fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
scalable_semilinear(R : pzSemiRingType)
(U : lSemiModType R) (V : nmodType) (s : Scale.preLaw R V) (f : U -> V) :
semilinear_for s f -> scalable_for s f.
Proof. by case. Qed.
HB.factory Record isSemilinear (R : pzSemiRingType) (U : lSemiModType R)
(V : nmodType) (s : Scale.semiLaw R V) (f : U -> V) := {
linear_subproof : semilinear_for s f;
}.
HB.builders Context R U V s f of isSemilinear R U V s f.
HB.instance Definition _ := isNmodMorphism.Build U V f
(additive_semilinear linear_subproof).
HB.instance Definition _ :=
isScalable.Build R U V s f (scalable_semilinear linear_subproof).
HB.end.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalable_semilinear
| |
linear_for(R : pzSemiRingType) (U : lSemiModType R) (V : nmodType)
(s : R -> V -> V) (f : U -> V) :=
forall a, {morph f : u v / a *: u + v >-> s a u + v}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linear_for
| |
zmod_morphism_linear(R : pzRingType) (U : lmodType R) V
(s : Scale.law R V) (f : U -> V) : linear_for s f -> zmod_morphism f.
Proof. by move=> Lsf x y; rewrite -scaleN1r addrC Lsf Scale.N1op addrC. Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `zmod_morphism_linear` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
zmod_morphism_linear
| |
additive_linear:= zmod_morphism_linear.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
additive_linear
| |
scalable_linear(R : pzRingType) (U : lmodType R) V
(s : Scale.law R V) (f : U -> V) : linear_for s f -> scalable_for s f.
Proof.
by move=> Lsf a v; rewrite -[a *:v](addrK v) (zmod_morphism_linear Lsf) Lsf addrK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalable_linear
| |
semilinear_linear(R : pzRingType) (U : lmodType R) V
(s : Scale.law R V) (f : U -> V) : linear_for s f -> semilinear_for s f.
Proof.
move=> Lsf; split=> [a x|x y]; first exact: (scalable_linear Lsf).
have f0: f 0 = 0 by rewrite -[0 in LHS]subr0 (zmod_morphism_linear Lsf) subrr.
by rewrite -[y in LHS]opprK -[- y]add0r !(zmod_morphism_linear Lsf) f0 sub0r opprK.
Qed.
HB.factory Record isLinear (R : pzRingType) (U : lmodType R) (V : zmodType)
(s : Scale.law R V) (f : U -> V) := {
linear_subproof : linear_for s f;
}.
HB.builders Context R U V s f of isLinear R U V s f.
HB.instance Definition _ := isZmodMorphism.Build U V f
(zmod_morphism_linear linear_subproof).
HB.instance Definition _ := isScalable.Build R U V s f
(scalable_linear linear_subproof).
HB.end.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semilinear_linear
| |
scalablef := (scalable_for *:%R f).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalable
| |
semilinearf := (semilinear_for *:%R f).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semilinear
| |
semiscalarf := (semilinear_for *%R f).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semiscalar
| |
linearf := (linear_for *:%R f).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linear
| |
scalarf := (linear_for *%R f).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalar
| |
map_class:= mapUV.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
map_class
| |
map_at(a : R) := mapUV.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
map_at
| |
map_fora s_a := MapFor {map_for_map : mapUV; _ : s a = s_a}.
|
Structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
map_for
| |
unify_map_ata (g : map_at a) := MapFor g (erefl (s a)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
unify_map_at
| |
wrapped:= Wrap {unwrap : mapUV}.
|
Structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
wrapped
| |
wrap(f : map_class) := Wrap f.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
wrap
| |
Linear.map_for_map : Linear.map_for >-> Linear.type.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Linear
| |
Linear.unify_map_at : Linear.map_at >-> Linear.map_for.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Linear
| |
Linear.unify_map_at.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Linear
| |
Linear.unwrap : Linear.wrapped >-> Linear.type.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Linear
| |
Linear.wrap : Linear.map_class >-> Linear.wrapped.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Linear
| |
Linear.wrap.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Linear
| |
linear0: f 0 = 0. Proof. exact: raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linear0
| |
linearD: {morph f : x y / x + y}. Proof. exact: raddfD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearD
| |
linearMnn : {morph f : x / x *+ n}. Proof. exact: raddfMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearMn
| |
linear_sumI r (P : pred I) E :
f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).
