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 |
|---|---|---|---|---|---|---|---|---|---|---|
nondegenerate | := radv == 0%VS. | Definition | nondegenerate | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"radv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_psymplectic | := [/\ nondegenerate, is_skew form &
2 \in [pchar F] -> forall u, '[u, u] = 0]. | Definition | is_psymplectic | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"is_skew",
"nondegenerate",
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_porthogonal | := [/\ nondegenerate, is_sym form &
2 \in [pchar F] -> forall u, '[u, u] = 0]. | Definition | is_porthogonal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"is_sym",
"nondegenerate",
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_unitary | := nondegenerate /\ (is_hermsym form). | Definition | is_unitary | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"is_hermsym",
"nondegenerate"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_symplectic | := is_psymplectic (only parsing). | Notation | is_symplectic | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"is_psymplectic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_orthogonal | := is_porthogonal (only parsing). | Notation | is_orthogonal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"is_porthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnorm_geiff0 u : 0 <= '[u] ?= iff (u == 0). | Proof.
by apply/leifP; have [->|uN0] := altP eqP; rewrite ?linear0r ?neq0_dnorm_gt0.
Qed. | Lemma | dnorm_geiff0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"leifP",
"linear0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnorm_ge0 u : 0 <= '[u]. | Proof. by rewrite dnorm_geiff0. Qed. | Lemma | dnorm_ge0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"dnorm_geiff0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnorm_eq0 u : ('[u] == 0) = (u == 0). | Proof. by rewrite -dnorm_geiff0 eq_sym. Qed. | Lemma | dnorm_eq0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"dnorm_geiff0",
"eq_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnorm_gt0 u : (0 < '[u]) = (u != 0). | Proof. by rewrite lt_def dnorm_eq0 dnorm_ge0 andbT. Qed. | Lemma | dnorm_gt0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"dnorm_eq0",
"dnorm_ge0",
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_dnorm_ge0 u : 0 <= sqrtC '[u]. | Proof. by rewrite sqrtC_ge0 dnorm_ge0. Qed. | Lemma | sqrt_dnorm_ge0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"dnorm_ge0",
"sqrtC",
"sqrtC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_dnorm_eq0 u : (sqrtC '[u] == 0) = (u == 0). | Proof. by rewrite sqrtC_eq0 dnorm_eq0. Qed. | Lemma | sqrt_dnorm_eq0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"dnorm_eq0",
"sqrtC",
"sqrtC_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_dnorm_gt0 u : (sqrtC '[u] > 0) = (u != 0). | Proof. by rewrite sqrtC_gt0 dnorm_gt0. Qed. | Lemma | sqrt_dnorm_gt0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"dnorm_gt0",
"sqrtC",
"sqrtC_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnormZ a u : '[a *: u]= `|a| ^+ 2 * '[u]. | Proof. by rewrite linearZl_LR linearZr_LR/= mulrA normCK. Qed. | Lemma | dnormZ | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"linearZl_LR",
"linearZr_LR",
"mulrA",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnormD u v : let d := '[u, v] in '[u + v] = '[u] + '[v] + (d + d^*). | Proof. by rewrite hnormD mul1r. Qed. | Lemma | dnormD | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hnormD",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dnormB u v : let d := '[u, v] in '[u - v] = '[u] + '[v] - (d + d^*). | Proof. by rewrite hnormB mul1r. Qed. | Lemma | dnormB | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hnormB",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairwise_orthogonalP S :
reflect (uniq (0 :: S)
/\ {in S &, forall phi psi, phi != psi -> '[phi, psi] = 0})
(pairwise_orthogonal form S). | Proof.
rewrite /pairwise_orthogonal /=; case notS0: (~~ _); last by right; case.
elim: S notS0 => [|phi S IH] /=; first by left.
rewrite inE eq_sym andbT => /norP[nz_phi {}/IH IH].
have [opS | not_opS] := allP; last first.
right=> [[/andP[notSp _] opS]]; case: not_opS => psi Spsi /=.
by rewrite opS ?mem_head 1?mem_... | Lemma | pairwise_orthogonalP | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"allP",
"apply",
"dnorm_eq0",
"eq_sym",
"form",
"hermC",
"inE",
"last",
"memPnC",
"mem_behead",
"mem_head",
"mulr0",
"notS0",
"oSS",
"pairwise_orthogonal",
"predU1P",
"rmorph0",
"split",
"uniq",
"uniqS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairwise_orthogonal_cat R S :
pairwise_orthogonal form (R ++ S) =
[&& pairwise_orthogonal form R, pairwise_orthogonal form S & orthogonal form R S]. | Proof.
rewrite /pairwise_orthogonal mem_cat negb_or -!andbA; do !bool_congr.
elim: R => [|phi R /= ->]; rewrite ?andbT// all_cat -!andbA /=.
by do !bool_congr.
