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 |
|---|---|---|---|---|---|---|---|---|---|---|
can2_semilinear f' : cancel f f' -> cancel f' f -> semilinear f'. | Proof.
by move=> fK f'K; split=> ? ?; apply: (canLR fK); rewrite semilinearPZ !f'K.
Qed. | Lemma | can2_semilinear | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"apply",
"fK",
"semilinear",
"semilinearPZ",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can2_linear f' : cancel f f' -> cancel f' f -> linear f'. | Proof. by move=> fK f'K a x y /=; apply: (canLR fK); rewrite linearP !f'K. Qed. | Lemma | can2_linear | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"apply",
"fK",
"linear",
"linearP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalarZ : scalable_for *%R f. | Proof. exact: linearZ_LR. Qed. | Lemma | scalarZ | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ_LR",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semiscalarP : semiscalar f. | Proof. exact: semilinearP. Qed. | Lemma | semiscalarP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"semilinearP",
"semiscalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalarP : scalar f. | Proof. exact: linearP. Qed. | Lemma | scalarP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearP",
"scalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
idfun_is_scalable : scalable (@idfun U). | Proof. by []. Qed. | Lemma | idfun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_is_scalable : scalable_for s (f \o g). | Proof. by move=> a v /=; rewrite !linearZ_LR. Qed. | Lemma | comp_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ_LR",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
null_fun_is_scalable : scalable_for s (\0 : U -> V). | Proof. by move=> a v /=; rewrite raddf0. Qed. | Lemma | null_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"raddf0",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_fun_is_scalable : scalable_for s (add_fun f g). | Proof. by move=> a u; rewrite /= !linearZ_LR raddfD. Qed. | Lemma | add_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"add_fun",
"linearZ_LR",
"raddfD",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_is_scalable : scalable (-%R : U -> U). | Proof. by move=> a v /=; rewrite scalerN. Qed. | Lemma | opp_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"scalable",
"scalerN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_fun_is_scalable : scalable_for s (f \- g). | Proof. by move=> a u; rewrite /= !linearZ_LR raddfB. Qed. | Lemma | sub_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ_LR",
"raddfB",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_fun_is_scalable : scalable_for s (\- f). | Proof. by move=> a u; rewrite /= linearZ_LR raddfN. Qed. | Lemma | opp_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ_LR",
"raddfN",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_fun_is_scalable : scalable (a \o* f). | Proof. by move=> k x /=; rewrite linearZ scalerAl. Qed. | Fact | mulr_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ",
"scalable",
"scalerAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'lrmorphism' A -> B | s }" | := (@LRMorphism.type _ A%type B%type s)
: type_scope. | Notation | { 'lrmorphism' A -> B | s } | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'lrmorphism' A -> B }" | := {lrmorphism A%type -> B%type | *:%R}
: type_scope. | Notation | { 'lrmorphism' A -> B } | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rmorph_alg a : f a%:A = a%:A. | Proof. by rewrite linearZ /= rmorph1. Qed. | Lemma | rmorph_alg | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr1 | := Monoid.mulC_id mulrC mul1r. | Definition | mulr1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mul1r",
"mulC_id",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrDr | := Monoid.mulC_dist mulrC mulrDl. | Definition | mulrDr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulC_dist",
"mulrC",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr0 : right_zero zero mul. | Proof. by move=> x; rewrite mulrC mul0r. Qed. | Lemma | mulr0 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mul",
"mul0r",
"mulrC",
"zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrCA : @left_commutative R R *%R. | Proof. exact: mulmCA. Qed. | Lemma | mulrCA | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulmCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrAC : @right_commutative R R *%R. | Proof. exact: mulmAC. Qed. | Lemma | mulrAC | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulmAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrACA : @interchange R *%R *%R. | Proof. exact: mulmACA. Qed. | Lemma | mulrACA | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulmACA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprMn n : {morph (fun x => x ^+ n) : x y / x * y}. | Proof. by move=> x y; exact/exprMn_comm/mulrC. Qed. | Lemma | exprMn | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"exprMn_comm",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodrXl n I r (P : pred I) (F : I -> R) :
\prod_(i <- r | P i) F i ^+ n = (\prod_(i <- r | P i) F i) ^+ n. | Proof. by rewrite (big_morph _ (exprMn n) (expr1n _ n)). Qed. | Lemma | prodrXl | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"big_morph",
"expr1n",
"exprMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodr_undup_exp_count (I : eqType) r (P : pred I) (F : I -> R) :
\prod_(i <- undup r | P i) F i ^+ count_mem i r = \prod_(i <- r | P i) F i. | Proof. exact: big_undup_iterop_count. Qed. | Lemma | prodr_undup_exp_count | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"big_undup_iterop_count",
"count_mem",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodrMl {I : finType} (A : pred I) (x : R) F :
\prod_(i in A) (x * F i) = x ^+ #|A| * \prod_(i in A) F i. | Proof. by rewrite big_split ?prodr_const. Qed. | Lemma | prodrMl | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"big_split",
"prodr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodrMr {I : finType} (A : pred I) (x : R) F :
\prod_(i in A) (F i * x) = \prod_(i in A) F i * x ^+ #|A|. | Proof. by rewrite big_split ?prodr_const. Qed. | Lemma | prodrMr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"big_split",
"prodr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprDn x y n :
(x + y) ^+ n = \sum_(i < n.+1) (x ^+ (n - i) * y ^+ i) *+ 'C(n, i). | Proof. by rewrite exprDn_comm //; apply: mulrC. Qed. | Lemma | exprDn | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"apply",
"exprDn_comm",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrrD x y : (x + y) ^+ 2 = x ^+ 2 + x * y *+ 2 + y ^+ 2. | Proof. by rewrite exprDn !big_ord_recr big_ord0 /= add0r mulr1 mul1r. Qed. | Lemma | sqrrD | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"add0r",
"big_ord0",
"big_ord_recr",
"exprDn",
"mul1r",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rmorph_comm (S : pzSemiRingType) (f : {rmorphism R -> S}) x y :
comm (f x) (f y). | Proof. by red; rewrite -!rmorphM mulrC. Qed. | Lemma | rmorph_comm | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"comm",
"mulrC",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_is_scalable : scalable ( *:%R b : V -> V). | Proof. by move=> a v /=; rewrite !scalerA mulrC. Qed. | Lemma | scale_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulrC",
"scalable",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_fun_is_scalable : scalable (b \*: f). | Proof. by move=> a v /=; rewrite !linearZ. Qed. | Lemma | scale_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pFrobenius_aut_is_nmod_morphism : nmod_morphism (pFrobenius_aut pcharRp). | Proof.
by split=> [|x y]; [exact: pFrobenius_aut0 | exact/pFrobenius_autD_comm/mulrC].
Qed. | Lemma | pFrobenius_aut_is_nmod_morphism | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulrC",
"nmod_morphism",
"pFrobenius_aut",
"pFrobenius_aut0",
"pFrobenius_autD_comm",
"pcharRp",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pFrobenius_aut_is_monoid_morphism : monoid_morphism (pFrobenius_aut pcharRp). | Proof.
by split=> [|x y]; [exact: pFrobenius_aut1 | exact/pFrobenius_autM_comm/mulrC].
Qed. | Lemma | pFrobenius_aut_is_monoid_morphism | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"monoid_morphism",
"mulrC",
"pFrobenius_aut",
"pFrobenius_aut1",
"pFrobenius_autM_comm",
"pcharRp",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprDn_pchar x y n : (pchar R).-nat n -> (x + y) ^+ n = x ^+ n + y ^+ n. | Proof.
pose p := pdiv n; have [|n_gt1 pcharRn] := leqP n 1; first by case: (n) => [|[]].
have pcharRp: p \in pchar R by rewrite (pnatPpi pcharRn) ?pi_pdiv.
have{pcharRn} /p_natP[e ->]: p.-nat n by rewrite -(eq_pnat _ (pcharf_eq pcharRp)).
by elim: e => // e IHe; rewrite !expnSr !exprM IHe -pFrobenius_autE rmorphD.
