_mx",
"is_trig_mxblockP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_mxblockP (B_ : forall i j, 'M[V]_(p_ i, p_ j)) :
reflect [/\ forall (i j : 'I_p), i != j -> B_ i j = 0 &
forall i, is_diag_mx (B_ i i)]
(is_diag_mx (\mxblock_(i, j) B_ i j)). | Proof.
apply: (iffP is_diag_mxP); last first.
move=> [Bneq1 /(_ _)/is_diag_mxP Bneq2]/= s s'; rewrite !mxE.
rewrite val_eqE -(can_eq sigK) /sig1 /sig2.
move: (sig s) (sig s') => -[/= i j] [/= i' j'].
rewrite -tag_eqE/= /tag_eq/= negb_and.
case: eqVneq => /= [ii'|/Bneq1->]; last by rewrite !mxE.
by rewrite -... | Lemma | is_diag_mxblockP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Rank",
"Rank2K",
"apply",
"can_eq",
"cast_ord_id",
"eqVneq",
"eq_Rank",
"is_diag_mx",
"is_diag_mxP",
"last",
"matrixP",
"mxE",
"sig",
"sig1",
"sig2",
"sigK",
"split",
"tag_eq",
"tag_eqE",
"tagged_asE",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_mxblock (B_ : forall i j, 'M[V]_(p_ i, p_ j)) :
is_diag_mx (\mxblock_(i, j) B_ i j) =
([forall i : 'I_p, forall j : 'I_p, (i != j) ==> (B_ i j == 0)] &&
[forall i, is_diag_mx (B_ i i)]). | Proof.
by apply/is_diag_mxblockP/andP => -[] => [/(_ _ _ _)/eqP|]
=> /'forall_'forall_implyP => [|/(_ _ _ _)/eqP] Blt /forallP.
Qed. | Lemma | is_diag_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"forallP",
"is_diag_mx",
"is_diag_mxblockP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiag (B_ : forall i, 'M[V]_(p_ i)) : 'M[V]_(\sum_i p_ i) | :=
\mxblock_(j, k) if j == k then conform_mx 0 (B_ j) else 0. | Definition | mxdiag | 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",
... | [
"conform_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxdiag_ i E" | := (mxdiag (fun i => E)) : ring_scope. | Notation | \mxdiag_ i E | 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",
... | [
"mxdiag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxblock_diag (B_ : forall i, 'M[V]_(p_ i)) i :
submxblock (\mxdiag_i B_ i) i i = B_ i. | Proof. by rewrite mxblockK conform_mx_id eqxx. Qed. | Lemma | submxblock_diag | 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",
... | [
"conform_mx_id",
"eqxx",
"mxblockK",
"submxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxdiagP (B_ B'_ : forall i, 'M[V]_(p_ i)) :
(forall i, B_ i = B'_ i) <-> (\mxdiag_i B_ i = \mxdiag_i B'_ i). | Proof.
rewrite /mxdiag; split; first by move=> e; apply/eq_mxblockP => i j; rewrite e.
by move=> + i => /eq_mxblockP/(_ i i); rewrite eqxx !conform_mx_id.
Qed. | Lemma | eq_mxdiagP | 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",
"conform_mx_id",
"eq_mxblockP",
"eqxx",
"mxdiag",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxdiag (B_ B'_ : forall i, 'M[V]_(p_ i)) :
(forall i, B_ i = B'_ i) -> (\mxdiag_i B_ i = \mxdiag_i B'_ i). | Proof. by move=> /eq_mxdiagP. Qed. | Lemma | eq_mxdiag | 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_mxdiagP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiagD (B_ B'_ : forall i, 'M[V]_(p_ i)) :
\mxdiag_i (B_ i + B'_ i) = \mxdiag_i (B_ i) + \mxdiag_i (B'_ i). | Proof.
rewrite /mxdiag -mxblockD; apply/eq_mxblock => i j.
by case: eqVneq => [->|]; rewrite ?conform_mx_id ?addr0.
Qed. | Lemma | mxdiagD | 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",
... | [
"addr0",
"apply",
"conform_mx_id",
"eqVneq",
"eq_mxblock",
"mxblockD",
"mxdiag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiag_sum (I : finType) (B_ : forall k i, 'M[V]_(p_ i)) (P : {pred I}) :
\mxdiag_i (\sum_(k | P k) B_ k i) = \sum_(k | P k) \mxdiag_i (B_ k i). | Proof.
rewrite /mxdiag -mxblock_sum; apply/eq_mxblock => i j.
case: eqVneq => [->|]; rewrite ?conform_mx_id//; last by rewrite big1.
by apply: eq_bigr => k; rewrite conform_mx_id.
Qed. | Lemma | mxdiag_sum | 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",
"big1",
"conform_mx_id",
"eqVneq",
"eq_bigr",
"eq_mxblock",
"last",
"mxblock_sum",
"mxdiag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_mxdiag (B_ : forall i, 'M[V]_(p_ i)) :
(\mxdiag_i B_ i)^T = \mxdiag_i (B_ i)^T. | Proof.
rewrite tr_mxblock; apply/eq_mxblock => i j.
by case: eqVneq => [->|]; rewrite ?trmx_conform ?trmx0.