Qed. | Lemma | pairwise_orthogonal_cat | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"all_cat",
"form",
"mem_cat",
"orthogonal",
"pairwise_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal_cat R S :
orthonormal form (R ++ S) =
[&& orthonormal form R, orthonormal form S & orthogonal form R S]. | Proof.
rewrite !orthonormalE pairwise_orthogonal_cat all_cat -!andbA.
by do !bool_congr.
Qed. | Lemma | orthonormal_cat | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"all_cat",
"form",
"orthogonal",
"orthonormal",
"orthonormalE",
"pairwise_orthogonal_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormalP S :
reflect (uniq S /\ {in S &, forall phi psi, '[phi, psi] = (phi == psi)%:R})
(orthonormal form S). | Proof.
rewrite orthonormalE; have [/= normS | not_normS] := allP; last first.
by right=> [[_ o1S]]; case: not_normS => phi Sphi; rewrite /= o1S ?eqxx.
apply: (iffP (pairwise_orthogonalP S)) => [] [uniqS oSS].
split=> // [|phi psi]; first by case/andP: uniqS.
by have [-> _ /normS/eqP | /oSS] := altP eqP.
split=> /... | Lemma | orthonormalP | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"allP",
"apply",
"eq_sym",
"eqxx",
"form",
"last",
"linear0r",
"oSS",
"oner_eq0",
"orthonormal",
"orthonormalE",
"pairwise_orthogonalP",
"split",
"uniq",
"uniqS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_orthonormal S1 S2 :
{subset S1 <= S2} -> uniq S1 -> orthonormal form S2 -> orthonormal form S1. | Proof.
move=> sS12 uniqS1 /orthonormalP[_ oS1].
by apply/orthonormalP; split; last apply: sub_in2 sS12 _ _.
Qed. | Lemma | sub_orthonormal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"S1",
"S2",
"apply",
"form",
"last",
"orthonormal",
"orthonormalP",
"split",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal2P phi psi :
reflect [/\ '[phi, psi] = 0, '[phi] = 1 & '[psi] = 1]
(orthonormal form [:: phi; psi]). | Proof.
rewrite /orthonormal /= !andbT andbC.
by apply: (iffP and3P) => [] []; do 3!move/eqP->.
Qed. | Lemma | orthonormal2P | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"form",
"orthonormal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_pairwise_orthogonal S1 S2 :
{subset S1 <= S2} -> uniq S1 ->
pairwise_orthogonal form S2 -> pairwise_orthogonal form S1. | Proof.
move=> sS12 uniqS1 /pairwise_orthogonalP[/andP[notS2_0 _] oS2].
apply/pairwise_orthogonalP; rewrite /= (contra (sS12 0)) //.
by split=> //; apply: sub_in2 oS2.
Qed. | Lemma | sub_pairwise_orthogonal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"S1",
"S2",
"apply",
"form",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"split",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_free S : pairwise_orthogonal form S -> free S. | Proof.
case/pairwise_orthogonalP=> [/=/andP[notS0 uniqS] oSS].
rewrite -(in_tupleE S); apply/freeP => a aS0 i.
have S_i: S`_i \in S by apply: mem_nth.
have /eqP: '[S`_i, 0] = 0 := linear0r _ _.
rewrite -{2}aS0 raddf_sum /= (bigD1 i) //= big1 => [j neq_ji|].
by rewrite linearZ /= oSS ?mulr0 ?mem_nth // eq_sym nth_uniq... | Lemma | orthogonal_free | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"addr0",
"apply",
"big1",
"bigD1",
"conjC_eq0",
"dnorm_eq0",
"eq_sym",
"form",
"free",
"freeP",
"in_tupleE",
"linear0r",
"linearZ",
"mem_nth",
"mulf_eq0",
"mulr0",
"notS0",
"nth_uniq",
"oSS",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"pred2P",
"raddf_sum",
"uniqS"
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_pairwise_orthogonal S p :
pairwise_orthogonal form S -> pairwise_orthogonal form (filter p S). | Proof.
move=> orthoS; apply: sub_pairwise_orthogonal (orthoS).
exact: mem_subseq (filter_subseq p S).
exact/filter_uniq/free_uniq/orthogonal_free.
Qed. | Lemma | filter_pairwise_orthogonal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"filter",
"filter_subseq",
"filter_uniq",
"form",
"free_uniq",
"mem_subseq",
"orthogonal_free",
"pairwise_orthogonal",
"sub_pairwise_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal_free S : orthonormal form S -> free S. | Proof. by move/orthonormal_orthogonal/orthogonal_free. Qed. | Lemma | orthonormal_free | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"free",
"orthogonal_free",
"orthonormal",
"orthonormal_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CauchySchwarz (u v : U) :
`|'[u, v]| ^+ 2 <= '[u] * '[v] ?= iff ~~ free [:: u; v]. | Proof.
rewrite free_cons span_seq1 seq1_free -negb_or negbK orbC.
have [-> | nz_v] /= := altP (v =P 0).
by apply/leifP; rewrite /= !linear0r normCK mul0r mulr0.
without loss ou: u / '[u, v] = 0.
move=> IHo; pose a := '[u, v] / '[v]; pose u1 := u - a *: v.