Qed. | Lemma | exprDn_pchar | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"eq_pnat",
"expnSr",
"exprM",
"leqP",
"nat",
"pFrobenius_autE",
"p_natP",
"pchar",
"pcharRp",
"pcharf_eq",
"pdiv",
"pi_pdiv",
"pnatPpi",
"rmorphD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprBn x y n :
(x - y) ^+ n =
\sum_(i < n.+1) ((-1) ^+ i * x ^+ (n - i) * y ^+ i) *+ 'C(n, i). | Proof. by rewrite exprBn_comm //; apply: mulrC. Qed. | Lemma | exprBn | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"apply",
"exprBn_comm",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subrXX x y n :
x ^+ n - y ^+ n = (x - y) * (\sum_(i < n) x ^+ (n.-1 - i) * y ^+ i). | Proof. by rewrite -subrXX_comm //; apply: mulrC. Qed. | Lemma | subrXX | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"apply",
"mulrC",
"subrXX_comm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrrB x y : (x - y) ^+ 2 = x ^+ 2 - x * y *+ 2 + y ^+ 2. | Proof. by rewrite sqrrD mulrN mulNrn sqrrN. Qed. | Lemma | sqrrB | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulNrn",
"mulrN",
"sqrrD",
"sqrrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_sqr x y : x ^+ 2 - y ^+ 2 = (x - y) * (x + y). | Proof. by rewrite subrXX !big_ord_recr big_ord0 /= add0r mulr1 mul1r. Qed. | Lemma | subr_sqr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"add0r",
"big_ord0",
"big_ord_recr",
"mul1r",
"mulr1",
"subrXX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_sqrDB x y : (x + y) ^+ 2 - (x - y) ^+ 2 = x * y *+ 4. | Proof.
rewrite sqrrD sqrrB -!(addrAC _ (y ^+ 2)) opprB.
by rewrite [LHS]addrC addrA subrK -mulrnDr.
Qed. | Lemma | subr_sqrDB | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"addrA",
"addrAC",
"addrC",
"mulrnDr",
"opprB",
"sqrrB",
"sqrrD",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalarAr k (x y : V) : k *: (x * y) = x * (k *: y). | Proof. by rewrite mulrC scalerAl mulrC. Qed. | Lemma | scalarAr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulrC",
"scalerAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalerCA k x y : k *: x * y = x * (k *: y). | Proof. by rewrite -scalerAl scalerAr. Qed. | Lemma | scalerCA | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"scalerAl",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_algr a x : x * a%:A = a *: x. | Proof. by rewrite -scalerAr mulr1. Qed. | Lemma | mulr_algr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulr1",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_alg a x : comm a%:A x. | Proof. by rewrite /comm mulr_algr mulr_algl. Qed. | Lemma | comm_alg | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"comm",
"mulr_algl",
"mulr_algr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprZn k x n : (k *: x) ^+ n = k ^+ n *: x ^+ n. | Proof.
elim: n => [|n IHn]; first by rewrite !expr0 scale1r.
by rewrite !exprS IHn -scalerA scalerAr scalerAl.
Qed. | Lemma | exprZn | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"expr0",
"exprS",
"scale1r",
"scalerA",
"scalerAl",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scaler_prod I r (P : pred I) (F : I -> R) (G : I -> A) :
\prod_(i <- r | P i) (F i *: G i) =
\prod_(i <- r | P i) F i *: \prod_(i <- r | P i) G i. | Proof.
elim/big_rec3: _ => [|i x a _ _ ->]; first by rewrite scale1r.
by rewrite -scalerAl -scalerAr scalerA.