Qed. | Lemma | tr_mxdiag | 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",
"eqVneq",
"eq_mxblock",
"tr_mxblock",
"trmx0",
"trmx_conform"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mxdiag (B_ : forall i, 'M[V]_(p_ i)) k :
let B'_ i := if sig1 k == i then conform_mx 0 (B_ i) else 0 in
row k (\mxdiag_ i B_ i) = row (sig2 k) (\mxrow_i B'_ i). | Proof.
rewrite /= row_mxblock row_mxrow; apply/eq_mxrow => i.
by case: eqVneq => // e; congr row; rewrite e.
Qed. | Lemma | row_mxdiag | 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",
"conform_mx",
"eqVneq",
"eq_mxrow",
"row",
"row_mxblock",
"row_mxrow",
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mxdiag (B_ : forall i, 'M[V]_(p_ i)) k :
let B'_ i := if sig1 k == i then conform_mx 0 (B_ i) else 0 in
col k (\mxdiag_ i B_ i) = col (sig2 k) (\mxcol_i B'_ i). | Proof.
by rewrite /= col_mxblock col_mxcol; apply/eq_mxcol => i; rewrite eq_sym.
Qed. | Lemma | col_mxdiag | 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_mxblock",
"col_mxcol",
"conform_mx",
"eq_mxcol",
"eq_sym",
"sig1",
"sig2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxdiag_ ( i < n ) E" | := (mxdiag (fun i : 'I_n => E))
(only parsing) : ring_scope. | Notation | \mxdiag_ ( i < n ) E | 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",
... | [
"mxdiag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\mxdiag_ i E" | := (\mxdiag_(i < _) E) : ring_scope. | Notation | \mxdiag_ i E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiagN (B_ : forall i, 'M[V]_(p_ i)) :
\mxdiag_i (- B_ i) = - \mxdiag_i (B_ i). | Proof.
rewrite /mxdiag -mxblockN; apply/eq_mxblock => i j.
by case: eqVneq => [->|]; rewrite ?conform_mx_id ?oppr0.
Qed. | Lemma | mxdiagN | 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",
"conform_mx_id",
"eqVneq",
"eq_mxblock",
"mxblockN",
"mxdiag",
"oppr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiagB (B_ B'_ : forall i, 'M[V]_(p_ i)) :
\mxdiag_i (B_ i - B'_ i) = \mxdiag_i (B_ i) - \mxdiag_i (B'_ i). | Proof. by rewrite mxdiagD mxdiagN. Qed. | Lemma | mxdiagB | 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",
... | [
"mxdiagD",
"mxdiagN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiag0 : \mxdiag_i (0 : 'M[V]_(p_ i)) = 0. | Proof. by under [LHS]eq_mxdiag do rewrite -[0]subr0; rewrite mxdiagB subrr. Qed. | Lemma | mxdiag0 | 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_mxdiag",
"mxdiagB",
"subr0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiag_recl {V : nmodType} {m : nat} {p_ : 'I_m.+1 -> nat}
(B_ : forall i, 'M[V]_(p_ i)) :
\mxdiag_i B_ i = castmx (mxsize_recl, mxsize_recl)
(block_mx (B_ 0) 0 0 (\mxdiag_i B_ (lift ord0 i))). | Proof.
rewrite /mxdiag mxblock_recul/= !conform_mx_id.
by congr (castmx _ (block_mx _ _ _ _)); rewrite ?mxrow0 ?mxcol0.
Qed. | Lemma | mxdiag_recl | 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",
"castmx",
"conform_mx_id",
"lift",
"mxblock_recul",
"mxcol0",
"mxdiag",
"mxrow0",
"mxsize_recl",
"nat",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_mxblock (B_ : forall i j, 'M[R]_(p_ i, p_ j)) :
\tr (\mxblock_(i, j) B_ i j) = \sum_i \tr (B_ i i). | Proof.
rewrite /mxtrace sig_big_dep (reindex _ sig_bij_on)/=.
by apply: eq_bigr => i _; rewrite !mxE.
Qed. | Lemma | mxtrace_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"mxE",
"mxtrace",
"reindex",
"sig_big_dep",
"sig_bij_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdiagZ a : \mxdiag_i (a%:M : 'M[R]_(p_ i)) = a%:M. | Proof.
apply/matrixP => s t; rewrite !mxE -(can_eq sigK) /sig1 /sig2.
case: (sig s) (sig t) => [/= i j] [/= i' j'].
case: eqP => [<-|ni] in j' *; last by rewrite !mxE; case: eqVneq => // -[].
by rewrite conform_mx_id eq_Tagged/= mxE.
