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 |
|---|---|---|---|---|---|---|---|---|---|---|
memmx_subP m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
reflect (forall A, A \in R1 -> A \in R2) (R1 <= R2)%MS. | Proof.
apply: (iffP idP) => [sR12 A R1_A | sR12]; first exact: submx_trans sR12.
by apply/rV_subP=> vA; rewrite -(vec_mxK vA); apply: sR12.
Qed. | Lemma | memmx_subP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"rV_subP",
"submx_trans",
"vA",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx_eqP m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
reflect (forall A, (A \in R1) = (A \in R2)) (R1 == R2)%MS. | Proof.
apply: (iffP eqmxP) => [eqR12 A | eqR12]; first by rewrite eqR12.
by apply/eqmxP/rV_eqP=> vA; rewrite -(vec_mxK vA) eqR12.
Qed. | Lemma | memmx_eqP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"eqmxP",
"rV_eqP",
"vA",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx_addsP m1 m2 n A (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
reflect (exists D, [/\ D.1 \in R1, D.2 \in R2 & A = D.1 + D.2])
(A \in R1 + R2)%MS. | Proof.
apply: (iffP sub_addsmxP) => [[u /(canRL mxvecK)->] | [D []]].
exists (vec_mx (u.1 *m R1), vec_mx (u.2 *m R2)).
by rewrite /= linearD !vec_mxK !submxMl.
case/submxP=> u1 defD1 /submxP[u2 defD2] ->.
by exists (u1, u2); rewrite linearD /= defD1 defD2.
Qed. | Lemma | memmx_addsP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"defD2",
"linearD",
"mxvecK",
"sub_addsmxP",
"submxMl",
"submxP",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx_sumsP (I : finType) (P : pred I) n (A : 'M_n) R_ :
reflect (exists2 A_, A = \sum_(i | P i) A_ i & forall i, A_ i \in R_ i)
(A \in \sum_(i | P i) R_ i)%MS. | Proof.
apply: (iffP sub_sumsmxP) => [[C defA] | [A_ -> R_A] {A}].
exists (fun i => vec_mx (C i *m R_ i)) => [|i].
by rewrite -linear_sum -defA /= mxvecK.
by rewrite vec_mxK submxMl.
exists (fun i => mxvec (A_ i) *m pinvmx (R_ i)).
by rewrite linear_sum; apply: eq_bigr => i _; rewrite mulmxKpV.
Qed. | Lemma | memmx_sumsP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eq_bigr",
"linear_sum",
"mulmxKpV",
"mxvec",
"mxvecK",
"pinvmx",
"sub_sumsmxP",
"submxMl",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_non_scalar_mxP m n (R : 'A_(m, n)) :
(1%:M \in R)%MS ->
reflect (exists2 A, A \in R & ~~ is_scalar_mx A)%MS (1 < \rank R). | Proof.
case: (posnP n) => [-> | n_gt0] in R *; set S := mxvec _ => sSR.
by rewrite [R]thinmx0 mxrank0; right; case; rewrite /is_scalar_mx ?insubF.
have rankS: \rank S = 1%N.
apply/eqP; rewrite eqn_leq rank_leq_row lt0n mxrank_eq0 mxvec_eq0.
by rewrite -mxrank_eq0 mxrank1 -lt0n.
rewrite -{2}rankS (ltn_leqif (mxran... | Lemma | has_non_scalar_mxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqn_leq",
"insubF",
"is_scalar_mx",
"lt0n",
"ltn_leqif",
"memmx1",
"mxrank0",
"mxrank1",
"mxrank_eq0",
"mxrank_leqif_sup",
"mxvec",
"mxvec_eq0",
"n_gt0",
"posnP",
"rank",
"rankS",
"rank_leq_row",
"row",
"row_sub",
"row_subPn",
"sSR",
"submx_trans",
"thinmx0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmx m1 m2 n (R1 : 'A[F]_(m1, n)) (R2 : 'A_(m2, n)) | :=
(\sum_i <<R1 *m lin_mx (mulmxr (vec_mx (row i R2)))>>)%MS. | Definition | mulsmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"lin_mx",
"mulmxr",
"row",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"R1 * R2" | := (mulsmx R1 R2) : matrix_set_scope. | Notation | R1 * R2 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"mulsmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_muls m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
<<(R1 * R2)%MS>>%MS = (R1 * R2)%MS. | Proof. by rewrite genmx_sums; apply: eq_bigr => i; rewrite genmx_id. Qed. | Lemma | genmx_muls | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"eq_bigr",
"genmx_id",
"genmx_sums"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_mulsmx m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) A1 A2 :
(A1 \in R1 -> A2 \in R2 -> A1 *m A2 \in R1 * R2)%MS. | Proof.
move=> R_A1 R_A2; rewrite -[A2]mxvecK; case/submxP: R_A2 => a ->{A2}.
rewrite mulmx_sum_row !linear_sum summx_sub // => i _.
rewrite 3!linearZ scalemx_sub {a}//= (sumsmx_sup i) // genmxE.
rewrite -[A1]mxvecK; case/submxP: R_A1 => a ->{A1}.
by apply/submxP; exists a; rewrite mulmxA mul_rV_lin.
Qed. | Lemma | mem_mulsmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"genmxE",
"linearZ",
"linear_sum",
"mul_rV_lin",
"mulmxA",
"mulmx_sum_row",
"mxvecK",
"scalemx_sub",
"submxP",
"summx_sub",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmx_subP m1 m2 m n
(R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) (R : 'A_(m, n)) :
reflect (forall A1 A2, A1 \in R1 -> A2 \in R2 -> A1 *m A2 \in R)
(R1 * R2 <= R)%MS. | Proof.
apply: (iffP memmx_subP) => [sR12R A1 A2 R_A1 R_A2 | sR12R A].
by rewrite sR12R ?mem_mulsmx.
case/memmx_sumsP=> A_ -> R_A; rewrite linear_sum summx_sub //= => j _.
rewrite (submx_trans (R_A _)) // genmxE; apply/row_subP=> i.
by rewrite row_mul mul_rV_lin sR12R ?vec_mxK ?row_sub.
