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 |
|---|---|---|---|---|---|---|---|---|---|---|
matrix_key : unit. | Proof. by []. Qed. | Fact | matrix_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"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 | |
matrix_unlockable | := Unlockable matrix_of_fun.unlock. | Canonical | matrix_unlockable | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
mxE k F : matrix_of_fun k F =2 F. | Proof. by move=> i j; rewrite unlock /fun_of_matrix /= ffunE. Qed. | Lemma | mxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"ffunE",
"fun_of_matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrixP (A B : matrix R m n) : A =2 B <-> A = B. | Proof.
rewrite /fun_of_matrix; split=> [/= eqAB | -> //].
by apply/val_inj/ffunP=> [[i j]]; apply: eqAB.
Qed. | Lemma | matrixP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"ffunP",
"fun_of_matrix",
"matrix",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mx k F1 F2 : (F1 =2 F2) -> matrix_of_fun k F1 = matrix_of_fun k F2. | Proof. by move=> eq_F; apply/matrixP => i j; rewrite !mxE eq_F. Qed. | Lemma | eq_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"F1",
"F2",
"apply",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M[' R ]_ ( m , n )" | := (matrix R m n) (only parsing): type_scope. | Notation | ''M[' R ]_ ( m , n ) | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''rV[' R ]_ n" | := 'M[R]_(1, n) (only parsing) : type_scope. | Notation | ''rV[' R ]_ n | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''cV[' R ]_ n" | := 'M[R]_(n, 1) (only parsing) : type_scope. | Notation | ''cV[' R ]_ n | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M[' R ]_ n" | := 'M[R]_(n, n) (only parsing) : type_scope. | Notation | ''M[' R ]_ n | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M[' R ]_ ( n )" | := 'M[R]_n (only parsing) : type_scope. | Notation | ''M[' R ]_ ( n ) | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M_' ( m , n )" | := 'M[_]_(m, n) : type_scope. | Notation | ''M_' ( m , n ) | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''rV_' n" | := 'M_(1, n) : type_scope. | Notation | ''rV_' n | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"ssrbool",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''cV_' n" | := 'M_(n, 1) : type_scope. | Notation | ''cV_' n | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M_' n" | := 'M_(n, n) : type_scope. | Notation | ''M_' n | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M_' ( n )" | := 'M_n (only parsing) : type_scope. | Notation | ''M_' ( n ) | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\matrix[ k ]_ ( i , j ) E" | := (matrix_of_fun k (fun i j => E)) :
ring_scope. | Notation | \matrix[ k ]_ ( i , j ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\matrix_ ( i < m , j < n ) E" | :=
(@matrix_of_fun _ m n matrix_key (fun i j => E)) (only parsing) : ring_scope. | Notation | \matrix_ ( i < m , j < n ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\matrix_ ( i , j < n ) E" | :=
(\matrix_(i < n, j < n) E) (only parsing) : ring_scope. | Notation | \matrix_ ( i , j < n ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\matrix_ ( i , j ) E" | := (\matrix_(i < _, j < _) E) : ring_scope. | Notation | \matrix_ ( i , j ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\matrix_ ( i < m ) E" | :=
(\matrix_(i < m, j < _) @fun_of_matrix _ 1 _ E 0 j)
(only parsing) : ring_scope. | Notation | \matrix_ ( i < m ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"fun_of_matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\matrix_ i E" | := (\matrix_(i < _) E) : ring_scope. | Notation | \matrix_ i E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\col_ ( i < n ) E" | := (@matrix_of_fun _ n 1 matrix_key (fun i _ => E))
(only parsing) : ring_scope. | Notation | \col_ ( i < n ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\col_ i E" | := (\col_(i < _) E) : ring_scope. | Notation | \col_ i E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\row_ ( j < n ) E" | := (@matrix_of_fun _ 1 n matrix_key (fun _ j => E))
(only parsing) : ring_scope. | Notation | \row_ ( j < n ) E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\row_ j E" | := (\row_(j < _) E) : ring_scope. | Notation | \row_ j E | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_mx (F : finType) m n : (#|{: 'M[F]_(m, n)}| = #|F| ^ (m * n))%N. | Proof. by rewrite card_sub card_ffun card_prod !card_ord. Qed. | Lemma | card_mx | algebra | algebra/matrix.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"ssrnat",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"card_ffun",
"card_ord",
"card_prod",
"card_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_mx_key : unit. | Proof. by []. Qed. | Fact | const_mx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Constant matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
const_mx m n a : 'M[R]_(m, n) | := \matrix[const_mx_key]_(i, j) a. | Definition | const_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"const_mx_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmx m' n' (eq_mn : (m = m') * (n = n')) A : 'M_(m', n') | :=
let: erefl in _ = m' := eq_mn.1 return 'M_(m', n') in
let: erefl in _ = n' := eq_mn.2 return 'M_(m, n') in A. | Definition | castmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"finfun",
