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 |
|---|---|---|---|---|---|---|---|---|---|---|
is_scalar_mx_is_trig A : is_scalar_mx A -> is_trig_mx A. | Proof. by move=> /is_scalar_mx_is_diag /is_diag_mx_is_trig. Qed. | Lemma | is_scalar_mx_is_trig | algebra | algebra/matrix.v | [
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... | [
"is_diag_mx_is_trig",
"is_scalar_mx",
"is_scalar_mx_is_diag",
"is_trig_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x %:M" | := (scalar_mx _ x) : ring_scope. | Notation | x %:M | algebra | algebra/matrix.v | [
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"finalg",
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... | [
"scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx11_scalar (A : 'M_1) : A = (A 0 0)%:M. | Proof. by apply/rowP=> j; rewrite ord1 mxE. Qed. | Lemma | mx11_scalar | algebra | algebra/matrix.v | [
"HB",
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... | [
"apply",
"mxE",
"ord1",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_block n1 n2 a : a%:M = block_mx a%:M 0 0 a%:M :> 'M_(n1 + n2). | Proof.
apply/matrixP=> i j; rewrite !mxE.
by do 2![case: split_ordP => ? ->; rewrite !mxE]; rewrite ?eq_shift.
Qed. | Lemma | scalar_mx_block | algebra | algebra/matrix.v | [
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"zmodp",
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... | [
"apply",
"block_mx",
"eq_shift",
"matrixP",
"mxE",
"split_ordP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace (A : 'M[V]_n) | := \sum_i A i i. | Definition | mxtrace | algebra | algebra/matrix.v | [
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... | [] | TODO: undergeneralize to monoid | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"'\tr' A" | := (mxtrace A) : ring_scope. | Notation | '\tr' A | algebra | algebra/matrix.v | [
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"finalg",
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... | [
"mxtrace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_tr A : \tr A^T = \tr A. | Proof. by apply: eq_bigr=> i _; rewrite mxE. Qed. | Lemma | mxtrace_tr | algebra | algebra/matrix.v | [
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... | [
"apply",
"eq_bigr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_is_nmod_morphism : nmod_morphism mxtrace. | Proof.
split=> [|A B]; first by apply: big1 => i; rewrite mxE.
by rewrite -big_split /=; apply: eq_bigr => i _; rewrite mxE.
Qed. | Fact | mxtrace_is_nmod_morphism | algebra | algebra/matrix.v | [
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... | [
"apply",
"big1",
"big_split",
"eq_bigr",
"mxE",
"mxtrace",
"nmod_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_is_semi_additive | := mxtrace_is_nmod_morphism. | Definition | mxtrace_is_semi_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
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"finalg",
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... | [
"mxtrace_is_nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace0 : \tr 0 = 0. | Proof. exact: raddf0. Qed. | Lemma | mxtrace0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"finalg",
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... | [
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtraceD A B : \tr (A + B) = \tr A + \tr B. | Proof. exact: raddfD. Qed. | Lemma | mxtraceD | algebra | algebra/matrix.v | [
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... | [
"raddfD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_diag D : \tr (diag_mx D) = \sum_j D 0 j. | Proof. by apply: eq_bigr => j _; rewrite mxE eqxx. Qed. | Lemma | mxtrace_diag | algebra | algebra/matrix.v | [
"HB",
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... | [
"apply",
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"eq_bigr",
"eqxx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_scalar a : \tr a%:M = a *+ n. | Proof.
rewrite -diag_const_mx mxtrace_diag; under eq_bigr do rewrite mxE.
by rewrite sumr_const card_ord.
Qed. | Lemma | mxtrace_scalar | algebra | algebra/matrix.v | [
"HB",
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... | [
"card_ord",
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"eq_bigr",
"mxE",
"mxtrace_diag",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trace_mx11 (A : 'M_1) : \tr A = A 0 0. | Proof. by rewrite [A in LHS]mx11_scalar mxtrace_scalar. Qed. | Lemma | trace_mx11 | algebra | algebra/matrix.v | [
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... | [
"mx11_scalar",
"mxtrace_scalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_block n1 n2 (Aul : 'M_n1) Aur Adl (Adr : 'M_n2) :
\tr (block_mx Aul Aur Adl Adr) = \tr Aul + \tr Adr. | Proof.
rewrite /(\tr _) big_split_ord /=.
by congr (_ + _); under eq_bigr do rewrite (block_mxEul, block_mxEdr).