Qed. | Lemma | mulsmx_subP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"genmxE",
"linear_sum",
"mem_mulsmx",
"memmx_subP",
"memmx_sumsP",
"mul_rV_lin",
"row_mul",
"row_sub",
"row_subP",
"submx_trans",
"summx_sub",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmxS m1 m2 m3 m4 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n))
(R3 : 'A_(m3, n)) (R4 : 'A_(m4, n)) :
(R1 <= R3 -> R2 <= R4 -> R1 * R2 <= R3 * R4)%MS. | Proof.
move=> sR13 sR24; apply/mulsmx_subP=> A1 A2 R_A1 R_A2.
by apply: mem_mulsmx; [apply: submx_trans sR13 | apply: submx_trans sR24].
Qed. | Lemma | mulsmxS | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"R3",
"apply",
"mem_mulsmx",
"mulsmx_subP",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muls_eqmx m1 m2 m3 m4 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n))
(R3 : 'A_(m3, n)) (R4 : 'A_(m4, n)) :
(R1 :=: R3 -> R2 :=: R4 -> R1 * R2 = R3 * R4)%MS. | Proof.
move=> eqR13 eqR24; rewrite -(genmx_muls R1 R2) -(genmx_muls R3 R4).
by apply/genmxP; rewrite !mulsmxS ?eqR13 ?eqR24.
Qed. | Lemma | muls_eqmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"R3",
"apply",
"genmxP",
"genmx_muls",
"mulsmxS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmxP m1 m2 n A (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
reflect (exists2 A1, forall i, A1 i \in R1
& exists2 A2, forall i, A2 i \in R2
& A = \sum_(i < n ^ 2) A1 i *m A2 i)
(A \in R1 * R2)%MS. | Proof.
apply: (iffP idP) => [R_A|[A1 R_A1 [A2 R_A2 ->{A}]]]; last first.
by rewrite linear_sum summx_sub // => i _; rewrite mem_mulsmx.
have{R_A}: (A \in R1 * <<R2>>)%MS.
by apply: memmx_subP R_A; rewrite mulsmxS ?genmxE.
case/memmx_sumsP=> A_ -> R_A; pose A2_ i := vec_mx (row i <<R2>>%MS).
pose A1_ i := mxvec (A_ ... | Lemma | mulsmxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"apply",
"eq_bigr",
"genmxE",
"last",
"lin_mx",
"linear_sum",
"mem_mulsmx",
"memmx_subP",
"memmx_sumsP",
"mulmxA",
"mulmxKpV",
"mulmxr",
"mulsmxS",
"mx_rV_lin",
"mxvec",
"mxvecK",
"pinvmx",
"row",
"row_sub",
"submxMl",
"summx_sub",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmxA m1 m2 m3 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) (R3 : 'A_(m3, n)) :
(R1 * (R2 * R3) = R1 * R2 * R3)%MS. | Proof.
rewrite -(genmx_muls (_ * _)%MS) -genmx_muls; apply/genmxP/andP; split.
apply/mulsmx_subP=> A1 A23 R_A1; case/mulsmxP=> A2 R_A2 [A3 R_A3 ->{A23}].
by rewrite !linear_sum summx_sub //= => i _; rewrite mulmxA !mem_mulsmx.
apply/mulsmx_subP=> _ A3 /mulsmxP[A1 R_A1 [A2 R_A2 ->]] R_A3.
rewrite mulmx_suml linear_s... | Lemma | mulsmxA | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"R3",
"apply",
"genmxP",
"genmx_muls",
"linear_sum",
"mem_mulsmx",
"mulmxA",
"mulmx_suml",
"mulsmxP",
"mulsmx_subP",
"split",
"summx_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmxDl m1 m2 m3 n
(R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) (R3 : 'A_(m3, n)) :
((R1 + R2) * R3 = R1 * R3 + R2 * R3)%MS. | Proof.
rewrite -(genmx_muls R2 R3) -(genmx_muls R1 R3) -genmx_muls -genmx_adds.
apply/genmxP; rewrite andbC addsmx_sub !mulsmxS ?addsmxSl ?addsmxSr //=.
apply/mulsmx_subP=> _ A3 /memmx_addsP[A [R_A1 R_A2 ->]] R_A3.
by rewrite mulmxDl linearD addmx_sub_adds ?mem_mulsmx.
Qed. | Lemma | mulsmxDl | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"R3",
"addmx_sub_adds",
"addsmxSl",
"addsmxSr",
"addsmx_sub",
"apply",
"genmxP",
"genmx_adds",
"genmx_muls",
"linearD",
"mem_mulsmx",
"memmx_addsP",
"mulmxDl",
"mulsmxS",
"mulsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmxDr m1 m2 m3 n
(R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) (R3 : 'A_(m3, n)) :
(R1 * (R2 + R3) = R1 * R2 + R1 * R3)%MS. | Proof.
rewrite -(genmx_muls R1 R3) -(genmx_muls R1 R2) -genmx_muls -genmx_adds.
apply/genmxP; rewrite andbC addsmx_sub !mulsmxS ?addsmxSl ?addsmxSr //=.
apply/mulsmx_subP=> A1 _ R_A1 /memmx_addsP[A [R_A2 R_A3 ->]].
by rewrite mulmxDr linearD addmx_sub_adds ?mem_mulsmx.