Qed. | Lemma | mxdiagZ | 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",
"can_eq",
"conform_mx_id",
"eqVneq",
"eq_Tagged",
"last",
"matrixP",
"mxE",
"sig",
"sig1",
"sig2",
"sigK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mxrow (B_ : forall j, 'rV[R]_(p_ j)) :
diag_mx (\mxrow_j B_ j) = \mxdiag_j (diag_mx (B_ j)). | Proof.
apply/matrixP => s s'; rewrite !mxE/= -(can_eq sigK) /sig1 /sig2.
case: (sig s) (sig s') => [/= i j] [/= i' j'].
rewrite -tag_eqE /tag_eq/=; case: (eqVneq i i') => ii'; rewrite ?mxE//=.
by case: _ / ii' in j' *; rewrite tagged_asE/= conform_mx_id mxE.
Qed. | Lemma | diag_mxrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"can_eq",
"conform_mx_id",
"diag_mx",
"eqVneq",
"matrixP",
"mxE",
"sig",
"sig1",
"sig2",
"sigK",
"tag_eq",
"tag_eqE",
"tagged_asE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_mxdiag (B_ : forall i, 'M[R]_(p_ i)) :
\tr (\mxdiag_i B_ i) = \sum_i \tr (B_ i). | Proof.
by rewrite mxtrace_mxblock; apply: eq_bigr => i _; rewrite eqxx/= conform_mx_id.
Qed. | Lemma | mxtrace_mxdiag | 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",
"conform_mx_id",
"eq_bigr",
"eqxx",
"mxtrace_mxblock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxdiag_mxcol m
(D_ : forall i, 'M[R]_(p_ i)) (C_ : forall i, 'M[R]_(p_ i, m)):
\mxdiag_i D_ i *m \mxcol_i C_ i = \mxcol_i (D_ i *m C_ i). | Proof.
rewrite /mxdiag mxblockEh mul_mxrow_mxcol.
under [LHS]eq_bigr do rewrite mxcol_mul; rewrite -mxcol_sum.
apply/eq_mxcol => i; rewrite (bigD1 i)//= eqxx conform_mx_id big1 ?addr0//.
by move=> j; case: eqVneq => //=; rewrite mul0mx.
Qed. | Lemma | mul_mxdiag_mxcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addr0",
"apply",
"big1",
"bigD1",
"conform_mx_id",
"eqVneq",
"eq_bigr",
"eq_mxcol",
"eqxx",
"mul0mx",
"mul_mxrow_mxcol",
"mxblockEh",
"mxcol_mul",
"mxcol_sum",
"mxdiag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxrow_mxdiag {R : pzSemiRingType} {p : nat} {p_ : 'I_p -> nat} m
(R_ : forall i, 'M[R]_(m, p_ i)) (D_ : forall i, 'M[R]_(p_ i)) :
\mxrow_i R_ i *m \mxdiag_i D_ i = \mxrow_i (R_ i *m D_ i). | Proof.
apply: trmx_inj; rewrite trmx_mul_rev !tr_mxrow tr_mxdiag mul_mxdiag_mxcol.
by apply/ eq_mxcol => i; rewrite trmx_mul_rev.
Qed. | Lemma | mul_mxrow_mxdiag | 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",
"eq_mxcol",
"mul_mxdiag_mxcol",
"nat",
"tr_mxdiag",
"tr_mxrow",
"trmx_inj",
"trmx_mul_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxblock_mxdiag {R : pzSemiRingType} {p q : nat}
{p_ : 'I_p -> nat} {q_ : 'I_q -> nat}
(B_ : forall i j, 'M[R]_(p_ i, q_ j)) (D_ : forall j, 'M[R]_(q_ j)) :
\mxblock_(i, j) B_ i j *m \mxdiag_j D_ j = \mxblock_(i, j) (B_ i j *m D_ j). | Proof.
by rewrite !mxblockEh mul_mxrow_mxdiag; under eq_mxrow do rewrite mxcol_mul.
Qed. | Lemma | mul_mxblock_mxdiag | 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_mxrow",
"mul_mxrow_mxdiag",
"mxblockEh",
"mxcol_mul",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mxdiag_mxblock {R : pzSemiRingType} {p q : nat}
{p_ : 'I_p -> nat} {q_ : 'I_q -> nat}
(D_ : forall j, 'M[R]_(p_ j)) (B_ : forall i j, 'M[R]_(p_ i, q_ j)):
\mxdiag_j D_ j *m \mxblock_(i, j) B_ i j = \mxblock_(i, j) (D_ i *m B_ i j). | Proof.
by rewrite !mxblockEv mul_mxdiag_mxcol; under eq_mxcol do rewrite mul_mxrow.
Qed. | Lemma | mul_mxdiag_mxblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"eq_mxcol",
"mul_mxdiag_mxcol",
"mul_mxrow",
"mxblockEv",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Vandermonde (R : pzRingType) (m n : nat) (a : 'rV[R]_n) | :=
\matrix_(i < m, j < n) a 0 j ^+ i. | Definition | Vandermonde | 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",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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