Qed. | Lemma | mxtrace_block | algebra | algebra/matrix.v | [
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... | [
"big_split_ord",
"block_mx",
"block_mxEdr",
"block_mxEul",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\tr A" | := (mxtrace A) : ring_scope. | Notation | \tr A | algebra | algebra/matrix.v | [
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"mathcomp",
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"finalg",
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... | [
"mxtrace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ^f" | := (map_mx f A) : ring_scope. | Notation | A ^f | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
"map_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx0 : 0^f = 0 :> 'M_(m, n). | Proof. by rewrite map_const_mx raddf0. Qed. | Lemma | map_mx0 | algebra | algebra/matrix.v | [
"HB",
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... | [
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"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxD A B : (A + B)^f = A^f + B^f. | Proof. by apply/matrixP=> i j; rewrite !mxE raddfD. Qed. | Lemma | map_mxD | algebra | algebra/matrix.v | [
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... | [
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"mxE",
"raddfD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_sum | := big_morph _ map_mxD map_mx0. | Definition | map_mx_sum | algebra | algebra/matrix.v | [
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... | [
"big_morph",
"map_mx0",
"map_mxD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppmx_key : unit. | Proof. by []. Qed. | Fact | oppmx_key | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppmx | := @map_mx V V -%R m n. | Definition | oppmx | algebra | algebra/matrix.v | [
"HB",
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... | [
"map_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addNmx : left_inverse (const_mx 0) oppmx (@addmx V m n). | Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE addNr. Qed. | Lemma | addNmx | algebra | algebra/matrix.v | [
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... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_mx_is_zmod_morphism : zmod_morphism const_mx. | Proof. exact: raddfB. Qed. | Fact | const_mx_is_zmod_morphism | algebra | algebra/matrix.v | [
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... | [
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"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_mx_is_additive | := const_mx_is_zmod_morphism. | Definition | const_mx_is_additive | algebra | algebra/matrix.v | [
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"mathcomp",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"const_mx_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
swizzle_mx_is_zmod_morphism k : zmod_morphism (swizzle_mx f g k). | Proof. exact: raddfB. Qed. | Fact | swizzle_mx_is_zmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"finalg",
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... | [
"raddfB",
"swizzle_mx",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
swizzle_mx_is_additive | := swizzle_mx_is_zmod_morphism. | Definition | swizzle_mx_is_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
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"ssrfun",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"swizzle_mx_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_row_mx m n1 n2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
- row_mx A1 A2 = row_mx (- A1) (- A2). | Proof. by split_mxE. Qed. | Lemma | opp_row_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"row_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_col_mx m1 m2 n (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
- col_mx A1 A2 = col_mx (- A1) (- A2). | Proof. by split_mxE. Qed. | Lemma | opp_col_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_block_mx m1 m2 n1 n2 (Aul : 'M_(m1, n1)) Aur Adl (Adr : 'M_(m2, n2)) :
- block_mx Aul Aur Adl Adr = block_mx (- Aul) (- Aur) (- Adl) (- Adr). | Proof. by rewrite opp_col_mx !opp_row_mx. Qed. | Lemma | opp_block_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"finalg",
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... | [
"block_mx",
"opp_col_mx",
"opp_row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_zmod_morphism n : zmod_morphism (@diag_mx V n). | Proof. exact: raddfB. Qed. | Fact | diag_mx_is_zmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
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"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"diag_mx",
"raddfB",
"zmod_morphism"
] | Diagonal matrices | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
diag_mx_is_additive | := diag_mx_is_zmod_morphism. | Definition | diag_mx_is_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
"diag_mx_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_zmod_morphism : zmod_morphism (@scalar_mx V n). | Proof. exact: raddfB. Qed. | Fact | scalar_mx_is_zmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"raddfB",
"scalar_mx",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_additive | := scalar_mx_is_zmod_morphism. | Definition | scalar_mx_is_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"scalar_mx_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_is_zmod_morphism : zmod_morphism (@mxtrace V n). | Proof. exact: raddfB. Qed. | Fact | mxtrace_is_zmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