have ou: '[u1, v] = 0.
rewrite linearBl/=.
rewrite... | Theorem | CauchySchwarz | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"add0r",
"apply",
"divfK",
"dnormZ",
"dnorm_eq0",
"dnorm_ge0",
"eq_sym",
"exprMn",
"free",
"free_cons",
"ger0_norm",
"hnormDd",
"last",
"leifBLR",
"leifP",
"linear0r",
"linearBl",
"linearDl",
"linearZl_LR",
"linearZr_LR",
"memv_line",
"mul0r",
"mulf_eq0",
"mulr0",
"mu... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CauchySchwarz_sqrt u v :
`|'[u, v]| <= sqrtC '[u] * sqrtC '[v] ?= iff ~~ free [:: u; v]. | Proof.
rewrite -(sqrCK (normr_ge0 _)) -sqrtCM ?nnegrE//.
rewrite (mono_in_leif (@ler_sqrtC _)) 1?rpredM//= ?nnegrE//=.
exact: CauchySchwarz.
Qed. | Lemma | CauchySchwarz_sqrt | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"CauchySchwarz",
"free",
"ler_sqrtC",
"mono_in_leif",
"nnegrE",
"normr_ge0",
"rpredM",
"sqrCK",
"sqrtC",
"sqrtCM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoP phi psi : reflect ('[phi, psi] = 0) (orthogonal form [:: phi] [:: psi]). | Proof. by rewrite /orthogonal /= !andbT; apply: eqP. Qed. | Lemma | orthoP | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"form",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoPl phi S :
reflect {in S, forall psi, '[phi, psi] = 0} (orthogonal form [:: phi] S). | Proof.
by rewrite [orthogonal form _ S]andbT /=; apply: (iffP allP) => ophiS ? /ophiS/eqP.
Qed. | Lemma | orthoPl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"allP",
"apply",
"form",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_sym : symmetric (orthogonal form). | Proof.
apply: symmetric_from_pre => R S /orthogonalP oRS.
by apply/orthogonalP=> phi psi Rpsi Sphi; rewrite hermC /= oRS ?rmorph0 ?mulr0.
Qed. | Lemma | orthogonal_sym | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"form",
"hermC",
"mulr0",
"orthogonal",
"orthogonalP",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoPr S psi :
reflect {in S, forall phi, '[phi, psi] = 0} (orthogonal form S [:: psi]). | Proof.
rewrite orthogonal_sym.
by apply: (iffP orthoPl) => oSpsi phi Sphi; rewrite hermC /= oSpsi //= conjC0 mulr0.
Qed. | Lemma | orthoPr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"conjC0",
"form",
"hermC",
"mulr0",
"orthoPl",
"orthogonal",
"orthogonal_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_catl R1 R2 S :
orthogonal form (R1 ++ R2) S = orthogonal form R1 S && orthogonal form R2 S. | Proof. exact: all_cat. Qed. | Lemma | orthogonal_catl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"R1",
"R2",
"all_cat",
"form",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_catr R S1 S2 :
orthogonal form R (S1 ++ S2) = orthogonal form R S1 && orthogonal form R S2. | Proof. by rewrite !(orthogonal_sym R) orthogonal_catl. Qed. | Lemma | orthogonal_catr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"S1",
"S2",
"form",
"orthogonal",
"orthogonal_catl",
"orthogonal_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_pairwise_orthogonal R S :
perm_eq R S -> pairwise_orthogonal form R = pairwise_orthogonal form S. | Proof.
apply: catCA_perm_subst R S => R S S'.
rewrite !pairwise_orthogonal_cat !orthogonal_catr (orthogonal_sym R S) -!andbA.
by do !bool_congr.
Qed. | Lemma | eq_pairwise_orthogonal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"catCA_perm_subst",
"form",
"orthogonal_catr",
"orthogonal_sym",
"pairwise_orthogonal",
"pairwise_orthogonal_cat",
"perm_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_orthonormal S0 S : perm_eq S0 S -> orthonormal form S0 = orthonormal form S. | Proof.
move=> eqRS; rewrite !orthonormalE (eq_all_r (perm_mem eqRS)).
by rewrite (eq_pairwise_orthogonal eqRS).
Qed. | Lemma | eq_orthonormal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"S0",
"eq_all_r",
"eq_pairwise_orthogonal",
"form",
"orthonormal",
"orthonormalE",
"perm_eq",
"perm_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_oppl S R : orthogonal form (map -%R S) R = orthogonal form S R. | Proof. by rewrite -!(orthogonal_sym R) orthogonal_oppr. Qed. | Lemma | orthogonal_oppl | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"map",
"orthogonal",
"orthogonal_oppr",
"orthogonal_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
triangle_lerif u v :
sqrtC '[u + v] <= sqrtC '[u] + sqrtC '[v]
?= iff ~~ free [:: u; v] && (0 <= coord [tuple v] 0 u). | Proof.
rewrite -(mono_in_leif ler_sqr) ?rpredD ?nnegrE ?sqrtC_ge0//.