Qed. | Lemma | scaler_prod | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"big_rec3",
"scale1r",
"scalerA",
"scalerAl",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scaler_prodl (I : finType) (S : pred I) (F : I -> A) k :
\prod_(i in S) (k *: F i) = k ^+ #|S| *: \prod_(i in S) F i. | Proof. by rewrite scaler_prod prodr_const. Qed. | Lemma | scaler_prodl | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"prodr_const",
"scaler_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scaler_prodr (I : finType) (S : pred I) (F : I -> R) x :
\prod_(i in S) (F i *: x) = \prod_(i in S) F i *: x ^+ #|S|. | Proof. by rewrite scaler_prod prodr_const. Qed. | Lemma | scaler_prodr | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"prodr_const",
"scaler_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mull_fun_is_scalable : scalable (a \*o f). | Proof. by move=> k x /=; rewrite linearZ scalerAr. Qed. | Lemma | mull_fun_is_scalable | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linearZ",
"scalable",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmod_closed | := nmod_closed. | Notation | nmod_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_closed | := oppr_closed. | Notation | oppr_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_closed | := zmod_closed. | Notation | zmod_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_closed | := mulr_closed. | Notation | mulr_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semiring_closed | := semiring_closed. | Notation | semiring_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
smulr_closed | := smulr_closed. | Notation | smulr_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subring_closed | := subring_closed. | Notation | subring_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scaler_closed | := scaler_closed. | Notation | scaler_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subsemimod_closed | := subsemimod_closed. | Notation | subsemimod_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_closed | := linear_closed. | Notation | linear_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submod_closed | := submod_closed. | Notation | submod_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subalg_closed | := subalg_closed. | Notation | subalg_closed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_closed0D : zmod_closed >-> nmod_closed. | Coercion | zmod_closed0D | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"nmod_closed",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
zmod_closedN : zmod_closed >-> oppr_closed. | Coercion | zmod_closedN | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"oppr_closed",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
semiring_closedD : semiring_closed >-> nmod_closed. | Coercion | semiring_closedD | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"nmod_closed",
"semiring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
semiring_closedM : semiring_closed >-> mulr_closed. | Coercion | semiring_closedM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulr_closed",
"semiring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
smulr_closedM : smulr_closed >-> mulr_closed. | Coercion | smulr_closedM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulr_closed",
"smulr_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
smulr_closedN : smulr_closed >-> oppr_closed. | Coercion | smulr_closedN | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"oppr_closed",
"smulr_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subring_closedB : subring_closed >-> zmod_closed. | Coercion | subring_closedB | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"subring_closed",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subring_closedM : subring_closed >-> smulr_closed. | Coercion | subring_closedM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"smulr_closed",
"subring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subring_closed_semi : subring_closed >-> semiring_closed. | Coercion | subring_closed_semi | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"semiring_closed",
"subring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subsemimod_closedD : subsemimod_closed >-> nmod_closed. | Coercion | subsemimod_closedD | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"nmod_closed",
"subsemimod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subsemimod_closedZ : subsemimod_closed >-> scaler_closed. | Coercion | subsemimod_closedZ | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"scaler_closed",
"subsemimod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
linear_closedB : linear_closed >-> subr_closed. | Coercion | linear_closedB | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"linear_closed",
"subr_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
submod_closedB : submod_closed >-> zmod_closed. | Coercion | submod_closedB | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"submod_closed",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
submod_closed_semi : submod_closed >-> subsemimod_closed. | Coercion | submod_closed_semi | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"submod_closed",
"subsemimod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subsemialg_closedZ : subsemialg_closed >-> subsemimod_closed. | Coercion | subsemialg_closedZ | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"subsemialg_closed",
"subsemimod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subsemialg_closedM : subsemialg_closed >-> semiring_closed. | Coercion | subsemialg_closedM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"semiring_closed",
"subsemialg_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subalg_closedZ : subalg_closed >-> submod_closed. | Coercion | subalg_closedZ | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"subalg_closed",