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"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxtrace",
"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_is_additive | := mxtrace_is_zmod_morphism. | Definition | mxtrace_is_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxtrace_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxN A : (- A)^f = - A^f. | Proof. exact: raddfN. Qed. | Lemma | map_mxN | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"finalg",
"zmodp",
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... | [
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxB A B : (A - B)^f = A^f - B^f. | Proof. exact: raddfB. Qed. | Lemma | map_mxB | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finalg",
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... | [
"raddfB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemx_key : unit. | Proof. by []. Qed. | Fact | scalemx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemx x A | := \matrix[scalemx_key]_(i, j) (x * A i j). | Definition | scalemx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"finalg",
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"GRing.Theory",
... | [
"matrix",
"scalemx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
delta_mx_key : unit. | Proof. by []. Qed. | Fact | delta_mx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Basis | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
delta_mx i0 j0 : 'M[R]_(m, n) | :=
\matrix[delta_mx_key]_(i, j) ((i == i0) && (j == j0))%:R. | Definition | delta_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"delta_mx_key",
"i0",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x *m: A" | := (scalemx x A) (at level 40) : ring_scope. | Notation | x *m: A | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"scalemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale0mx A : 0 *m: A = 0. | Proof. by apply/matrixP=> i j; rewrite !mxE mul0r. Qed. | Fact | scale0mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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... | [
"apply",
"matrixP",
"mul0r",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale1mx A : 1 *m: A = A. | Proof. by apply/matrixP=> i j; rewrite !mxE mul1r. Qed. | Fact | scale1mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mul1r",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemxDl A x y : (x + y) *m: A = x *m: A + y *m: A. | Proof. by apply/matrixP=> i j; rewrite !mxE mulrDl. Qed. | Fact | scalemxDl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mulrDl",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemxDr x A B : x *m: (A + B) = x *m: A + x *m: B. | Proof. by apply/matrixP=> i j; rewrite !mxE mulrDr. Qed. | Fact | scalemxDr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
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... | [
"apply",
"matrixP",
"mulrDr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemxA x y A : x *m: (y *m: A) = (x * y) *m: A. | Proof. by apply/matrixP=> i j; rewrite !mxE mulrA. Qed. | Fact | scalemxA | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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... | [
"apply",
"matrixP",
"mulrA",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemx_const a b : a *: const_mx b = const_mx (a * b). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | scalemx_const | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
"apply",
"const_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_sum_delta A : A = \sum_(i < m) \sum_(j < n) A i j *: delta_mx i j. | Proof.
apply/matrixP=> i j.
rewrite summxE (bigD1_ord i) // summxE (bigD1_ord j) //= !mxE !eqxx mulr1.
rewrite !big1 ?addr0 //= => [j' | i'] _.
by rewrite !mxE eqxx eq_liftF mulr0.
by rewrite summxE big1// => j' _; rewrite !mxE eq_liftF mulr0.
Qed. | Lemma | matrix_sum_delta | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"eqtype",
"ssrnat",
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... | [
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"delta_mx",
"eq_liftF",
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"matrixP",
"mulr0",
"mulr1",
"mxE",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_delta m n i j : (delta_mx i j)^T = delta_mx j i :> 'M[R]_(n, m). | Proof. by apply/matrixP=> i' j'; rewrite !mxE andbC. Qed. | Lemma | trmx_delta | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
"apply",
"delta_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
delta_mx_lshift m n1 n2 i j :
delta_mx i (lshift n2 j) = row_mx (delta_mx i j) 0 :> 'M_(m, n1 + n2). | Proof.
apply/matrixP=> i' j'; rewrite !mxE -(can_eq splitK) (unsplitK (inl _ _)).
by case: split => ?; rewrite mxE ?andbF.
Qed. | Lemma | delta_mx_lshift | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"lshift",
"matrixP",
"mxE",
"row_mx",
"split",
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"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
delta_mx_rshift m n1 n2 i j :
delta_mx i (rshift n1 j) = row_mx 0 (delta_mx i j) :> 'M_(m, n1 + n2). | Proof.
apply/matrixP=> i' j'; rewrite !mxE -(can_eq splitK) (unsplitK (inr _ _)).
by case: split => ?; rewrite mxE ?andbF.