rewrite andbC sqrrD !sqrtCK addrAC dnormD (mono_leif (lerD2l _))/=.
rewrite -mulr_natr -[_ + _](divfK (negbT (pnatr_eq0 C 2))) -/('Re _).
rewrite (mono_leif (ler_pM2r _)) ?ltr0n//.
have := leif_trans (leif_Re_Creal '[u, v]) (CauchySchwarz_sqrt u v).
r... | Lemma | triangle_lerif | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"CauchySchwarz_sqrt",
"Re",
"ReE",
"addrAC",
"apply",
"coord",
"coord0",
"coord_free",
"divfK",
"dnormD",
"dnorm_gt0",
"eqxx",
"free",
"free_cons",
"leif_Re_Creal",
"leif_trans",
"lerD2l",
"ler_pM2r",
"ler_sqr",
"linear0",
"linearZ",
"linearZl",
"ltr0n",
"mono_in_leif",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
span_orthogonal S1 S2 phi1 phi2 :
orthogonal form S1 S2 -> phi1 \in <<S1>>%VS -> phi2 \in <<S2>>%VS ->
'[phi1, phi2] = 0. | Proof.
move/orthogonalP=> oS12; do 2!move/(@coord_span _ _ _ (in_tuple _))->.
rewrite linear_sumlz big1 // => i _; rewrite linear_sumr big1 // => j _.
by rewrite linearZlr/= oS12 ?mem_nth ?mulr0.
Qed. | Lemma | span_orthogonal | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"S1",
"S2",
"big1",
"coord_span",
"form",
"in_tuple",
"linearZlr",
"linear_sumlz",
"linear_sumr",
"mem_nth",
"mulr0",
"orthogonal",
"orthogonalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_split S beta :
{X : U & X \in <<S>>%VS &
{Y :U | [/\ beta = X + Y, '[X, Y] = 0 & orthogonal form [:: Y] S]}}. | Proof.
suffices [X S_X [Y -> oYS]]:
{X : _ & X \in <<S>>%VS & {Y | beta = X + Y & orthogonal form [:: Y] S}}.
- exists X => //; exists Y.
by rewrite hermC /= (span_orthogonal oYS) ?memv_span1 ?conjC0 // mulr0.
elim: S beta => [|phi S IHS] beta.
by exists 0; last exists beta; rewrite ?mem0v ?add0r.
have [[UU S_U [... | Lemma | orthogonal_split | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"add0r",
"addKr",
"addrA",
"addrC",
"apply",
"conjC0",
"divfK",
"dnorm_eq0",
"eqVneq",
"form",
"hermC",
"inE",
"last",
"linear0r",
"linearBl",
"linearDr",
"linearZl_LR",
"mem0v",
"mem_head",
"memvB",
"memvZ",
"memv_add",
"memv_line",
"memv_span",
"memv_span1",
"mul0... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normf1 | := fun u => form1 u u. | Definition | normf1 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normf2 | := fun u => form2 u u. | Definition | normf2 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_of_dnorm S tauS :
pairwise_orthogonal form1 S -> pairwise_orthogonal form2 tauS ->
map normf2 tauS = map normf1 S ->
{tau : {linear U1 -> U2} | map tau S = tauS
& {in <<S>>%VS &, isometry form2 form1 tau}}. | Proof.
move=> oS oT eq_nST; have freeS := orthogonal_free oS.
have eq_sz: size tauS = size S by have:= congr1 size eq_nST; rewrite !size_map.
have [tau defT] := linear_of_free S tauS; rewrite -[S]/(tval (in_tuple S)).
exists tau => [|u v /coord_span-> /coord_span->]; rewrite ?raddf_sum ?defT //=.
apply: eq_bigr => i _ ... | Lemma | isometry_of_dnorm | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"coord_span",
"eqVneq",
"eq_bigr",
"freeS",
"in_tuple",
"isometry",
"linear",
"linearZ",
"linearZl",
"linear_of_free",
"linear_sumlz",
"map",
"mem_nth",
"normf1",
"normf2",
"nth_map",
"nth_uniq",
"orthogonal_free",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"ra... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_of_free S f :
free S -> {in S &, isometry form2 form1 f} ->
{tau : {linear U1 -> U2} |
{in S, tau =1 f} & {in <<S>>%VS &, isometry form2 form1 tau}}. | Proof.
move=> freeS If; have defS := free_span freeS.
have [tau /(_ freeS (size_map f S))Dtau] := linear_of_free S (map f S).
have {}Dtau: {in S, tau =1 f}.
by move=> _ /(nthP 0)[i ltiS <-]; rewrite -!(nth_map 0 0) ?Dtau.
exists tau => // _ _ /defS[a -> _] /defS[b -> _] /=.
rewrite 2!{1}linear_sum /= !{1}linear_suml... | Lemma | isometry_of_free | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"If",
"apply",
"eq_big_seq",
"free",
"freeS",
"free_span",
"isometry",
"linear",
"linearZ",
"linearZl",
"linear_of_free",
"linear_sum",
"linear_sumlz",
"linear_sumr",
"map",
"nthP",
"nth_map",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_raddf_inj (tau : {additive U1 -> U2}) :
{in U1 &, isometry form2 form1 tau} ->
{in U1 &, forall u v, u - v \in U1} ->
{in U1 &, injective tau}. | Proof.
move=> Itau linU phi psi Uphi Upsi /eqP; rewrite -subr_eq0 -raddfB.
by rewrite -(dnorm_eq0 form2) Itau ?linU // dnorm_eq0 subr_eq0 => /eqP.