Qed. | Lemma | mulsmxDr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"R3",
"addmx_sub_adds",
"addsmxSl",
"addsmxSr",
"addsmx_sub",
"apply",
"genmxP",
"genmx_adds",
"genmx_muls",
"linearD",
"mem_mulsmx",
"memmx_addsP",
"mulmxDr",
"mulsmxS",
"mulsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulsmx0 m1 m2 n (R1 : 'A_(m1, n)) : (R1 * (0 : 'A_(m2, n)) = 0)%MS. | Proof.
apply/eqP; rewrite -submx0; apply/mulsmx_subP=> A1 A0 _.
by rewrite [A0 \in 0]eqmx0 => /memmx0->; rewrite mulmx0 mem0mx.
Qed. | Lemma | mulsmx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"apply",
"eqmx0",
"mem0mx",
"memmx0",
"mulmx0",
"mulsmx_subP",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muls0mx m1 m2 n (R2 : 'A_(m2, n)) : ((0 : 'A_(m1, n)) * R2 = 0)%MS. | Proof.
apply/eqP; rewrite -submx0; apply/mulsmx_subP=> A0 A2.
by rewrite [A0 \in 0]eqmx0 => /memmx0->; rewrite mul0mx mem0mx.
Qed. | Lemma | muls0mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R2",
"apply",
"eqmx0",
"mem0mx",
"memmx0",
"mul0mx",
"mulsmx_subP",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
left_mx_ideal m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) | :=
(R1 * R2 <= R2)%MS. | Definition | left_mx_ideal | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
right_mx_ideal m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) | :=
(R2 * R1 <= R2)%MS. | Definition | right_mx_ideal | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_ideal m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) | :=
left_mx_ideal R1 R2 && right_mx_ideal R1 R2. | Definition | mx_ideal | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"left_mx_ideal",
"right_mx_ideal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxring_id m n (R : 'A_(m, n)) e | :=
[/\ e != 0,
e \in R,
forall A, A \in R -> e *m A = A
& forall A, A \in R -> A *m e = A]%MS. | Definition | mxring_id | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_mxring_id m n (R : 'A[F]_(m , n)) | :=
(R != 0) &&
(row_mx 0 (row_mx (mxvec R) (mxvec R))
<= row_mx (cokermx R) (row_mx (lin_mx (mulmx R \o lin_mulmx))
(lin_mx (mulmx R \o lin_mulmxr))))%MS. | Definition | has_mxring_id | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cokermx",
"lin_mulmx",
"lin_mulmxr",
"lin_mx",
"mulmx",
"mxvec",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxring m n (R : 'A_(m, n)) | :=
left_mx_ideal R R && has_mxring_id R. | Definition | mxring | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"has_mxring_id",
"left_mx_ideal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxring_idP m n (R : 'A_(m, n)) :
reflect (exists e, mxring_id R e) (has_mxring_id R). | Proof.
apply: (iffP andP) => [[nzR] | [e [nz_e Re ideR idRe]]].
case/submxP=> v; rewrite -[v]vec_mxK; move/vec_mx: v => e.
rewrite !mul_mx_row; case/eq_row_mx => /eqP.
rewrite eq_sym -submxE => Re.
case/eq_row_mx; rewrite !{1}mul_rV_lin1 /= mxvecK.
set u := (_ *m _) => /(can_inj mxvecK) idRe /(can_inj mxvecK)... | Lemma | mxring_idP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Re",
"apply",
"contraNneq",
"eq_row_mx",
"eq_sym",
"eqmx0",
"has_mxring_id",
"linear0",
"memmx0",
"mul_mx_row",
"mul_rV_lin1",
"mulmxA",
"mx_rV_lin",
"mxring_id",
"mxvec",
"mxvecK",
"row_matrixP",
"row_mul",
"row_mx",
"row_sub",
"split",
"submxE",
"submxP",
"vec_mx",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mx_fun (B : 'M[F]_n) | := R *m lin_mx (mulmxr B \- mulmx B). | Definition | cent_mx_fun | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"lin_mx",
"mulmx",
"mulmxr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mx_fun_is_linear : linear cent_mx_fun. | Proof.
move=> a A B; apply/row_matrixP=> i; rewrite linearP row_mul mul_rV_lin.
rewrite /= [row i _ as v in a *: v]row_mul mul_rV_lin row_mul mul_rV_lin.
by rewrite -linearP -(linearP (mulmx (vec_mx (row i R)) \- mulmxr _)).
Qed. | Lemma | cent_mx_fun_is_linear | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"cent_mx_fun",
"linear",
"linearP",
"mul_rV_lin",
"mulmx",
"mulmxr",
"row",
"row_matrixP",
"row_mul",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mx | := kermx (lin_mx cent_mx_fun). | Definition | cent_mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cent_mx_fun",
"kermx",
"lin_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
center_mx | := (R :&: cent_mx)%MS. | Definition | center_mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cent_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C' ( R )" | := (cent_mx R) : matrix_set_scope. | Notation | ''C' ( R ) | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cent_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Z' ( R )" | := (center_mx R) : matrix_set_scope. | Notation | ''Z' ( R ) | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"center_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_rowP m n B (R : 'A_(m, n)) :
reflect (forall i (A := vec_mx (row i R)), A *m B = B *m A) (B \in 'C(R))%MS. | Proof.
apply: (iffP sub_kermxP); rewrite mul_vec_lin => cBE.
move/(canRL mxvecK): cBE => cBE i A /=; move/(congr1 (row i)): cBE.
rewrite row_mul mul_rV_lin -/A; move/(canRL mxvecK).
by move/(canRL (subrK _)); rewrite !linear0 add0r.