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"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"n'"
] | Reshape a matrix, to accommodate the block functions for instance. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
conform_mx m' n' B A | :=
match m =P m', n =P n' with
| ReflectT eq_m, ReflectT eq_n => castmx (eq_m, eq_n) A
| _, _ => B
end. | Definition | conform_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"finfun",
"bigop",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"castmx",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_key : unit. | Proof. by []. Qed. | Fact | trmx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Transpose a matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
trmx A | := \matrix[trmx_key]_(i, j) A j i. | Definition | trmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix",
"trmx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_perm_key : unit. | Proof. by []. Qed. | Fact | row_perm_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Permute a matrix vertically (rows) or horizontally (columns) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
row_perm (s : 'S_m) A | := \matrix[row_perm_key]_(i, j) A (s i) j. | Definition | row_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix",
"row_perm_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_perm_key : unit. | Proof. by []. Qed. | Fact | col_perm_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_perm (s : 'S_n) A | := \matrix[col_perm_key]_(i, j) A i (s j). | Definition | col_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_perm_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xrow i1 i2 | := row_perm (tperm i1 i2). | Definition | xrow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"row_perm",
"tperm"
] | Exchange two rows/columns of a matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
xcol j1 j2 | := col_perm (tperm j1 j2). | Definition | xcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_perm",
"tperm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row i0 A | := \row_j A i0 j. | Definition | row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"i0"
] | Row/Column sub matrices of a matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
col j0 A | := \col_i A i j0. | Definition | col | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row' i0 A | := \matrix_(i, j) A (lift i0 i) j. | Definition | row' | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"seq",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"i0",
"lift"
] | Removing a row/column from a matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
col' j0 A | := \matrix_(i, j) A i (lift j0 j). | Definition | col' | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub m' n' f g A | := \matrix_(i < m', j < n') A (f i) (g j). | Definition | mxsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"n'"
] | reindexing/subindex a matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
colsub g | := (mxsub id g). | Notation | colsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"id",
"mxsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowsub f | := (mxsub f id). | Notation | rowsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"id",
"mxsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmx_const m' n' (eq_mn : (m = m') * (n = n')) a :
castmx eq_mn (const_mx a) = const_mx a. | Proof. by case: eq_mn; case: m' /; case: n' /. Qed. | Lemma | castmx_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"castmx",
"const_mx",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_const a : trmx (const_mx a) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | trmx_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"zmodp",
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... | [
"apply",
"const_mx",
"matrixP",
"mxE",
"trmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_perm_const s a : row_perm s (const_mx a) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | row_perm_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"matrixP",
"mxE",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_perm_const s a : col_perm s (const_mx a) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | col_perm_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col_perm",
"const_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xrow_const i1 i2 a : xrow i1 i2 (const_mx a) = const_mx a. | Proof. exact: row_perm_const. Qed. | Lemma | xrow_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"const_mx",
"row_perm_const",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcol_const j1 j2 a : xcol j1 j2 (const_mx a) = const_mx a. | Proof. exact: col_perm_const. Qed. | Lemma | xcol_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_perm_const",
"const_mx",
"xcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowP (u v : 'rV[R]_n) : u 0 =1 v 0 <-> u = v. | Proof. by split=> [eq_uv | -> //]; apply/matrixP=> i; rewrite ord1. Qed. | Lemma | rowP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"ord1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowK u_ i0 : row i0 (\matrix_i u_ i) = u_ i0. | Proof. by apply/rowP=> i'; rewrite !mxE. Qed. | Lemma | rowK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"i0",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_matrixP A B : (forall i, row i A = row i B) <-> A = B. | Proof.
split=> [eqAB | -> //]; apply/matrixP=> i j.
by move/rowP/(_ j): (eqAB i); rewrite !mxE.