Qed. | Lemma | isometry_raddf_inj | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"additive",
"dnorm_eq0",
"isometry",
"raddfB",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_of_form (form : 'rV[R]_n -> 'rV[R]_n -> R) : 'M[R]_n | :=
\matrix_(i, j) form 'e_i 'e_j. | Definition | matrix_of_form | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_of_formE form i j : matrix_of_form form i j = form 'e_i 'e_j. | Proof. by rewrite mxE. Qed. | Lemma | matrix_of_formE | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form",
"matrix_of_form",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' U , V ]" | := (form_of_matrix theta M U%R V%R) : ring_scope. | Notation | ''[' U , V ] | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' U ]" | := '[U, U]%R : ring_scope. | Notation | ''[' U ] | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form_of_matrix_is_linear U :
linear_for (theta \; *%R) (form_of_matrix theta M U). | Proof.
rewrite unlock => k v w; rewrite -linearP/=.
by rewrite linearP map_mxD map_mxZ !mulmxDr !scalemxAr.
Qed. | Let | form_of_matrix_is_linear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"linearP",
"linear_for",
"map_mxD",
"map_mxZ",
"mulmxDr",
"scalemxAr",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form_of_matrixr U | := (form_of_matrix theta M)^~U. | Definition | form_of_matrixr | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form_of_matrixr_is_linear U : linear_for *%R (form_of_matrixr U). | Proof.
rewrite /form_of_matrixr unlock => k v w.
by rewrite -linearP /= !mulmxDl -!scalemxAl.
Qed. | Let | form_of_matrixr_is_linear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"form_of_matrixr",
"linearP",
"linear_for",
"mulmxDl",
"scalemxAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form_of_matrix_is_bilinear :
bilinear_for
(GRing.Scale.Law.clone _ _ ( *%R ) _) (GRing.Scale.Law.clone _ _ (theta \; *%R ) _)
(@form_of_matrix _ _ theta m M). | Proof.
split=> [u'|u] a x y /=.
- by rewrite unlock !mulmxDl linearD/= -!scalemxAl linearZ.
- rewrite unlock -linearZ/= -linearD/= [in LHS]linearD/= map_mxD.
rewrite mulmxDr; congr (\tr (_ + _)).
rewrite scalemxAr; congr (_ *m _).
by rewrite linearZ/= map_mxZ.
Qed. | Lemma | form_of_matrix_is_bilinear | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"bilinear_for",
"clone",
"linearD",
"linearZ",
"map_mxD",
"map_mxZ",
"mulmxDl",
"mulmxDr",
"scalemxAl",
"scalemxAr",
"split",
"theta"
] | TODO
Canonical form_of_matrixr_rev :=
[revop form_of_matrixr of form_of_matrix theta M]. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"''[' u , v ]" | := (form_of_matrix theta M u%R v%R) : ring_scope. | Notation | ''[' u , v ] | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV_formee i j : '['e_i :'rV__, 'e_j] = M i j. | Proof.
rewrite unlock -rowE -map_trmx map_delta_mx -[M in LHS]trmxK.
by rewrite -tr_col -trmx_mul -rowE trace_mx11 !mxE.
Qed. | Lemma | rV_formee | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_delta_mx",
"map_trmx",
"mxE",
"rowE",
"tr_col",
"trace_mx11",
"trmxK",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form_of_matrixK : matrix_of_form (form_of_matrix theta M) = M. | Proof. by apply/matrixP => i j; rewrite !mxE rV_formee. Qed. | Lemma | form_of_matrixK | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"matrixP",
"matrix_of_form",
"mxE",
"rV_formee",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV_form0_eq0 : M = 0 -> forall u v, '[u, v] = 0. | Proof.
by rewrite unlock => -> u v; rewrite mulmx0 mul0mx trace_mx11 mxE.
Qed. | Lemma | rV_form0_eq0 | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"mul0mx",
"mulmx0",
"mxE",
"trace_mx11"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_of_formK : form_of_matrix theta (matrix_of_form form) =2 form. | Proof.
set f := (X in X =2 _); have f_eq i j : f 'e_i 'e_j = form 'e_i 'e_j.
by rewrite /f rV_formee mxE.
move=> u v; rewrite [u]row_sum_delta [v]row_sum_delta /f.
rewrite !linear_sum/=; apply: eq_bigr => j _.
rewrite !linear_sumlz/=; apply: eq_bigr => i _.
by rewrite !linearZlr/= -f_eq.