apply: (canLR vec_mxK); apply/row_matrixP=> i.
by rewrite row_mul mul_rV_lin /= c... | Lemma | cent_rowP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"add0r",
"apply",
"linear0",
"mul_rV_lin",
"mul_vec_lin",
"mxvecK",
"row",
"row_matrixP",
"row_mul",
"sub_kermxP",
"subrK",
"subrr",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mxP m n B (R : 'A_(m, n)) :
reflect (forall A, A \in R -> A *m B = B *m A) (B \in 'C(R))%MS. | Proof.
apply: (iffP cent_rowP) => cEB => [A sAE | i A].
rewrite -[A]mxvecK -(mulmxKpV sAE); move: (mxvec A *m _) => u.
rewrite !mulmx_sum_row !linear_sum mulmx_suml; apply: eq_bigr => i _ /=.
by rewrite 2!linearZ -scalemxAl /= cEB.
by rewrite cEB // vec_mxK row_sub.
Qed. | Lemma | cent_mxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"cent_rowP",
"eq_bigr",
"linearZ",
"linear_sum",
"mulmxKpV",
"mulmx_sum_row",
"mulmx_suml",
"mxvec",
"mxvecK",
"row_sub",
"scalemxAl",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_cent m n a (R : 'A_(m, n)) : (a%:M \in 'C(R))%MS. | Proof. by apply/cent_mxP=> A _; apply: scalar_mxC. Qed. | Lemma | scalar_mx_cent | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"cent_mxP",
"scalar_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
center_mx_sub m n (R : 'A_(m, n)) : ('Z(R) <= R)%MS. | Proof. exact: capmxSl. Qed. | Lemma | center_mx_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"capmxSl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
center_mxP m n A (R : 'A_(m, n)) :
reflect (A \in R /\ forall B, B \in R -> B *m A = A *m B)
(A \in 'Z(R))%MS. | Proof.
rewrite sub_capmx; case R_A: (A \in R); last by right; case.
by apply: (iffP cent_mxP) => [cAR | [_ cAR]].
Qed. | Lemma | center_mxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"cent_mxP",
"last",
"sub_capmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxring_id_uniq m n (R : 'A_(m, n)) e1 e2 :
mxring_id R e1 -> mxring_id R e2 -> e1 = e2. | Proof.
by case=> [_ Re1 idRe1 _] [_ Re2 _ ide2R]; rewrite -(idRe1 _ Re2) ide2R.
Qed. | Lemma | mxring_id_uniq | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Re2",
"mxring_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mx_ideal m n (R : 'A_(m, n)) : left_mx_ideal 'C(R)%MS 'C(R)%MS. | Proof.
apply/mulsmx_subP=> A1 A2 C_A1 C_A2; apply/cent_mxP=> B R_B.
by rewrite mulmxA (cent_mxP C_A1) // -!mulmxA (cent_mxP C_A2).
Qed. | Lemma | cent_mx_ideal | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"cent_mxP",
"left_mx_ideal",
"mulmxA",
"mulsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mx_ring m n (R : 'A_(m, n)) : n > 0 -> mxring 'C(R)%MS. | Proof.
move=> n_gt0; rewrite /mxring cent_mx_ideal; apply/mxring_idP.
exists 1%:M; split=> [||A _|A _]; rewrite ?mulmx1 ?mul1mx ?scalar_mx_cent //.
by rewrite -mxrank_eq0 mxrank1 -lt0n.
Qed. | Lemma | cent_mx_ring | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"cent_mx_ideal",
"lt0n",
"mul1mx",
"mulmx1",
"mxrank1",
"mxrank_eq0",
"mxring",
"mxring_idP",
"n_gt0",
"scalar_mx_cent",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_adds_center m1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
mx_ideal (R1 + R2)%MS R1 -> mx_ideal (R1 + R2)%MS R2 ->
mxdirect (R1 + R2) ->
('Z((R1 + R2)%MS) :=: 'Z(R1) + 'Z(R2))%MS. | Proof.
case/andP=> idlR1 idrR1 /andP[idlR2 idrR2] /mxdirect_addsP dxR12.
apply/eqmxP/andP; split.
apply/memmx_subP=> z0; rewrite sub_capmx => /andP[].
case/memmx_addsP=> z [R1z1 R2z2 ->{z0}] Cz.
rewrite linearD addmx_sub_adds //= ?sub_capmx ?R1z1 ?R2z2 /=.
apply/cent_mxP=> A R1_A; have R_A := submx_trans R1_A... | Lemma | mxdirect_adds_center | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"R1",
"R2",
"addKr",
"addmx_sub_adds",
"addrK",
"addsmxSl",
"addsmxSr",
"addsmx_sub",
"apply",
"cent_mxP",
"eqmxP",
"linearD",
"memmx0",
"memmx_addsP",
"memmx_subP",
"mulmxBr",
"mulmxDl",
"mulmxDr",
"mulmxN",
"mulsmx_subP",
"mx_ideal",
"mxdirect",
"mxdirect_addsP",
"spl... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sums_center (I : finType) m n (R : 'A_(m, n)) R_ :
(\sum_i R_ i :=: R)%MS -> mxdirect (\sum_i R_ i) ->
(forall i : I, mx_ideal R (R_ i)) ->
('Z(R) :=: \sum_i 'Z(R_ i))%MS. | Proof.
move=> defR dxR idealR.
have sR_R: (R_ _ <= R)%MS by move=> i; rewrite -defR (sumsmx_sup i).
have anhR i j A B : i != j -> A \in R_ i -> B \in R_ j -> A *m B = 0.
move=> ne_ij RiA RjB; apply: memmx0.
have [[_ idRiR] [idRRj _]] := (andP (idealR i), andP (idealR j)).