Qed. | Lemma | delta_mx_rshift | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"eqtype",
"ssrnat",
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... | [
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"can_eq",
"delta_mx",
"matrixP",
"mxE",
"row_mx",
"rshift",
"split",
"splitK",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
delta_mx_ushift m1 m2 n i j :
delta_mx (lshift m2 i) j = col_mx (delta_mx i j) 0 :> 'M_(m1 + m2, n). | Proof.
apply/matrixP=> i' j'; rewrite !mxE -(can_eq splitK) (unsplitK (inl _ _)).
by case: split => ?; rewrite mxE.
Qed. | Lemma | delta_mx_ushift | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
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"lshift",
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"split",
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"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
delta_mx_dshift m1 m2 n i j :
delta_mx (rshift m1 i) j = col_mx 0 (delta_mx i j) :> 'M_(m1 + m2, n). | Proof.
apply/matrixP=> i' j'; rewrite !mxE -(can_eq splitK) (unsplitK (inr _ _)).
by case: split => ?; rewrite mxE.
Qed. | Lemma | delta_mx_dshift | algebra | algebra/matrix.v | [
"HB",
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"mathcomp",
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... | [
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"can_eq",
"col_mx",
"delta_mx",
"matrixP",
"mxE",
"rshift",
"split",
"splitK",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vec_mx_delta m n i j :
vec_mx (delta_mx 0 (mxvec_index i j)) = delta_mx i j :> 'M_(m, n). | Proof.
by apply/matrixP=> i' j'; rewrite !mxE /= [_ == _](inj_eq enum_rank_inj).
Qed. | Lemma | vec_mx_delta | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"delta_mx",
"enum_rank_inj",
"inj_eq",
"matrixP",
"mxE",
"mxvec_index",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_delta m n i j :
mxvec (delta_mx i j) = delta_mx 0 (mxvec_index i j) :> 'rV_(m * n). | Proof. by rewrite -vec_mx_delta vec_mxK. Qed. | Lemma | mxvec_delta | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"delta_mx",
"mxvec",
"mxvec_index",
"vec_mxK",
"vec_mx_delta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx1 n : (1%:M)^T = 1%:M :> 'M[R]_n. | Proof. exact: tr_scalar_mx. Qed. | Lemma | trmx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"tr_scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row1 n i : row i (1%:M : 'M_n) = delta_mx 0 i. | Proof. by apply/rowP=> j; rewrite !mxE eq_sym. Qed. | Lemma | row1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"bigop",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"delta_mx",
"eq_sym",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col1 n i : col i (1%:M : 'M_n) = delta_mx i 0. | Proof. by apply/colP => j; rewrite !mxE eqxx andbT. Qed. | Lemma | col1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"col",
"colP",
"delta_mx",
"eqxx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_key : unit. | Proof. by []. Qed. | Fact | mulmx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Matrix multiplication using bigops. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulmx {m n p} (A : 'M_(m, n)) (B : 'M_(n, p)) : 'M[R]_(m, p) | :=
\matrix[mulmx_key]_(i, k) \sum_j (A i j * B j k). | Definition | mulmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix",
"mulmx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A *m B" | := (mulmx A B) : ring_scope. | Notation | A *m B | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxA m n p q (A : 'M_(m, n)) (B : 'M_(n, p)) (C : 'M_(p, q)) :
A *m (B *m C) = A *m B *m C. | Proof.
apply/matrixP=> i l /[!mxE]; under eq_bigr do rewrite mxE big_distrr/=.
rewrite exchange_big; apply: eq_bigr => j _; rewrite mxE big_distrl /=.
by under eq_bigr do rewrite mulrA.
Qed. | Lemma | mulmxA | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_distrl",
"big_distrr",
"eq_bigr",
"exchange_big",
"matrixP",
"mulrA",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul0mx m n p (A : 'M_(n, p)) : 0 *m A = 0 :> 'M_(m, p). | Proof.
by apply/matrixP=> i k; rewrite !mxE big1 //= => j _; rewrite mxE mul0r.
Qed. | Lemma | mul0mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big1",
"matrixP",
"mul0r",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx0 m n p (A : 'M_(m, n)) : A *m 0 = 0 :> 'M_(m, p). | Proof.
by apply/matrixP=> i k; rewrite !mxE big1 // => j _; rewrite mxE mulr0.
