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 |
|---|---|---|---|---|---|---|
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
|
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
|
rpredZnat
| |
subsemimodClosedP(modS : submodClosed V) : subsemimod_closed modS.
Proof. by split; [exact: rpred0D | exact: rpredZ]. 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
|
subsemimodClosedP
| |
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
|
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
|
rpredZsign
| |
submodClosedP(modS : submodClosed V) : submod_closed modS.
Proof. exact/subsemimod_closed_submod/subsemimodClosedP. 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
|
submodClosedP
| |
subsemialgClosedP(algS : subalgClosed A) : subsemialg_closed algS.
Proof.
split; [ exact: rpred1 | exact: rpred0D | exact: rpredZ | exact: rpredM ].
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
|
subsemialgClosedP
| |
subalgClosedP(algS : subalgClosed A) : subalg_closed algS.
Proof. exact/subsemialg_closed_subalg/subsemialgClosedP. 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
|
subalgClosedP
| |
rpredVx : (x^-1 \in S) = (x \in S).
Proof. by apply/idP/idP=> /rpredVr; rewrite ?invrK. 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
|
rpredV
| |
rpred_div: {in S &, forall x y, x / y \in S}.
Proof. by move=> x y Sx Sy; rewrite /= rpredM ?rpredV. 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
|
rpred_div
| |
rpredXNn : {in S, forall x, x ^- n \in S}.
Proof. by move=> x Sx; rewrite /= rpredV rpredX. 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
|
rpredXN
| |
rpredMlx y : x \in S -> x \is a unit-> (x * y \in S) = (y \in S).
Proof.
move=> Sx Ux; apply/idP/idP=> [Sxy | /(rpredM _ _ Sx)-> //].
by rewrite -(mulKr Ux y); rewrite rpredM ?rpredV.
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
|
rpredMl
| |
rpredMrx y : x \in S -> x \is a unit -> (y * x \in S) = (y \in S).
Proof.
move=> Sx Ux; apply/idP/idP=> [Sxy | /rpredM-> //].
by rewrite -(mulrK Ux y); rewrite rpred_div.
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
|
rpredMr
| |
rpred_divrx y : x \in S -> x \is a unit -> (y / x \in S) = (y \in S).
Proof. by rewrite -rpredV -unitrV; apply: rpredMr. 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
|
rpred_divr
| |
rpred_divlx y : x \in S -> x \is a unit -> (x / y \in S) = (y \in S).
Proof. by rewrite -(rpredV y); apply: rpredMl. 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
|
rpred_divl
| |
divringClosedP(divS : divringClosed R) : divring_closed divS.
Proof. split; [ exact: rpred1 | exact: rpredB | exact: rpred_div ]. Qed.
Fact unitr_sdivr_closed : @sdivr_closed R unit.
Proof. by split=> [|x y Ux Uy]; rewrite ?unitrN1 // unitrMl ?unitrV. Qed.
#[export]
HB.instance Definition _ := isSdivClosed.Build R unit_pred unitr_sdivr_closed.
Implicit Type x : R.
|
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
|
divringClosedP
| |
unitrNx : (- x \is a unit) = (x \is a unit). Proof. exact: rpredN. 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
|
unitrN
| |
invrNx : (- x)^-1 = - x^-1.
Proof.
have [Ux | U'x] := boolP (x \is a unit); last by rewrite !invr_out ?unitrN.
by rewrite -mulN1r invrM ?unitrN1 // invrN1 mulrN1.
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
|
invrN
| |
divrNNx y : (- x) / (- y) = x / y.
Proof. by rewrite invrN mulrNN. 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
|
divrNN
| |
divrNx y : x / (- y) = - (x / y).
Proof. by rewrite invrN mulrN. 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
|
divrN
| |
invr_signMn x : ((-1) ^+ n * x)^-1 = (-1) ^+ n * x^-1.
Proof. by rewrite -signr_odd !mulr_sign; case: ifP => // _; rewrite invrN. 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
|
invr_signM
| |
divr_signM(b1 b2 : bool) x1 x2:
((-1) ^+ b1 * x1) / ((-1) ^+ b2 * x2) = (-1) ^+ (b1 (+) b2) * (x1 / x2).
Proof. by rewrite invr_signM mulr_signM. 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
|
divr_signM
| |
rpredZeq(S : submodClosed V) a v :
(a *: v \in S) = (a == 0) || (v \in S).
Proof.
have [-> | nz_a] := eqVneq; first by rewrite scale0r rpred0.
by apply/idP/idP; first rewrite -{2}(scalerK nz_a v); apply: rpredZ.
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
|
rpredZeq
| |
fpredMlx y : x \in S -> x != 0 -> (x * y \in S) = (y \in S).
Proof. by rewrite -!unitfE; apply: rpredMl. 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
|
fpredMl
| |
fpredMrx y : x \in S -> x != 0 -> (y * x \in S) = (y \in S).