rewrite -(mxdirect_sumsP dxR j) // sub_ca... | Lemma | mxdirect_sums_center | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addr0",
"apply",
"big1",
"bigD1",
"cent_mxP",
"defR",
"eq_sym",
"eqmxP",
"last",
"memmx0",
"memmx_subP",
"memmx_sumsP",
"mulmxDl",
"mulmxDr",
"mulmx_suml",
"mulmx_sumr",
"mulsmx_subP",
"mx_ideal",
"mxdirect",
"mxdirect_sumsP",
"split",
"sub_capmx",
"sumsmx_subP",
"sums... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A \in R" | := (submx (mxvec A) R) : matrix_set_scope. | Notation | A \in R | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxvec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"R * S" | := (mulsmx R S) : matrix_set_scope. | Notation | R * S | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mulsmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C_' R ( S )" | := (R :&: 'C(S))%MS : matrix_set_scope. | Notation | ''C_' R ( S ) | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C_' ( R ) ( S )" | := ('C_R(S))%MS (only parsing) : matrix_set_scope. | Notation | ''C_' ( R ) ( S ) | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gaussian_elimination_map m n (A : 'M_(m, n)) :
Gaussian_elimination_ A^f = ((col_ebase A)^f, (row_ebase A)^f, \rank A). | Proof.
rewrite mxrankE /row_ebase /col_ebase unlock.
elim: m n A => [|m IHm] [|n] A /=; rewrite ?map_mx1 //.
set pAnz := [pred k | A k.1 k.2 != 0].
rewrite (@eq_pick _ _ pAnz) => [k|]; first by rewrite /= mxE fmorph_eq0.
case: {+}(pick _) => [[i j]|]; last by rewrite !map_mx1.
rewrite mxE -fmorphV -map_xcol -map_xrow ... | Lemma | Gaussian_elimination_map | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Gaussian_elimination_",
"col_ebase",
"eq_pick",
"fmorphV",
"fmorph_eq0",
"last",
"map_block_mx",
"map_dlsubmx",
"map_drsubmx",
"map_mx0",
"map_mx1",
"map_mxB",
"map_mxM",
"map_mxZ",
"map_scalar_mx",
"map_ursubmx",
"map_xcol",
"map_xrow",
"mxE",
"mxrankE",
"pick",
"rank",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_map m n (A : 'M_(m, n)) : \rank A^f = \rank A. | Proof. by rewrite mxrankE Gaussian_elimination_map. Qed. | Lemma | mxrank_map | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Gaussian_elimination_map",
"mxrankE",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_free_map m n (A : 'M_(m, n)) : row_free A^f = row_free A. | Proof. by rewrite /row_free mxrank_map. Qed. | Lemma | row_free_map | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxrank_map",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_full_map m n (A : 'M_(m, n)) : row_full A^f = row_full A. | Proof. by rewrite /row_full mxrank_map. Qed. | Lemma | row_full_map | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxrank_map",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_row_ebase m n (A : 'M_(m, n)) : (row_ebase A)^f = row_ebase A^f. | Proof. by rewrite {2}/row_ebase unlock Gaussian_elimination_map. Qed. | Lemma | map_row_ebase | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Gaussian_elimination_map",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_col_ebase m n (A : 'M_(m, n)) : (col_ebase A)^f = col_ebase A^f. | Proof. by rewrite {2}/col_ebase unlock Gaussian_elimination_map. Qed. | Lemma | map_col_ebase | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Gaussian_elimination_map",
"col_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_row_base m n (A : 'M_(m, n)) :
(row_base A)^f = castmx (mxrank_map A, erefl n) (row_base A^f). | Proof.
move: (mxrank_map A); rewrite {2}/row_base mxrank_map => eqrr.
by rewrite castmx_id map_mxM map_pid_mx map_row_ebase.
Qed. | Lemma | map_row_base | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"castmx",
"castmx_id",
"map_mxM",
"map_pid_mx",
"map_row_ebase",
"mxrank_map",
"row_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_col_base m n (A : 'M_(m, n)) :
(col_base A)^f = castmx (erefl m, mxrank_map A) (col_base A^f). | Proof.
move: (mxrank_map A); rewrite {2}/col_base mxrank_map => eqrr.
by rewrite castmx_id map_mxM map_pid_mx map_col_ebase.
Qed. | Lemma | map_col_base | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"castmx",
"castmx_id",
"col_base",
"map_col_ebase",
"map_mxM",
"map_pid_mx",
"mxrank_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_pinvmx m n (A : 'M_(m, n)) : (pinvmx A)^f = pinvmx A^f. | Proof.
rewrite !map_mxM !map_invmx map_row_ebase map_col_ebase.
by rewrite map_pid_mx -mxrank_map.
Qed. | Lemma | map_pinvmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"map_col_ebase",
"map_invmx",
"map_mxM",
"map_pid_mx",
"map_row_ebase",
"mxrank_map",
"pinvmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_kermx m n (A : 'M_(m, n)) : (kermx A)^f = kermx A^f. | Proof.
by rewrite !map_mxM map_invmx map_col_ebase -mxrank_map map_copid_mx.
Qed. | Lemma | map_kermx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"kermx",
"map_col_ebase",
"map_copid_mx",
"map_invmx",
"map_mxM",
"mxrank_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_cokermx m n (A : 'M_(m, n)) : (cokermx A)^f = cokermx A^f. | Proof.
by rewrite !map_mxM map_invmx map_row_ebase -mxrank_map map_copid_mx.