"submod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subalg_closedBM : subalg_closed >-> subring_closed. | Coercion | subalg_closedBM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"subalg_closed",
"subring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subalg_closed_semi : subalg_closed >-> subsemialg_closed. | Coercion | subalg_closed_semi | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"subalg_closed",
"subsemialg_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
addrClosed | := addrClosed. | Notation | addrClosed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | Structures for stability properties | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
opprClosed | := opprClosed. | Notation | opprClosed | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred1M : mulr_closed S. | Proof. exact: (conj rpred1 rpredM). Qed. | Lemma | rpred1M | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"conj",
"mulr_closed",
"rpred1",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_prod I r (P : pred I) F :
(forall i, P i -> F i \in S) -> \prod_(i <- r | P i) F i \in S. | Proof. by move=> IH; elim/big_ind: _; [apply: rpred1 | apply: rpredM |]. Qed. | Lemma | rpred_prod | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"apply",
"big_ind",
"rpred1",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredX n : {in S, forall u, u ^+ n \in S}. | Proof. by move=> u Su; rewrite -(card_ord n) -prodr_const rpred_prod. Qed. | Lemma | rpredX | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"card_ord",
"prodr_const",
"rpred_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_nat (S : semiringClosed R) n : n%:R \in S. | Proof. by rewrite rpredMn ?rpred1. Qed. | Lemma | rpred_nat | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpred1",
"rpredMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semiringClosedP (rngS : semiringClosed R) : semiring_closed rngS. | Proof. split; [ exact: rpred0D | exact: rpred1M ]. Qed. | Lemma | semiringClosedP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpred0D",
"rpred1M",
"semiring_closed",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredMsign (S : opprClosed R) n x : ((-1) ^+ n * x \in S) = (x \in S). | Proof. by rewrite -signr_odd mulr_sign; case: ifP => // _; rewrite rpredN. Qed. | Lemma | rpredMsign | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"mulr_sign",
"opprClosed",
"rpredN",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredN1 (S : smulClosed R) : -1 \in S. | Proof. by rewrite rpredN rpred1. Qed. | Lemma | rpredN1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpred1",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_sign (S : smulClosed R) n : (-1) ^+ n \in S. | Proof. by rewrite rpredX ?rpredN1. Qed. | Lemma | rpred_sign | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpredN1",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subringClosedP (rngS : subringClosed R) : subring_closed rngS. | Proof. split; [ exact: rpred1 | exact: rpredB | exact: rpredM ]. Qed. | Lemma | subringClosedP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpred1",
"rpredB",
"rpredM",
"split",
"subring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredZnat (S : addrClosed V) n : {in S, forall u, n%:R *: u \in S}. | Proof. by move=> u Su; rewrite /= scaler_nat rpredMn. Qed. | Lemma | rpredZnat | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"addrClosed",
"rpredMn",
"scaler_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subsemimodClosedP (modS : submodClosed V) : subsemimod_closed modS. | Proof. by split; [exact: rpred0D | exact: rpredZ]. Qed. | Lemma | subsemimodClosedP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpred0D",
"rpredZ",
"split",
"subsemimod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredZsign (S : opprClosed V) n u : ((-1) ^+ n *: u \in S) = (u \in S). | Proof. by rewrite -signr_odd scaler_sign fun_if if_arg rpredN if_same. Qed. | Lemma | rpredZsign | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"opprClosed",
"rpredN",
"scaler_sign",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submodClosedP (modS : submodClosed V) : submod_closed modS. | Proof. exact/subsemimod_closed_submod/subsemimodClosedP. Qed. | Lemma | submodClosedP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"submod_closed",
"subsemimodClosedP",
"subsemimod_closed_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subsemialgClosedP (algS : subalgClosed A) : subsemialg_closed algS. | Proof.
split; [ exact: rpred1 | exact: rpred0D | exact: rpredZ | exact: rpredM ].
Qed. | Lemma | subsemialgClosedP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rpred0D",
"rpred1",
"rpredM",
"rpredZ",
"split",
"subsemialg_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subalgClosedP (algS : subalgClosed A) : subalg_closed algS. | Proof. exact/subsemialg_closed_subalg/subsemialgClosedP. Qed. | Lemma | subalgClosedP | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"subalg_closed",
"subsemialgClosedP",
"subsemialg_closed_subalg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val | := (val : U -> R). | Notation | val | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val1 : val 1 = 1. | Proof. exact: rmorph1. Qed. | Lemma | val1 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rmorph1",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valM : {morph val : x y / x * y}. | Proof. exact: rmorphM. Qed. | Lemma | valM | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/rings_modules_and_algebras.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"Algebra",
"Monoid.Theory",
"Scale.Exports",
"AllExports",
"ClosedExports"
] | [
"rmorphM",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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