Qed. | Lemma | matrix_of_formK | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"eq_bigr",
"form",
"linearZlr",
"linear_sum",
"linear_sumlz",
"matrix_of_form",
"mxE",
"rV_formee",
"row_sum_delta",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitianmx | :=
[qualify M : 'M_n | M == ((-1) ^+ eps) *: M ^t theta]. | Definition | hermitianmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitianmx_key : pred_key hermitianmx. | Proof. by []. Qed. | Fact | hermitianmx_key | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitianmx_keyed | := KeyedQualifier hermitianmx_key. | Canonical | hermitianmx_keyed | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitian_matrix | := HermitianMx {
mx_of_hermitian :> 'M[R]_n;
_ : mx_of_hermitian \is hermitianmx }. | Structure | hermitian_matrix | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_hermitianmxE M :
(M \is hermitianmx) = (M == (-1) ^+ eps *: M ^t theta). | Proof. by rewrite qualifE. Qed. | Lemma | is_hermitianmxE | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmx",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_hermitianmxP M :
reflect (M = (-1) ^+ eps *: M ^t theta) (M \is hermitianmx). | Proof. by rewrite is_hermitianmxE; apply/eqP. Qed. | Lemma | is_hermitianmxP | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"hermitianmx",
"is_hermitianmxE",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitianmxE (M : hermitian_matrix) :
M = ((-1) ^+ eps) *: M ^t theta :> 'M__. | Proof. by apply/eqP; case: M. Qed. | Lemma | hermitianmxE | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"hermitian_matrix",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_hermitian (M : hermitian_matrix) :
M^T = ((-1) ^+ eps) *: M ^ theta :> 'M__. | Proof. by rewrite {1}hermitianmxE linearZ /= map_trmx trmxK. Qed. | Lemma | trmx_hermitian | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitian_matrix",
"hermitianmxE",
"linearZ",
"map_trmx",
"theta",
"trmxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maptrmx_hermitian : M^t theta = (-1) ^+ eps *: (M : 'M__). | Proof.
rewrite trmx_hermitian map_mxZ rmorph_sign -map_mx_comp.
by rewrite (map_mx_id (rmorphK _)).
Qed. | Lemma | maptrmx_hermitian | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"map_mxZ",
"map_mx_comp",
"map_mx_id",
"rmorphK",
"rmorph_sign",
"theta",
"trmx_hermitian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form_of_matrix_is_hermitian m x y :
(@form_of_matrix _ _ theta m M) x y =
(-1) ^+ eps * theta ((@form_of_matrix _ _ theta m M) y x). | Proof.
rewrite {1}hermitianmxE unlock.
rewrite -!(scalemxAr, scalemxAl) linearZ/=; congr (_ * _).
rewrite -mxtrace_tr -trace_map_mx !(trmx_mul, map_mxM, map_trmx, trmxK).
by rewrite -mulmxA -!map_mx_comp !(map_mx_id (rmorphK _)).
Qed. | Lemma | form_of_matrix_is_hermitian | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmxE",
"linearZ",
"map_mxM",
"map_mx_comp",
"map_mx_id",
"map_trmx",
"mulmxA",
"mxtrace_tr",
"rmorphK",
"scalemxAl",
"scalemxAr",
"theta",
"trace_map_mx",
"trmxK",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"B ^!" | := (orthomx theta M B) : matrix_set_scope. | Notation | B ^! | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"orthomx",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A '_|_ B" | := (A%MS <= B%MS^!)%MS : matrix_set_scope. | Notation | A '_|_ B | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomxE u v : (u '_|_ v)%MS = ('[u, v] == 0). | Proof.
rewrite (sameP sub_kermxP eqP) mulmxA.
by rewrite [_ *m _^t _]mx11_scalar -trace_mx11 fmorph_eq0 unlock.
Qed. | Lemma | orthomxE | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"fmorph_eq0",
"mulmxA",
"mx11_scalar",
"sub_kermxP",
"trace_mx11"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermmx_eq0P {u v} : reflect ('[u, v] = 0) (u '_|_ v)%MS. | Proof. by rewrite orthomxE; apply/eqP. Qed. | Lemma | hermmx_eq0P | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"orthomxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomxP p q (A : 'M_(p, n)) (B :'M_(q, n)) :
reflect (forall (u v : 'rV_n), u <= A -> v <= B -> u '_|_ v)%MS
(A '_|_ B)%MS. | Proof.
apply: (iffP idP) => AnB.
move=> u v uA vB; rewrite (submx_trans uA) // (submx_trans AnB) //.
apply/sub_kermxP; have /submxP [w ->] := vB.
rewrite trmx_mul map_mxM !mulmxA -[kermx _ *m _ *m _]mulmxA.
by rewrite [kermx _ *m _](sub_kermxP _) // mul0mx.