Qed. | Lemma | map_cokermx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cokermx",
"map_copid_mx",
"map_invmx",
"map_mxM",
"map_row_ebase",
"mxrank_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_submx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A^f <= B^f)%MS = (A <= B)%MS. | Proof. by rewrite !submxE -map_cokermx -map_mxM map_mx_eq0. Qed. | Lemma | map_submx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"map_cokermx",
"map_mxM",
"map_mx_eq0",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_ltmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A^f < B^f)%MS = (A < B)%MS. | Proof. by rewrite /ltmx !map_submx. Qed. | Lemma | map_ltmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx",
"map_submx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_eqmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A^f :=: B^f)%MS <-> (A :=: B)%MS. | Proof.
split=> [/eqmxP|eqAB]; first by rewrite !map_submx => /eqmxP.
by apply/eqmxP; rewrite !map_submx !eqAB !submx_refl.
Qed. | Lemma | map_eqmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"map_submx",
"split",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_genmx m n (A : 'M_(m, n)) : (<<A>>^f :=: <<A^f>>)%MS. | Proof. by apply/eqmxP; rewrite !(genmxE, map_submx) andbb. Qed. | Lemma | map_genmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"genmxE",
"map_submx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_addsmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(((A + B)%MS)^f :=: A^f + B^f)%MS. | Proof.
by apply/eqmxP; rewrite !addsmxE -map_col_mx !map_submx !addsmxE andbb.
Qed. | Lemma | map_addsmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxE",
"apply",
"eqmxP",
"map_col_mx",
"map_submx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_capmx_gen m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(capmx_gen A B)^f = capmx_gen A^f B^f. | Proof. by rewrite map_mxM map_lsubmx map_kermx map_col_mx. Qed. | Lemma | map_capmx_gen | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"capmx_gen",
"map_col_mx",
"map_kermx",
"map_lsubmx",
"map_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_capmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
((A :&: B)^f :=: A^f :&: B^f)%MS. | Proof.
by apply/eqmxP; rewrite !capmxE -map_capmx_gen !map_submx -!capmxE andbb.
Qed. | Lemma | map_capmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"capmxE",
"eqmxP",
"map_capmx_gen",
"map_submx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_complmx m n (A : 'M_(m, n)) : (A^C^f = A^f^C)%MS. | Proof. by rewrite map_mxM map_row_ebase -mxrank_map map_copid_mx. Qed. | Lemma | map_complmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"map_copid_mx",
"map_mxM",
"map_row_ebase",
"mxrank_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_diffmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
((A :\: B)^f :=: A^f :\: B^f)%MS. | Proof.
apply/eqmxP; rewrite !diffmxE -map_capmx_gen -map_complmx.
by rewrite -!map_capmx !map_submx -!diffmxE andbb.
Qed. | Lemma | map_diffmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"diffmxE",
"eqmxP",
"map_capmx",
"map_capmx_gen",
"map_complmx",
"map_submx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_eigenspace n (g : 'M_n) a : (eigenspace g a)^f = eigenspace g^f (f a). | Proof. by rewrite map_kermx map_mxB ?map_scalar_mx. Qed. | Lemma | map_eigenspace | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eigenspace",
"map_kermx",
"map_mxB",
"map_scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue_map n (g : 'M_n) a : eigenvalue g^f (f a) = eigenvalue g a. | Proof. by rewrite /eigenvalue -map_eigenspace map_mx_eq0. Qed. | Lemma | eigenvalue_map | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eigenvalue",
"map_eigenspace",
"map_mx_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx_map m n A (E : 'A_(m, n)) : (A^f \in E^f)%MS = (A \in E)%MS. | Proof. by rewrite -map_mxvec map_submx. Qed. | Lemma | memmx_map | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"map_mxvec",
"map_submx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mulsmx m1 m2 n (E1 : 'A_(m1, n)) (E2 : 'A_(m2, n)) :
((E1 * E2)%MS^f :=: E1^f * E2^f)%MS. | Proof.
rewrite /mulsmx; elim/big_rec2: _ => [|i A Af _ eqA]; first by rewrite map_mx0.
apply: (eqmx_trans (map_addsmx _ _)); apply: adds_eqmx {A Af}eqA.
apply/eqmxP; rewrite !map_genmx !genmxE map_mxM.
apply/rV_eqP=> u; congr (u <= _ *m _)%MS.
by apply: map_lin_mx => //= A; rewrite map_mxM // map_vec_mx map_row.
Qed. | Lemma | map_mulsmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"adds_eqmx",
"apply",
"big_rec2",
"eqmxP",
"eqmx_trans",
"genmxE",
"map_addsmx",
"map_genmx",
"map_lin_mx",
"map_mx0",
"map_mxM",
"map_row",
"map_vec_mx",
"mulsmx",
"rV_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_cent_mx m n (E : 'A_(m, n)) : ('C(E)%MS)^f = 'C(E^f)%MS. | Proof.
rewrite map_kermx; congr kermx; apply: map_lin_mx => A; rewrite map_mxM.
by congr (_ *m _); apply: map_lin_mx => B; rewrite map_mxB ?map_mxM.
Qed. | Lemma | map_cent_mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"kermx",
"map_kermx",
"map_lin_mx",
"map_mxB",
"map_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_center_mx m n (E : 'A_(m, n)) : (('Z(E))^f :=: 'Z(E^f))%MS. | Proof. by rewrite /center_mx -map_cent_mx; apply: map_capmx. Qed. | Lemma | map_center_mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"center_mx",
"map_capmx",
"map_cent_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_col {m} (V_ : forall i, 'M[F]_(p_ i, m)) :
(\mxcol_i V_ i :=: \sum_i <<V_ i>>)%MS. | Proof.
apply/eqmxP/andP; split.
apply/row_subP => i; rewrite row_mxcol.
by rewrite (sumsmx_sup (sig1 i))// genmxE row_sub.
apply/sumsmx_subP => i0 _; rewrite genmxE; apply/row_subP => j.
apply: (eq_row_sub (Rank _ j)); apply/rowP => k.
by rewrite !mxE Rank2K; case: _ / esym; rewrite cast_ord_id.