Qed. | Lemma | row_matrixP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"zmodp",
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... | [
"apply",
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"mxE",
"row",
"rowP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colP (u v : 'cV[R]_m) : u^~ 0 =1 v^~ 0 <-> u = v. | Proof. by split=> [eq_uv | -> //]; apply/matrixP=> i j; rewrite ord1. Qed. | Lemma | colP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"divalg",
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"finalg",
"zmodp",
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... | [
"apply",
"matrixP",
"ord1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_const i0 a : row i0 (const_mx a) = const_mx a. | Proof. by apply/rowP=> j; rewrite !mxE. Qed. | Lemma | row_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"finalg",
"zmodp",
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... | [
"apply",
"const_mx",
"i0",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_const j0 a : col j0 (const_mx a) = const_mx a. | Proof. by apply/colP=> i; rewrite !mxE. Qed. | Lemma | col_const | algebra | algebra/matrix.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"finalg",
"zmodp",
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... | [
"apply",
"col",
"colP",
"const_mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'_const i0 a : row' i0 (const_mx a) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | row'_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
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"finalg",
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... | [
"apply",
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"i0",
"matrixP",
"mxE",
"row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col'_const j0 a : col' j0 (const_mx a) = const_mx a. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | col'_const | algebra | algebra/matrix.v | [
"HB",
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"mathcomp",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col'",
"const_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_perm1 A : col_perm 1 A = A. | Proof. by apply/matrixP=> i j; rewrite mxE perm1. Qed. | Lemma | col_perm1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col_perm",
"matrixP",
"mxE",
"perm1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_perm1 A : row_perm 1 A = A. | Proof. by apply/matrixP=> i j; rewrite mxE perm1. Qed. | Lemma | row_perm1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"perm1",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_permM s t A : col_perm (s * t) A = col_perm s (col_perm t A). | Proof. by apply/matrixP=> i j; rewrite !mxE permM. Qed. | Lemma | col_permM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col_perm",
"matrixP",
"mxE",
"permM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_permM s t A : row_perm (s * t) A = row_perm s (row_perm t A). | Proof. by apply/matrixP=> i j; rewrite !mxE permM. Qed. | Lemma | row_permM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"permM",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_row_permC s t A :
col_perm s (row_perm t A) = row_perm t (col_perm s A). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | col_row_permC | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col_perm",
"matrixP",
"mxE",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowEsub i : row i = rowsub (fun=> i). | Proof. by []. Qed. | Lemma | rowEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"row",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colEsub j : col j = colsub (fun=> j). | Proof. by []. Qed. | Lemma | colEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"colsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'Esub i : row' i = rowsub (lift i). | Proof. by []. Qed. | Lemma | row'Esub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"row'",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col'Esub j : col' j = colsub (lift j). | Proof. by []. Qed. | Lemma | col'Esub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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'",
"colsub",
"lift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_permEsub s : row_perm s = rowsub s. | Proof. by rewrite /row_perm /mxsub !unlock. Qed. | Lemma | row_permEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"row_perm",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_permEsub s : col_perm s = colsub s. | Proof. by rewrite /col_perm /mxsub !unlock. Qed. | Lemma | col_permEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_perm",