apply/rV_subP => u /AnB /(_ _) /sub_kermxP uMv; apply/... | Lemma | orthomxP | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"kermx",
"map_delta_mx",
"map_mxM",
"map_trmx",
"mul0mx",
"mulmxA",
"rV_subP",
"row0",
"rowE",
"row_matrixP",
"sub_kermxP",
"submxP",
"submx_trans",
"theta",
"trmx0",
"trmxK",
"trmx_mul",
"vB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx_sym p q (A : 'M_(p, n)) (B :'M_(q, n)) :
(A '_|_ B)%MS = (B '_|_ A)%MS. | Proof.
gen have nC : p q A B / (A '_|_ B -> B '_|_ A)%MS; last by apply/idP/idP; apply/nC.
move=> AnB; apply/orthomxP => u v ? ?; rewrite orthomxE.
rewrite hermC mulf_eq0 ?fmorph_eq0 ?signr_eq0 /=.
by rewrite -orthomxE (orthomxP _ _ AnB).
Qed. | Lemma | orthomx_sym | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"fmorph_eq0",
"gen",
"hermC",
"last",
"mulf_eq0",
"orthomxE",
"orthomxP",
"signr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ortho_ortho_mx p (A : 'M_(p, n)) : (A^! '_|_ A)%MS. | Proof. by []. Qed. | Lemma | ortho_ortho_mx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ortho_mx_ortho p (A : 'M_(p, n)) : (A '_|_ A^!)%MS. | Proof. by rewrite orthomx_sym. Qed. | Lemma | ortho_mx_ortho | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"orthomx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_orthomx u : (\rank (u ^!) >= n.-1)%N. | Proof.
rewrite mxrank_ker -subn1 leq_sub2l //.
by rewrite (leq_trans (mxrankM_maxr _ _)) // rank_leq_col.
Qed. | Lemma | rank_orthomx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"leq_sub2l",
"leq_trans",
"mxrankM_maxr",
"mxrank_ker",
"rank",
"rank_leq_col",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
radmx | := (1%:M^!)%MS. | Notation | radmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
radmxE : radmx = kermx M. | Proof. by rewrite /orthomx /orthomx trmx1 map_mx1 mulmx1. Qed. | Lemma | radmxE | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"kermx",
"map_mx1",
"mulmx1",
"orthomx",
"radmx",
"trmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoNmx k m (A : 'M[R]_(k, n)) (B : 'M[R]_(m, n)) :
((- A) '_|_ B)%MS = (A '_|_ B)%MS. | Proof. by rewrite eqmx_opp. Qed. | Lemma | orthoNmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"eqmx_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomxN k m (A : 'M[R]_(k, n)) (B : 'M[R]_(m, n)) :
(A '_|_ (- B))%MS = (A '_|_ B)%MS. | Proof. by rewrite ![(A '_|_ _)%MS]orthomx_sym orthoNmx. Qed. | Lemma | orthomxN | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"orthoNmx",
"orthomx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoDmx k m p (A : 'M[R]_(k, n)) (B : 'M[R]_(m, n)) (C : 'M[R]_(p, n)) :
(A + B '_|_ C)%MS = (A '_|_ C)%MS && (B '_|_ C)%MS. | Proof. by rewrite addsmxE !(sameP sub_kermxP eqP) mul_col_mx col_mx_eq0. Qed. | Lemma | orthoDmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"addsmxE",
"col_mx_eq0",
"mul_col_mx",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomxD k m p (A : 'M[R]_(k, n)) (B : 'M[R]_(m, n)) (C : 'M[R]_(p, n)) :
(A '_|_ B + C)%MS = (A '_|_ B)%MS && (A '_|_ C)%MS. | Proof. by rewrite ![(A '_|_ _)%MS]orthomx_sym orthoDmx. Qed. | Lemma | orthomxD | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"orthoDmx",
"orthomx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoZmx p m a (A : 'M[R]_(p, n)) (B : 'M[R]_(m, n)) : a != 0 ->
(a *: A '_|_ B)%MS = (A '_|_ B)%MS. | Proof. by move=> a_neq0; rewrite eqmx_scale. Qed. | Lemma | orthoZmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"eqmx_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomxZ p m a (A : 'M[R]_(p, n)) (B : 'M[R]_(m, n)) : a != 0 ->
(A '_|_ (a *: B))%MS = (A '_|_ B)%MS. | Proof. by move=> a_neq0; rewrite ![(A '_|_ _)%MS]orthomx_sym orthoZmx. Qed. | Lemma | orthomxZ | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"orthoZmx",
"orthomx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_ortho p m (A : 'M[R]_(p, n)) (B : 'M[R]_(m, n)) :
(A :=: B)%MS -> (A^! :=: B^!)%MS. | Proof.
move=> eqAB; apply/eqmxP.
by rewrite orthomx_sym -eqAB ortho_mx_ortho orthomx_sym eqAB ortho_mx_ortho.