Qed. | Lemma | eqmx_col | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Rank",
"Rank2K",
"apply",
"cast_ord_id",
"eq_row_sub",
"eqmxP",
"genmxE",
"i0",
"mxE",
"rowP",
"row_mxcol",
"row_sub",
"row_subP",
"sig1",
"split",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_mxdiag (V_ : forall i, 'M[F]_(p_ i)) :
(\rank (\mxdiag_i V_ i) = \sum_i \rank (V_ i))%N. | Proof.
elim: {+}n {+}p_ V_ => [|m IHm] q_ V_.
by move: (\mxdiag__ _); rewrite !big_ord0 => M; rewrite flatmx0 mxrank0.
rewrite mxdiag_recl [RHS]big_ord_recl/= -IHm.
by case: _ / mxsize_recl; rewrite ?castmx_id rank_diag_block_mx.
Qed. | Lemma | rank_mxdiag | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"big_ord0",
"big_ord_recl",
"castmx_id",
"flatmx0",
"mxdiag_recl",
"mxrank0",
"mxsize_recl",
"rank",
"rank_diag_block_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVpoly v | := \poly_(k < d) (if insub k is Some i then v 0 i else 0). | Definition | rVpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"insub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_rV p | := \row_(i < d) p`_i. | Definition | poly_rV | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef_rVpoly v k : (rVpoly v)`_k = if insub k is Some i then v 0 i else 0. | Proof. by rewrite coef_poly; case: insubP => [i ->|]; rewrite ?if_same. Qed. | Lemma | coef_rVpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"coef_poly",
"insub",
"insubP",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef_rVpoly_ord v (i : 'I_d) : (rVpoly v)`_i = v 0 i. | Proof. by rewrite coef_rVpoly valK. Qed. | Lemma | coef_rVpoly_ord | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"coef_rVpoly",
"rVpoly",
"valK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVpoly_delta i : rVpoly (delta_mx 0 i) = 'X^i. | Proof.
apply/polyP=> j; rewrite coef_rVpoly coefXn.
case: insubP => [k _ <- | j_ge_d]; first by rewrite mxE.
by case: eqP j_ge_d => // ->; rewrite ltn_ord.
Qed. | Lemma | rVpoly_delta | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coefXn",
"coef_rVpoly",
"delta_mx",
"insubP",
"ltn_ord",
"mxE",
"polyP",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVpolyK : cancel rVpoly poly_rV. | Proof. by move=> u; apply/rowP=> i; rewrite mxE coef_rVpoly_ord. Qed. | Lemma | rVpolyK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coef_rVpoly_ord",
"mxE",
"poly_rV",
"rVpoly",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_rV_K p : size p <= d -> rVpoly (poly_rV p) = p. | Proof.
move=> le_p_d; apply/polyP=> k; rewrite coef_rVpoly.
case: insubP => [i _ <- | ]; first by rewrite mxE.
by rewrite -ltnNge => le_d_l; rewrite nth_default ?(leq_trans le_p_d).
Qed. | Lemma | poly_rV_K | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coef_rVpoly",
"insubP",
"leq_trans",
"ltnNge",
"mxE",
"nth_default",
"polyP",
"poly_rV",
"rVpoly",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_rV_is_semilinear : semilinear poly_rV. | Proof. by split=> [a p|p q]; apply/rowP=> i; rewrite !mxE (coefZ, coefD). Qed. | Lemma | poly_rV_is_semilinear | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coefD",
"coefZ",
"mxE",
"poly_rV",
"rowP",
"semilinear",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_rV_is_linear : linear poly_rV. | Proof. exact: linearP. Qed. | Lemma | poly_rV_is_linear | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"linear",
"linearP",
"poly_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVpoly_is_semilinear : semilinear rVpoly. | Proof.
split=> [a u|u v]; apply/polyP=> k; rewrite (coefZ, coefD) !coef_rVpoly.
by case: insubP => [i _ _|_]; rewrite ?mxE // mulr0.
by case: insubP=> [i _ _|_]; rewrite ?mxE ?addr0.
Qed. | Lemma | rVpoly_is_semilinear | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addr0",
"apply",
"coefD",
"coefZ",
"coef_rVpoly",
"insubP",
"mulr0",
"mxE",
"polyP",
"rVpoly",
"semilinear",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rvPoly_is_linear : linear rVpoly. | Proof. exact: linearP. Qed. | Lemma | rvPoly_is_linear | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"linear",
"linearP",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dS | := ((size q).-1 + (size p).-1)%N. | Let | dS | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
band r | := (lin1_mx (poly_rV \o r \o* rVpoly)). | Notation | band | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"lin1_mx",
"poly_rV",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylvester_mx : 'M[R]_dS | := col_mx (band p) (band q). | Definition | Sylvester_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"band",
"col_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylvester_mxE (i j : 'I_dS) :
let S_ r k := r`_(j - k) *+ (k <= j) in
Sylvester_mx i j = match split i with inl k => S_ p k | inr k => S_ q k end. | Proof.
move=> S_ /[1!mxE]; case: {i}(split i) => i /[!mxE]/=;
by rewrite rVpoly_delta coefXnM ltnNge if_neg -mulrb.