"colsub",
"mxsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xrowEsub i1 i2 : xrow i1 i2 = rowsub (tperm i1 i2). | Proof. exact: row_permEsub. Qed. | Lemma | xrowEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"row_permEsub",
"rowsub",
"tperm",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcolEsub j1 j2 : xcol j1 j2 = colsub (tperm j1 j2). | Proof. exact: col_permEsub. Qed. | Lemma | xcolEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_permEsub",
"colsub",
"tperm",
"xcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_id : mxsub id id =1 id. | Proof. by move=> A; apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | mxsub_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",
"id",
"matrixP",
"mxE",
"mxsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mxsub m' n' f f' g g' : f =1 f' -> g =1 g' ->
@mxsub m' n' f g =1 mxsub f' g'. | Proof. by move=> eq_f eq_g A; apply/matrixP => i j; rewrite !mxE eq_f eq_g. Qed. | Lemma | eq_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",
"eq_f",
"matrixP",
"mxE",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_rowsub m' (f f' : 'I_m' -> 'I_m) : f =1 f' -> rowsub f =1 rowsub f'. | Proof. by move=> /eq_mxsub; apply. Qed. | Lemma | eq_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",
... | [
"apply",
"eq_mxsub",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_colsub n' (g g' : 'I_n' -> 'I_n) : g =1 g' -> colsub g =1 colsub g'. | Proof. by move=> /eq_mxsub; apply. Qed. | Lemma | eq_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",
... | [
"apply",
"colsub",
"eq_mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_eq_id f g : f =1 id -> g =1 id -> mxsub f g =1 id. | Proof. by move=> fid gid A; rewrite (eq_mxsub fid gid) mxsub_id. Qed. | Lemma | mxsub_eq_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",
... | [
"eq_mxsub",
"id",
"mxsub",
"mxsub_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_eq_colsub n' f g : f =1 id -> @mxsub _ n' f g =1 colsub g. | Proof. by move=> f_id; apply: eq_mxsub. Qed. | Lemma | mxsub_eq_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",
... | [
"apply",
"colsub",
"eq_mxsub",
"id",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_eq_rowsub m' f g : g =1 id -> @mxsub m' _ f g =1 rowsub f. | Proof. exact: eq_mxsub. Qed. | Lemma | mxsub_eq_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",
... | [
"eq_mxsub",
"id",
"mxsub",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_ffunl m' n' f g : @mxsub m' n' (finfun f) g =1 mxsub f g. | Proof. by apply: eq_mxsub => // i; rewrite ffunE. Qed. | Lemma | mxsub_ffunl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_mxsub",
"ffunE",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_ffunr m' n' f g : @mxsub m' n' f (finfun g) =1 mxsub f g. | Proof. by apply: eq_mxsub => // i; rewrite ffunE. Qed. | Lemma | mxsub_ffunr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_mxsub",
"ffunE",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_ffun m' n' f g : @mxsub m' n' (finfun f) (finfun g) =1 mxsub f g. | Proof. by move=> A; rewrite mxsub_ffunl mxsub_ffunr. Qed. | Lemma | mxsub_ffun | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"mxsub_ffunl",
"mxsub_ffunr",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_const m' n' f g a : @mxsub m' n' f g (const_mx a) = const_mx a. | Proof. by apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | mxsub_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",
"mxsub",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ^T" | := (trmx A) : ring_scope. | Notation | A ^T | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"trmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmx_id m n erefl_mn (A : 'M_(m, n)) : castmx erefl_mn A = A. | Proof. by case: erefl_mn => e_m e_n; rewrite [e_m]eq_axiomK [e_n]eq_axiomK. Qed. | Lemma | castmx_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",
... | [
"castmx",
"eq_axiomK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmx_comp m1 n1 m2 n2 m3 n3 (eq_m1 : m1 = m2) (eq_n1 : n1 = n2)
(eq_m2 : m2 = m3) (eq_n2 : n2 = n3) A :
castmx (eq_m2, eq_n2) (castmx (eq_m1, eq_n1) A)
= castmx (etrans eq_m1 eq_m2, etrans eq_n1 eq_n2) A. | Proof.
by case: m2 / eq_m1 eq_m2; case: m3 /; case: n2 / eq_n1 eq_n2; case: n3 /.