Proof. exact: raddf_sum. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linear_sum
| |
linearZ_LR: scalable_for s f. Proof. exact: semi_linear_subproof. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearZ_LR
| |
semilinearP: semilinear_for s f.
Proof. split; [exact: linearZ_LR | exact: linearD]. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semilinearP
| |
linearP: linear_for s f.
Proof. by move=> a u v /=; rewrite !semilinearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearP
| |
linearN: {morph f : x / - x}. Proof. exact: raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearN
| |
linearB: {morph f : x y / x - y}. Proof. exact: raddfB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearB
| |
linearMNnn : {morph f : x / x *- n}. Proof. exact: raddfMNn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearMNn
| |
linearZ(R : pzSemiRingType) (U : lSemiModType R) (V : nmodType)
(s : R -> V -> V) (S : pzSemiRingType) (h : Scale.preLaw S V)
(c : S) (a : R) (h_c := h c) (f : Linear.map_for U s a h_c) (u : U) :
f (a *: u) = h_c (Linear.wrap f u).
Proof. by rewrite linearZ_LR; case: f => f /= ->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearZ
| |
linearZZ: scalable f. Proof. exact: linearZ_LR. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearZZ
| |
semilinearPZ: semilinear f. Proof. exact: semilinearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semilinearPZ
| |
linearPZ: linear f. Proof. exact: linearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linearPZ
| |
can2_scalablef' : cancel f f' -> cancel f' f -> scalable f'.
Proof. by move=> fK f'K a x; apply: (canLR fK); rewrite linearZZ f'K. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
can2_scalable
| |
can2_semilinearf' : cancel f f' -> cancel f' f -> semilinear f'.
Proof.
by move=> fK f'K; split=> ? ?; apply: (canLR fK); rewrite semilinearPZ !f'K.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
can2_semilinear
| |
can2_linearf' : 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
can2_linear
| |
scalarZ: scalable_for *%R f. Proof. exact: linearZ_LR. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalarZ
| |
semiscalarP: semiscalar f. Proof. exact: semilinearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semiscalarP
| |
scalarP: scalar f. Proof. exact: linearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalarP
| |
idfun_is_scalable: scalable (@idfun U). Proof. by []. Qed.
#[export]
HB.instance Definition _ := isScalable.Build R U U *:%R idfun idfun_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
idfun_is_scalable
| |
comp_is_scalable: scalable_for s (f \o g).
Proof. by move=> a v /=; rewrite !linearZ_LR. Qed.
#[export]
HB.instance Definition _ := isScalable.Build R W V s (f \o g) comp_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
comp_is_scalable
| |
null_fun_is_scalable: scalable_for s (\0 : U -> V).
Proof. by move=> a v /=; rewrite raddf0. Qed.
#[export]
HB.instance Definition _ :=
isScalable.Build R U V s \0 null_fun_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
null_fun_is_scalable
| |
add_fun_is_scalable: scalable_for s (add_fun f g).
Proof. by move=> a u; rewrite /= !linearZ_LR raddfD. Qed.
#[export]
HB.instance Definition _ :=
isScalable.Build R U V s (f \+ g) add_fun_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
add_fun_is_scalable
| |
opp_is_scalable: scalable (-%R : U -> U).
Proof. by move=> a v /=; rewrite scalerN. Qed.
#[export]
HB.instance Definition _ := isScalable.Build R U U *:%R -%R opp_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
opp_is_scalable
| |
sub_fun_is_scalable: scalable_for s (f \- g).
Proof. by move=> a u; rewrite /= !linearZ_LR raddfB. Qed.
#[export]
HB.instance Definition _ :=
isScalable.Build R U V s (f \- g) sub_fun_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sub_fun_is_scalable
| |
opp_fun_is_scalable: scalable_for s (\- f).