Qed. | Lemma | mulmx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big1",
"matrixP",
"mulr0",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxDl m n p (A1 A2 : 'M_(m, n)) (B : 'M_(n, p)) :
(A1 + A2) *m B = A1 *m B + A2 *m B. | Proof.
apply/matrixP=> i k; rewrite !mxE -big_split /=.
by apply: eq_bigr => j _; rewrite !mxE -mulrDl.
Qed. | Lemma | mulmxDl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_split",
"eq_bigr",
"matrixP",
"mulrDl",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxDr m n p (A : 'M_(m, n)) (B1 B2 : 'M_(n, p)) :
A *m (B1 + B2) = A *m B1 + A *m B2. | Proof.
apply/matrixP=> i k; rewrite !mxE -big_split /=.
by apply: eq_bigr => j _; rewrite mxE mulrDr.
Qed. | Lemma | mulmxDr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_split",
"eq_bigr",
"matrixP",
"mulrDr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemxAl m n p a (A : 'M_(m, n)) (B : 'M_(n, p)) :
a *: (A *m B) = (a *: A) *m B. | Proof.
apply/matrixP=> i k; rewrite !mxE big_distrr /=.
by apply: eq_bigr => j _; rewrite mulrA mxE.
Qed. | Lemma | scalemxAl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_distrr",
"eq_bigr",
"matrixP",
"mulrA",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_suml m n p (A : 'M_(n, p)) I r P (B_ : I -> 'M_(m, n)) :
(\sum_(i <- r | P i) B_ i) *m A = \sum_(i <- r | P i) B_ i *m A. | Proof.
by apply: (big_morph (mulmx^~ A)) => [B C|]; rewrite ?mul0mx ?mulmxDl.
Qed. | Lemma | mulmx_suml | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_morph",
"mul0mx",
"mulmx",
"mulmxDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_sumr m n p (A : 'M_(m, n)) I r P (B_ : I -> 'M_(n, p)) :
A *m (\sum_(i <- r | P i) B_ i) = \sum_(i <- r | P i) A *m B_ i. | Proof. exact: raddf_sum. Qed. | Lemma | mulmx_sumr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finfun",
"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"raddf_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowE m n i (A : 'M_(m, n)) : row i A = delta_mx 0 i *m A. | Proof.
apply/rowP=> j; rewrite !mxE (bigD1_ord i) //= mxE !eqxx mul1r.
by rewrite big1 ?addr0 // => i'; rewrite mxE /= lift_eqF mul0r.
Qed. | Lemma | rowE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
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"GRing.Theory",
... | [
"addr0",
"apply",
"big1",
"bigD1_ord",
"delta_mx",
"eqxx",
"lift_eqF",
"mul0r",
"mul1r",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colE m n i (A : 'M_(m, n)) : col i A = A *m delta_mx i 0. | Proof.
apply/colP=> j; rewrite !mxE (bigD1_ord i) //= mxE !eqxx mulr1.
by rewrite big1 ?addr0 // => i'; rewrite mxE /= lift_eqF mulr0.
Qed. | Lemma | colE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
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"finalg",
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... | [
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"bigD1_ord",
"col",
"colP",
"delta_mx",
"eqxx",
"lift_eqF",
"mulr0",
"mulr1",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_rVP m n A B : ((@mulmx 1 m n)^~ A =1 mulmx^~ B) <-> (A = B). | Proof. by split=> [eqAB|->//]; apply/row_matrixP => i; rewrite !rowE eqAB. Qed. | Lemma | mul_rVP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"mulmx",
"rowE",
"row_matrixP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mul m n p (i : 'I_m) A (B : 'M_(n, p)) :
row i (A *m B) = row i A *m B. | Proof. by rewrite !rowE mulmxA. Qed. | Lemma | row_mul | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmxA",
"row",
"rowE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsub_mul m n m' n' p f g (A : 'M_(m, p)) (B : 'M_(p, n)) :
mxsub f g (A *m B) = rowsub f A *m colsub g B :> 'M_(m', n'). | Proof. by split_mxE; under [RHS]eq_bigr do rewrite !mxE. Qed. | Lemma | mxsub_mul | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"colsub",
"eq_bigr",
"mxE",
"mxsub",
"n'",
"rowsub",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_rowsub_mx m n m' p f (A : 'M_(m, p)) (B : 'M_(p, n)) :
rowsub f A *m B = rowsub f (A *m B) :> 'M_(m', n). | Proof. by rewrite mxsub_mul mxsub_id. Qed. | Lemma | mul_rowsub_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxsub_id",
"mxsub_mul",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_colsub m n n' p g (A : 'M_(m, p)) (B : 'M_(p, n)) :
A *m colsub g B = colsub g (A *m B) :> 'M_(m, n'). | Proof. by rewrite mxsub_mul mxsub_id. Qed. | Lemma | mulmx_colsub | 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",
... | [
"colsub",
"mxsub_id",
"mxsub_mul",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_delta_mx_cond m n p (j1 j2 : 'I_n) (i1 : 'I_m) (k2 : 'I_p) :
delta_mx i1 j1 *m delta_mx j2 k2 = delta_mx i1 k2 *+ (j1 == j2). | Proof.