Proof. by rewrite -!unitfE; apply: rpredMr. 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
|
fpredMr
| |
fpred_divlx y : x \in S -> x != 0 -> (x / y \in S) = (y \in S).
Proof. by rewrite -!unitfE; apply: rpred_divl. 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
|
fpred_divl
| |
fpred_divrx y : x \in S -> x != 0 -> (y / x \in S) = (y \in S).
Proof. by rewrite -!unitfE; apply: rpred_divr. 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
|
fpred_divr
| |
RecordisSubPzSemiRing (R : pzSemiRingType) (S : pred R) U
of SubNmodule R S U & PzSemiRing U := {
valM_subproof : monoid_morphism (val : U -> 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 S U := (isSubPzSemiRing.Build R S 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
|
Build
| |
isSubSemiRingR S U := (isSubPzSemiRing R S U) (only parsing).
#[short(type="subPzSemiRingType")]
HB.structure Definition SubPzSemiRing (R : pzSemiRingType) (S : pred R) :=
{ U of SubNmodule R S U & PzSemiRing U & isSubPzSemiRing R S U }.
#[short(type="subNzSemiRingType")]
HB.structure Definition SubNzSemiRing (R : nzSemiRingType) (S : pred R) :=
{ U of SubNmodule R S U & NzSemiRing U & isSubPzSemiRing R S U }.
#[deprecated(since="mathcomp 2.4.0",
note="Use SubNzSemiRing 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
|
isSubSemiRing
| |
SubSemiRingR := (SubNzSemiRing 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
|
SubSemiRing
| |
sort:= (SubNzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubNzSemiRing.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 := (SubNzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubNzSemiRing.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 := (SubNzSemiRing.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
| |
val:= (val : U -> R).
#[export]
HB.instance Definition _ := isMonoidMorphism.Build U R val valM_subproof.
|
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
|
val
| |
val1: val 1 = 1. Proof. exact: 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
|
val1
| |
valM: {morph val : x y / x * y}. Proof. exact: rmorphM. 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
|
valM
| |
valM1: monoid_morphism val. Proof. exact: valM_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
|
valM1
| |
RecordSubNmodule_isSubPzSemiRing (R : pzSemiRingType) S U
of SubNmodule R S U := {
mulr_closed_subproof : mulr_closed S
}.
HB.builders Context R S U of SubNmodule_isSubPzSemiRing R S U.
HB.instance Definition _ := isMulClosed.Build R S mulr_closed_subproof.
Let inU v Sv : U := Sub v Sv.
Let oneU : U := inU (@rpred1 _ (MulClosed.clone R S _)).
Let mulU (u1 u2 : U) := inU (rpredM _ _ (valP u1) (valP u2)).
|
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
| |
mulrA: associative mulU.
Proof. by move=> x y z; apply: val_inj; rewrite !SubK mulrA. 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
|
mulrA
| |
mul1r: left_id oneU mulU.
Proof. by move=> x; apply: val_inj; rewrite !SubK 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
|
mul1r
| |
mulr1: right_id oneU mulU.
Proof. by move=> x; apply: val_inj; rewrite !SubK mulr1. 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
|
mulr1
| |
mulrDl: left_distributive mulU +%R.
Proof.
by move=> x y z; apply: val_inj; rewrite !(SubK, raddfD)/= !SubK mulrDl.
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
|
mulrDl
| |
mulrDr: right_distributive mulU +%R.
Proof.
by move=> x y z; apply: val_inj; rewrite !(SubK, raddfD)/= !SubK mulrDr.
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
|
mulrDr
| |
mul0r: left_zero 0%R mulU.
Proof. by move=> x; apply: val_inj; rewrite SubK val0 mul0r. 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
|
mul0r
| |
mulr0: right_zero 0%R mulU.
Proof. by move=> x; apply: val_inj; rewrite SubK val0 mulr0. Qed.
HB.instance Definition _ := Nmodule_isPzSemiRing.Build U
mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0.
|
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
| |
valM: monoid_morphism (val : U -> R).
Proof. by split=> [|x y] /=; rewrite !SubK. Qed.
HB.instance Definition _ := isSubPzSemiRing.Build R S U valM.
HB.end.
HB.factory Record SubPzSemiRing_isNonZero (R : nzSemiRingType) S U
of SubPzSemiRing R S U := {}.
HB.builders Context R S U of SubPzSemiRing_isNonZero R S U.
|
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
|
valM
| |
oner_neq0: (1 : U) != 0.
Proof. by rewrite -(inj_eq val_inj) rmorph0 rmorph1 oner_neq0. Qed.
HB.instance Definition _ := PzSemiRing_isNonZero.Build U oner_neq0.
HB.end.
HB.factory Record SubNmodule_isSubNzSemiRing (R : nzSemiRingType) S U
of SubNmodule R S U := {
mulr_closed_subproof : mulr_closed S
}.
|
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
|
oner_neq0
| |
BuildR S U := (SubNmodule_isSubNzSemiRing.Build R S 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
|
Build
| |
SubNmodule_isSubSemiRingR S U :=
(SubNmodule_isSubNzSemiRing R S U) (only parsing).