Qed. | Lemma | castmx_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",
... | [
"castmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmxK m1 n1 m2 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) :
cancel (castmx (eq_m, eq_n)) (castmx (esym eq_m, esym eq_n)). | Proof. by case: m2 / eq_m; case: n2 / eq_n. Qed. | Lemma | castmxK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmxKV m1 n1 m2 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) :
cancel (castmx (esym eq_m, esym eq_n)) (castmx (eq_m, eq_n)). | Proof. by case: m2 / eq_m; case: n2 / eq_n. Qed. | Lemma | castmxKV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmx_sym m1 n1 m2 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) A1 A2 :
A1 = castmx (eq_m, eq_n) A2 -> A2 = castmx (esym eq_m, esym eq_n) A1. | Proof. by move/(canLR (castmxK _ _)). Qed. | Lemma | castmx_sym | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"castmxK"
] | This can be use to reverse an equation that involves a cast. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_castmx m1 n1 m2 n2 (eq_mn eq_mn' : (m1 = m2) * (n1 = n2)) :
castmx eq_mn =1 castmx eq_mn'. | Proof.
case: eq_mn eq_mn' => [em en] [em' en'] A.
by apply: (canRL (castmxKV _ _)); rewrite castmx_comp castmx_id.
Qed. | Lemma | eq_castmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"castmx",
"castmxKV",
"castmx_comp",
"castmx_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castmxE m1 n1 m2 n2 (eq_mn : (m1 = m2) * (n1 = n2)) A i j :
castmx eq_mn A i j =
A (cast_ord (esym eq_mn.1) i) (cast_ord (esym eq_mn.2) j). | Proof.
by do [case: eq_mn; case: m2 /; case: n2 /] in A i j *; rewrite !cast_ord_id.
Qed. | Lemma | castmxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"cast_ord",
"cast_ord_id",
"castmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conform_mx_id m n (B A : 'M_(m, n)) : conform_mx B A = A. | Proof. by rewrite /conform_mx; do 2!case: eqP => // *; rewrite castmx_id. Qed. | Lemma | conform_mx_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",
... | [
"castmx_id",
"conform_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonconform_mx m m' n n' (B : 'M_(m', n')) (A : 'M_(m, n)) :
(m != m') || (n != n') -> conform_mx B A = B. | Proof. by rewrite /conform_mx; do 2!case: eqP. Qed. | Lemma | nonconform_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",
... | [
"conform_mx",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conform_castmx m1 n1 m2 n2 m3 n3
(e_mn : (m2 = m3) * (n2 = n3)) (B : 'M_(m1, n1)) A :
conform_mx B (castmx e_mn A) = conform_mx B A. | Proof. by do [case: e_mn; case: m3 /; case: n3 /] in A *. Qed. | Lemma | conform_castmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"castmx",
"conform_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmxK m n : cancel (@trmx m n) (@trmx n m). | Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | trmxK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"trmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_inj m n : injective (@trmx m n). | Proof. exact: can_inj (@trmxK m n). Qed. | Lemma | trmx_inj | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"trmx",
"trmxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_cast m1 n1 m2 n2 (eq_mn : (m1 = m2) * (n1 = n2)) A :
(castmx eq_mn A)^T = castmx (eq_mn.2, eq_mn.1) A^T. | Proof.
by case: eq_mn => eq_m eq_n; apply/matrixP=> i j; rewrite !(mxE, castmxE).
Qed. | Lemma | trmx_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",
"castmx",
"castmxE",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_conform m' n' m n (B : 'M_(m', n')) (A : 'M_(m, n)) :
(conform_mx B A)^T = conform_mx B^T A^T. | Proof.
rewrite /conform_mx; do !case: eqP; rewrite ?mxE// => en em.
by rewrite trmx_cast.
Qed. | Lemma | trmx_conform | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"mxE",
"n'",
"trmx_cast"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_row_perm m n s (A : 'M_(m, n)) : (row_perm s A)^T = col_perm s A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | tr_row_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col_perm",
"matrixP",
"mxE",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_col_perm m n s (A : 'M_(m, n)) : (col_perm s A)^T = row_perm s A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | tr_col_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col_perm",
"matrixP",
"mxE",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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