Proof. by move=> a u; rewrite /= linearZ_LR raddfN. Qed.
#[export]
HB.instance Definition _ := isScalable.Build R U V s (\- f) opp_fun_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
opp_fun_is_scalable
| |
Definition_ :=
isScalable.Build R U A *:%R (a \o* f) mulr_fun_is_scalable.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
DefinitionLRMorphism (R : pzSemiRingType) (A : lSemiAlgType R)
(B : pzSemiRingType) (s : R -> B -> B) :=
{f of @RMorphism A B f & isScalable R A B s f}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
rmorph_alga : f a%:A = a%:A.
Proof. by rewrite linearZ /= rmorph1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_alg
| |
RecordPzSemiRing_hasCommutativeMul R of PzSemiRing R := {
mulrC : commutative (@mul R)
}.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
BuildR := (PzSemiRing_hasCommutativeMul.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Build
| |
SemiRing_hasCommutativeMulR :=
(PzSemiRing_hasCommutativeMul R) (only parsing).
#[short(type="comPzSemiRingType")]
HB.structure Definition ComPzSemiRing :=
{R of PzSemiRing R & PzSemiRing_hasCommutativeMul R}.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
SemiRing_hasCommutativeMul
| |
RecordNmodule_isComPzSemiRing R of Nmodule R := {
one : R;
mul : R -> R -> R;
mulrA : associative mul;
mulrC : commutative mul;
mul1r : left_id one mul;
mulrDl : left_distributive mul add;
mul0r : left_zero zero mul;
}.
HB.builders Context R of Nmodule_isComPzSemiRing R.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
mulr1:= Monoid.mulC_id mulrC mul1r.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr1
| |
mulrDr:= Monoid.mulC_dist mulrC mulrDl.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrDr
| |
mulr0: right_zero zero mul.
Proof. by move=> x; rewrite mulrC mul0r. Qed.
HB.instance Definition _ := Nmodule_isPzSemiRing.Build R
mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0.
HB.instance Definition _ := PzSemiRing_hasCommutativeMul.Build R mulrC.
HB.end.
#[short(type="comNzSemiRingType")]
HB.structure Definition ComNzSemiRing :=
{R of NzSemiRing R & PzSemiRing_hasCommutativeMul R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use ComNzSemiRing instead.")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr0
| |
ComSemiRingR := (ComNzSemiRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
ComSemiRing
| |
sort:= (ComNzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use ComNzSemiRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sort
| |
onR := (ComNzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use ComNzSemiRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
on
| |
copyT U := (ComNzSemiRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
copy
| |
RecordNmodule_isComNzSemiRing R of Nmodule R := {
one : R;
mul : R -> R -> R;
mulrA : associative mul;
mulrC : commutative mul;
mul1r : left_id one mul;
mulrDl : left_distributive mul add;
mul0r : left_zero zero mul;
oner_neq0 : one != zero
}.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
BuildR := (Nmodule_isComNzSemiRing.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Build
| |
Nmodule_isComSemiRingR := (Nmodule_isComNzSemiRing R) (only parsing).
HB.builders Context R of Nmodule_isComNzSemiRing R.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Nmodule_isComSemiRing
| |
mulr1:= Monoid.mulC_id mulrC mul1r.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr1
| |
mulrDr:= Monoid.mulC_dist mulrC mulrDl.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrDr
| |
mulr0: right_zero zero mul.
Proof. by move=> x; rewrite mulrC mul0r. Qed.
HB.instance Definition _ := Nmodule_isNzSemiRing.Build R
mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0 oner_neq0.
HB.instance Definition _ := PzSemiRing_hasCommutativeMul.Build R mulrC.
HB.end.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr0
| |
Definition_ := SemiGroup.isCommutativeLaw.Build R *%R mulrC.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
mulrCA: @left_commutative R R *%R. Proof. exact: mulmCA. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrCA
| |
mulrAC: @right_commutative R R *%R. Proof. exact: mulmAC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrAC
| |
mulrACA: @interchange R *%R *%R. Proof. exact: mulmACA. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrACA
| |
exprMnn : {morph (fun x => x ^+ n) : x y / x * y}.