apply/matrixP => i k; rewrite !mxE (bigD1_ord j1) //=.
rewrite mulmxnE !mxE !eqxx andbT -natrM -mulrnA !mulnb !andbA andbAC.
by rewrite big1 ?addr0 // => j; rewrite !mxE andbC -natrM lift_eqF.
Qed. | Lemma | mul_delta_mx_cond | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addr0",
"apply",
"big1",
"bigD1_ord",
"delta_mx",
"eqxx",
"lift_eqF",
"matrixP",
"mulmxnE",
"mulnb",
"mulrnA",
"mxE",
"natrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_delta_mx m n p (j : 'I_n) (i : 'I_m) (k : 'I_p) :
delta_mx i j *m delta_mx j k = delta_mx i k. | Proof. by rewrite mul_delta_mx_cond eqxx. Qed. | Lemma | mul_delta_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"delta_mx",
"eqxx",
"mul_delta_mx_cond"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_delta_mx_0 m n p (j1 j2 : 'I_n) (i1 : 'I_m) (k2 : 'I_p) :
j1 != j2 -> delta_mx i1 j1 *m delta_mx j2 k2 = 0. | Proof. by rewrite mul_delta_mx_cond => /negPf->. Qed. | Lemma | mul_delta_mx_0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"delta_mx",
"mul_delta_mx_cond"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_diag_mx m n d (A : 'M_(m, n)) :
diag_mx d *m A = \matrix_(i, j) (d 0 i * A i j). | Proof.
apply/matrixP=> i j; rewrite !mxE (bigD1 i) //= mxE eqxx big1 ?addr0 // => i'.
by rewrite mxE eq_sym mulrnAl => /negPf->.
Qed. | Lemma | mul_diag_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",
... | [
"addr0",
"apply",
"big1",
"bigD1",
"diag_mx",
"eq_sym",
"eqxx",
"matrixP",
"mulrnAl",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mx_diag m n (A : 'M_(m, n)) d :
A *m diag_mx d = \matrix_(i, j) (A i j * d 0 j). | Proof.
apply/matrixP=> i j; rewrite !mxE (bigD1 j) //= mxE eqxx big1 ?addr0 // => i'.
by rewrite mxE eq_sym mulrnAr; move/negPf->.
Qed. | Lemma | mul_mx_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",
... | [
"addr0",
"apply",
"big1",
"bigD1",
"diag_mx",
"eq_sym",
"eqxx",
"matrixP",
"mulrnAr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_diag n (d e : 'rV_n) :
diag_mx d *m diag_mx e = diag_mx (\row_j (d 0 j * e 0 j)). | Proof. by apply/matrixP=> i j; rewrite mul_diag_mx !mxE mulrnAr. Qed. | Lemma | mulmx_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",
... | [
"apply",
"diag_mx",
"matrixP",
"mul_diag_mx",
"mulrnAr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mxM n a b : (a * b)%:M = a%:M *m b%:M :> 'M_n. | Proof.
rewrite -[in RHS]diag_const_mx mul_diag_mx.
by apply/matrixP => i j; rewrite !mxE mulrnAr.
Qed. | Lemma | scalar_mxM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"diag_const_mx",
"matrixP",
"mul_diag_mx",
"mulrnAr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1mx m n (A : 'M_(m, n)) : 1%:M *m A = A. | Proof.
by rewrite -diag_const_mx mul_diag_mx; apply/matrixP => i j; rewrite !mxE mul1r.