Qed. | Lemma | Sylvester_mxE | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Sylvester_mx",
"coefXnM",
"ltnNge",
"mulrb",
"mxE",
"rVpoly_delta",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
resultant | := \det Sylvester_mx. | Definition | resultant | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Sylvester_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
resultant_in_ideal (R : comNzRingType) (p q : {poly R}) :
size p > 1 -> size q > 1 ->
{uv : {poly R} * {poly R} | size uv.1 < size q /\ size uv.2 < size p
& (resultant p q)%:P = uv.1 * p + uv.2 * q}. | Proof.
move=> p_nc q_nc; pose dp := (size p).-1; pose dq := (size q).-1.
pose S := Sylvester_mx p q; pose dS := (dq + dp)%N.
have dS_gt0: dS > 0 by rewrite /dS /dq -(subnKC q_nc).
pose j0 := Ordinal dS_gt0.
pose Ss0 := col_mx (p *: \col_(i < dq) 'X^i) (q *: \col_(i < dp) 'X^i).
pose Ss := \matrix_(i, j) (if j == j0 the... | Lemma | resultant_in_ideal | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Sylvester_mx",
"Sylvester_mxE",
"add0r",
"add1n",
"addnC",
"addnS",
"addr0",
"addrC",
"apply",
"big1",
"big_distrl",
"big_ord0",
"big_split_ord",
"bigmax_leqP",
"coefC",
"coefD",
"coefMXn",
"coefXnM",
"coef_poly",
"cofactor",
"col_mx",
"col_mxEd",
"col_mxEu",
"dS",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
resultant_eq0 (R : idomainType) (p q : {poly R}) :
(resultant p q == 0) = (size (gcdp p q) > 1). | Proof.
have dvdpp := dvdpp; set r := gcdp p q.
pose dp := (size p).-1; pose dq := (size q).-1.
have /andP[r_p r_q]: (r %| p) && (r %| q) by rewrite -dvdp_gcd.
apply/det0P/idP=> [[uv nz_uv] | r_nonC].
have [p0 _ | p_nz] := eqVneq p 0.
have: dq + dp > 0.
rewrite lt0n; apply: contraNneq nz_uv => dqp0.
by... | Lemma | resultant_eq0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"add0r",
"addNr",
"addn0",
"addnA",
"addnC",
"addnCA",
"addrC",
"addr_eq0",
"apply",
"contraNneq",
"contra_eq",
"def_r",
"det0P",
"dvdpN0",
"dvdpP",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_leq",
"dvdp_mull",
"dvdp_mulr",
"dvdpp",
"eqVneq",
"eq_row_mx",
"gcd0p",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx | := horner_morph (comm_mx_scalar^~ A). | Definition | horner_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"comm_mx_scalar",
"horner_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_C a : horner_mx a%:P = a%:M. | Proof. exact: horner_morphC. Qed. | Lemma | horner_mx_C | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_morphC",
"horner_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_X : horner_mx 'X = A. | Proof. exact: horner_morphX. Qed. | Lemma | horner_mx_X | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_morphX",
"horner_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mxZ : scalable horner_mx. | Proof.
move=> a p /=; rewrite -mul_polyC rmorphM /=.
by rewrite horner_mx_C [_ * _]mul_scalar_mx.
Qed. | Lemma | horner_mxZ | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx",
"horner_mx_C",
"mul_polyC",
"mul_scalar_mx",
"rmorphM",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
powers_mx d | := \matrix_(i < d) mxvec (A ^+ i). | Definition | powers_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"mxvec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_rVpoly m (u : 'rV_m) :
horner_mx (rVpoly u) = vec_mx (u *m powers_mx m). | Proof.
rewrite mulmx_sum_row [rVpoly u]poly_def 2!linear_sum; apply: eq_bigr => i _.
by rewrite valK /= 2!linearZ rmorphXn/= horner_mx_X rowK mxvecK.
Qed. | Lemma | horner_rVpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"eq_bigr",
"horner_mx",
"horner_mx_X",
"linearZ",
"linear_sum",
"mulmx_sum_row",
"mxvecK",
"poly_def",
"powers_mx",
"rVpoly",
"rmorphXn",
"rowK",
"valK",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_diag (d : 'rV[R]_n) (p : {poly R}) :
horner_mx (diag_mx d) p = diag_mx (map_mx (horner p) d). | Proof.
apply/matrixP => i j; rewrite !mxE.
elim/poly_ind: p => [|p c ihp]; first by rewrite rmorph0 horner0 mxE mul0rn.
rewrite !hornerE mulrnDl rmorphD rmorphM /= horner_mx_X horner_mx_C !mxE.
rewrite (bigD1 j)//= ihp mxE eqxx mulr1n -mulrnAl big1 ?addr0.
by move=> k /negPf nkF; rewrite mxE nkF mulr0.
by have [->|_]... | Lemma | horner_mx_diag | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addr0",
"apply",
"big1",
"bigD1",
"diag_mx",
"eqVneq",
"eqxx",
"horner",
"horner0",
"hornerE",
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"map_mx",
"matrixP",
"mul0r",
"mul0rn",
"mulr0",
"mulr1n",
"mulrnAl",
"mulrnDl",
"mxE",
"poly",
"poly_ind",
"rmorph0",
"rmorphD... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_horner A B p : comm_mx A B -> comm_mx A (horner_mx B p). | Proof.
move=> fg; apply: commr_horner => // i.
by rewrite coef_map; apply/comm_scalar_mx.
Qed. | Lemma | comm_mx_horner | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coef_map",
"comm_mx",
"comm_scalar_mx",
"commr_horner",
"horner_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_horner_mx A B p : comm_mx A B -> comm_mx (horner_mx A p) B. | Proof. by move=> ?; apply/comm_mx_sym/comm_mx_horner/comm_mx_sym. Qed. | Lemma | comm_horner_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"comm_mx",
"comm_mx_horner",
"comm_mx_sym",
"horner_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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