Proof. by move=> x y; exact/exprMn_comm/mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprMn
| |
prodrXln 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
prodrXl
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
prodr_undup_exp_count
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
prodrMl
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
prodrMr
| |
exprDnx 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprDn
| |
sqrrDx 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sqrrD
| |
rmorph_comm(S : pzSemiRingType) (f : {rmorphism R -> S}) x y :
comm (f x) (f y).
Proof. by red; rewrite -!rmorphM mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_comm
| |
scale_is_scalable: scalable ( *:%R b : V -> V).
Proof. by move=> a v /=; rewrite !scalerA mulrC. Qed.
#[export]
HB.instance Definition _ :=
isScalable.Build R V V *:%R ( *:%R b) scale_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scale_is_scalable
| |
scale_fun_is_scalable: scalable (b \*: f).
Proof. by move=> a v /=; rewrite !linearZ. Qed.
#[export]
HB.instance Definition _ :=
isScalable.Build R U V *:%R (b \*: f) scale_fun_is_scalable.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scale_fun_is_scalable
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_aut_is_nmod_morphism
| |
pFrobenius_aut_is_monoid_morphism: monoid_morphism (pFrobenius_aut pcharRp).
Proof.
by split=> [|x y]; [exact: pFrobenius_aut1 | exact/pFrobenius_autM_comm/mulrC].
Qed.
#[export]
HB.instance Definition _ := isNmodMorphism.Build R R (pFrobenius_aut pcharRp)
pFrobenius_aut_is_nmod_morphism.
#[export]
HB.instance Definition _ := isMonoidMorphism.Build R R (pFrobenius_aut pcharRp)
pFrobenius_aut_is_monoid_morphism.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_aut_is_monoid_morphism
| |
exprDn_pcharx 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprDn_pchar
| |
DefinitionComPzRing := {R of PzRing R & ComPzSemiRing R}.
HB.factory Record PzRing_hasCommutativeMul R of PzRing R := {
mulrC : commutative (@mul R)
}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
BuildR := (PzRing_hasCommutativeMul.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Build
| |
Ring_hasCommutativeMulR :=
(PzRing_hasCommutativeMul R) (only parsing).
HB.builders Context R of PzRing_hasCommutativeMul R.
HB.instance Definition _ := PzSemiRing_hasCommutativeMul.Build R mulrC.
HB.end.
HB.factory Record Zmodule_isComPzRing R of Zmodule R := {
one : R;
mul : R -> R -> R;
mulrA : associative mul;
mulrC : commutative mul;
mul1r : left_id one mul;
mulrDl : left_distributive mul add;
}.
HB.builders Context R of Zmodule_isComPzRing R.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Ring_hasCommutativeMul
| |
mulr1:= Monoid.mulC_id mulrC mul1r.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr1
| |
mulrDr:= Monoid.mulC_dist mulrC mulrDl.
HB.instance Definition _ := Zmodule_isPzRing.Build R
mulrA mul1r mulr1 mulrDl mulrDr.
HB.instance Definition _ := PzRing_hasCommutativeMul.Build R mulrC.
HB.end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrDr
| |
pFrobenius_aut_is_multiplicative:=
(fun p => (p.2, p.1) \o pFrobenius_aut_is_monoid_morphism) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use exprDn_pchar instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_aut_is_multiplicative
| |
exprDn_char:= exprDn_pchar (only parsing).
#[short(type="comNzRingType")]
HB.structure Definition ComNzRing := {R of NzRing R & ComNzSemiRing R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use ComNzRing instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprDn_char
| |
ComRingR := (ComNzRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
ComRing
| |
sort:= (ComNzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use ComNzRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sort
| |
onR := (ComNzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use ComNzRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
on
| |
copyT U := (ComNzRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
copy
| |
RecordZmodule_isComNzRing R of Zmodule R := {
one : R;
mul : R -> R -> R;
mulrA : associative mul;
mulrC : commutative mul;
mul1r : left_id one mul;
mulrDl : left_distributive mul add;
oner_neq0 : one != zero
}.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
BuildR := (Zmodule_isComNzRing.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Build
|
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