Qed. | Lemma | mul1mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"diag_const_mx",
"matrixP",
"mul1r",
"mul_diag_mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx1 m n (A : 'M_(m, n)) : A *m 1%:M = A. | Proof.
by rewrite -diag_const_mx mul_mx_diag; apply/matrixP=> i j; rewrite !mxE mulr1.
Qed. | Lemma | mulmx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"diag_const_mx",
"matrixP",
"mul_mx_diag",
"mulr1",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowsubE m m' n f (A : 'M_(m, n)) :
rowsub f A = rowsub f 1%:M *m A :> 'M_(m', n). | Proof. by rewrite mul_rowsub_mx mul1mx. Qed. | Lemma | rowsubE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mul1mx",
"mul_rowsub_mx",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_col_perm m n p s (A : 'M_(m, n)) (B : 'M_(n, p)) :
col_perm s A *m B = A *m row_perm s^-1 B. | Proof.
apply/matrixP=> i k; rewrite !mxE (reindex_perm s^-1).
by apply: eq_bigr => j _ /=; rewrite !mxE permKV.
Qed. | Lemma | mul_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",
"eq_bigr",
"matrixP",
"mxE",
"permKV",
"reindex_perm",
"row_perm"
] | mulmx and col_perm, row_perm, xcol, xrow | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mul_row_perm m n p s (A : 'M_(m, n)) (B : 'M_(n, p)) :
A *m row_perm s B = col_perm s^-1 A *m B. | Proof. by rewrite mul_col_perm invgK. Qed. | Lemma | mul_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",
... | [
"col_perm",
"invgK",
"mul_col_perm",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_xcol m n p j1 j2 (A : 'M_(m, n)) (B : 'M_(n, p)) :
xcol j1 j2 A *m B = A *m xrow j1 j2 B. | Proof. by rewrite mul_col_perm tpermV. Qed. | Lemma | mul_xcol | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mul_col_perm",
"tpermV",
"xcol",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mx n s : 'M_n | := row_perm s (1%:M : 'M[R]_n). | Definition | perm_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",
... | [
"row_perm"
] | Permutation matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tperm_mx n i1 i2 : 'M_n | := perm_mx (tperm i1 i2). | Definition | tperm_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",
... | [
"perm_mx",
"tperm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_permE m n s (A : 'M_(m, n)) : col_perm s A = A *m perm_mx s^-1. | Proof. by rewrite mul_row_perm mulmx1 invgK. Qed. | Lemma | col_permE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"invgK",
"mul_row_perm",
"mulmx1",
"perm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_permE m n s (A : 'M_(m, n)) : row_perm s A = perm_mx s *m A. | Proof.
by rewrite -[perm_mx _]mul1mx mul_row_perm mulmx1 -mul_row_perm mul1mx.
Qed. | Lemma | row_permE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mul1mx",
"mul_row_perm",
"mulmx1",
"perm_mx",
"row_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcolE m n j1 j2 (A : 'M_(m, n)) : xcol j1 j2 A = A *m tperm_mx j1 j2. | Proof. by rewrite /xcol col_permE tpermV. Qed. | Lemma | xcolE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_permE",
"tpermV",
"tperm_mx",
"xcol"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xrowE m n i1 i2 (A : 'M_(m, n)) : xrow i1 i2 A = tperm_mx i1 i2 *m A. | Proof. exact: row_permE. Qed. | Lemma | xrowE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_permE",
"tperm_mx",
"xrow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mxEsub n s : @perm_mx n s = rowsub s 1%:M. | Proof. by rewrite /perm_mx row_permEsub. Qed. | Lemma | perm_mxEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"perm_mx",
"row_permEsub",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tperm_mxEsub n i1 i2 : @tperm_mx n i1 i2 = rowsub (tperm i1 i2) 1%:M. | Proof. by rewrite /tperm_mx perm_mxEsub. Qed. | Lemma | tperm_mxEsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"perm_mxEsub",
"rowsub",
"tperm",
"tperm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_perm_mx n (s : 'S_n) : (perm_mx s)^T = perm_mx s^-1. | Proof. by rewrite -[_^T]mulmx1 tr_row_perm mul_col_perm trmx1 mul1mx. Qed. | Lemma | tr_perm_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",
... | [
"mul1mx",
"mul_col_perm",
"mulmx1",
"perm_mx",
"tr_row_perm",
"trmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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