Qed. | Lemma | eqmx_ortho | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"apply",
"eqmxP",
"ortho_mx_ortho",
"orthomx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_ortho p (A : 'M[R]_(p, n)) : (<<A>>^! :=: A^!)%MS. | Proof. exact: (eqmx_ortho (genmxE _)). Qed. | Lemma | genmx_ortho | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"eqmx_ortho",
"genmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
symmetricmx | := (hermitianmx _ false idfun). | Notation | symmetricmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
skewmx | := (hermitianmx _ true idfun). | Notation | skewmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitianmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermsymmx | := (hermitianmx _ false conjC). | Notation | hermsymmx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"conjC",
"hermitianmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitian1mx_subproof {C : numClosedFieldType} n : (1%:M : 'M[C]_n) \is hermsymmx. | Proof.
by rewrite qualifE /= expr0 scale1r tr_scalar_mx map_scalar_mx/= conjC1.
Qed. | Lemma | hermitian1mx_subproof | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"conjC1",
"expr0",
"hermsymmx",
"map_scalar_mx",
"scale1r",
"tr_scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitian1mx {C : numClosedFieldType} n | :=
HermitianMx (@hermitian1mx_subproof C n). | Canonical | hermitian1mx | algebra | algebra/sesquilinear.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"orderedzmod",
"numdomain",
"numfield",
"matrix",
"mxalgebra",
"vector",
"GRin... | [
"hermitian1mx_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue_closed {C : numClosedFieldType} n (A : 'M[C]_n) : (n > 0)%N ->
exists a, eigenvalue A a. | Proof.
move=> n_gt0; have /closed_rootP [a rAa] : size (char_poly A) != 1%N.
by rewrite size_char_poly; case: (n) n_gt0.
by exists a; rewrite eigenvalue_root_char.
Qed. | Lemma | eigenvalue_closed | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"char_poly",
"closed_rootP",
"eigenvalue",
"eigenvalue_root_char",
"n_gt0",
"size",
"size_char_poly"
] | TODO: move? * | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
common_eigenvector {C : numClosedFieldType} n (As : seq 'M[C]_n) :
(n > 0)%N -> {in As &, forall A B, comm_mx A B} ->
exists2 v : 'rV_n, v != 0 & all (fun A => stablemx v A) As. | Proof.
move=> n_gt0 /all_comm_mxP; have [k sAsk] := ubnP (size As).
elim: k n n_gt0 As sAsk => [//|k IHk] n n_gt0 [|A As].
exists (const_mx 1) => //; apply/negP => /eqP/rowP/(_ (Ordinal n_gt0)).
by rewrite !mxE => /eqP; rewrite oner_eq0.
rewrite ltnS all_comm_mx_cons => sAsk /andP[].
move=> /allP/(_ _ _)/eqP/= A_c... | Lemma | common_eigenvector | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"all",
"allP",
"all_comm_mxP",
"all_comm_mx_cons",
"apply",
"comm_mx",
"comm_mx_stable_eigenspace",
"conjmxM",
"const_mx",
"eigenspace",
"eigenvalue_closed",
"eigenvectorP",
"eq_row_base",
"inE",
"last",
"lt0n",
"ltnS",
"mapP",
"mulmx_free_eq0",
"mulmx_sub",
"mxE",
"mxrank_... | TODO: move? * | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
common_eigenvector2 {C : numClosedFieldType}n (A B : 'M[C]_n) :
(n > 0)%N -> A *m B = B *m A ->
exists2 v : 'rV_n, v != 0 & (stablemx v A) && (stablemx v B). | Proof.
move=> n_gt0 AB_comm; have [] := @common_eigenvector _ _ [:: A; B] n_gt0.
by move=> A' B'; rewrite !inE => /orP [] /eqP-> /orP [] /eqP->.
by move=> v v_neq0 /allP vP; exists v; rewrite ?vP ?(mem_head, in_cons, orbT).
Qed. | Lemma | common_eigenvector2 | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"A'",
"allP",
"common_eigenvector",
"inE",
"in_cons",
"mem_head",
"n_gt0",
"stablemx"
] | TODO: move? * | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"M ^t*" | := (M ^t conjC) (at level 29, left associativity)
: sesquilinear_scope. | Notation | M ^t* | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realmx | := (mxOver Num.real). | Notation | realmx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"mxOver",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmxCK {C : numClosedFieldType} m n (A : 'M[C]_(m, n)) : A ^t* ^t* = A. | Proof. by apply/matrixP=> i j; rewrite !mxE conjCK. Qed. | Lemma | trmxCK | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"conjCK",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realmxC A : A \is a realmx -> A ^ conjC = A. | Proof.
by move=> ?; apply/matrixP => x y; rewrite mxE; exact/CrealP/mxOverP.
Qed. | Lemma | realmxC | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"CrealP",
"apply",
"conjC",
"matrixP",
"mxE",
"mxOverP",
"realmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realmxD A B : A \is a realmx -> B \is a realmx -> A + B \is a realmx. | Proof.
rewrite !qualifE/= => /'forall_forallP realA /'forall_forallP realB.
by apply/'forall_forallP => i j; rewrite mxE realD.
Qed. | Lemma | realmxD | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"mxE",
"realB",
"realD",
"realmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.