HB.builders Context R S U of SubNmodule_isSubNzSemiRing R S U.
HB.instance Definition _ := SubNmodule_isSubPzSemiRing.Build R S U
mulr_closed_subproof.
HB.instance Definition _ := SubPzSemiRing_isNonZero.Build R S U.
HB.end.
#[short(type="subComPzSemiRingType")]
HB.structure Definition SubComPzSemiRing (R : pzSemiRingType) S :=
{U of SubPzSemiRing R S U & ComPzSemiRing U}.
HB.factory Record SubPzSemiRing_isSubComPzSemiRing (R : comPzSemiRingType) S U
of SubPzSemiRing R S U := {}.
HB.builders Context R S U of SubPzSemiRing_isSubComPzSemiRing R S U.
|
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
|
SubNmodule_isSubSemiRing
| |
mulrC: @commutative U U *%R.
Proof. by move=> x y; apply: val_inj; rewrite !rmorphM mulrC. Qed.
HB.instance Definition _ := PzSemiRing_hasCommutativeMul.Build U mulrC.
HB.end.
#[short(type="subComNzSemiRingType")]
HB.structure Definition SubComNzSemiRing (R : nzSemiRingType) S :=
{U of SubNzSemiRing R S U & ComNzSemiRing U}.
#[deprecated(since="mathcomp 2.4.0",
note="Use SubComNzSemiRing 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
|
mulrC
| |
SubComSemiRingR := (SubComNzSemiRing 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
|
SubComSemiRing
| |
sort:= (SubComNzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubComNzSemiRing.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 := (SubComNzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubComNzSemiRing.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 := (SubComNzSemiRing.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
| |
RecordSubNzSemiRing_isSubComNzSemiRing (R : comNzSemiRingType) S U
of SubNzSemiRing R S U := {}.
|
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 S U :=
(SubNzSemiRing_isSubComNzSemiRing.Build R S 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
|
Build
| |
SubSemiRing_isSubComSemiRingR S U :=
(SubNzSemiRing_isSubComNzSemiRing R S U) (only parsing).
HB.builders Context R S U of SubNzSemiRing_isSubComNzSemiRing R S U.
HB.instance Definition _ := SubPzSemiRing_isSubComPzSemiRing.Build R S U.
HB.end.
#[short(type="subPzRingType")]
HB.structure Definition SubPzRing (R : pzRingType) (S : pred R) :=
{ U of SubPzSemiRing R S U & PzRing U & isSubZmodule R S U }.
HB.factory Record SubZmodule_isSubPzRing (R : pzRingType) S U
of SubZmodule R S U := {
subring_closed_subproof : subring_closed S
}.
HB.builders Context R S U of SubZmodule_isSubPzRing R S U.
HB.instance Definition _ := SubNmodule_isSubPzSemiRing.Build R S U
(smulr_closedM (subring_closedM subring_closed_subproof)).
HB.end.
#[short(type="subNzRingType")]
HB.structure Definition SubNzRing (R : nzRingType) (S : pred R) :=
{ U of SubNzSemiRing R S U & NzRing U & isSubBaseAddUMagma R S U }.
#[deprecated(since="mathcomp 2.4.0",
note="Use SubNzRing 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
|
SubSemiRing_isSubComSemiRing
| |
SubRingR := (SubNzRing 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
|
SubRing
| |
sort:= (SubNzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubNzRing.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 := (SubNzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubNzRing.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 := (SubNzRing.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
| |
RecordSubZmodule_isSubNzRing (R : nzRingType) S U
of SubZmodule R S U := {
subring_closed_subproof : subring_closed S
}.
|
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 S U := (SubZmodule_isSubNzRing.Build R S 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
|
Build
| |
SubZmodule_isSubRingR S U :=
(SubZmodule_isSubNzRing R S U) (only parsing).
HB.builders Context R S U of SubZmodule_isSubNzRing R S U.
HB.instance Definition _ := SubNmodule_isSubNzSemiRing.Build R S U
(smulr_closedM (subring_closedM subring_closed_subproof)).
HB.end.
#[short(type="subComPzRingType")]
HB.structure Definition SubComPzRing (R : pzRingType) S :=
{U of SubPzRing R S U & ComPzRing U}.
HB.factory Record SubPzRing_isSubComPzRing (R : comPzRingType) S U
of SubPzRing R S U := {}.
HB.builders Context R S U of SubPzRing_isSubComPzRing R S U.
HB.instance Definition _ := SubPzSemiRing_isSubComPzSemiRing.Build R S U.
HB.end.
#[short(type="subComNzRingType")]
HB.structure Definition SubComNzRing (R : nzRingType) S :=
{U of SubNzRing R S U & ComNzRing U}.
#[deprecated(since="mathcomp 2.4.0",
note="Use SubComNzRing 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
|
SubZmodule_isSubRing
| |
SubComRingR := (SubComNzRing 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
|
SubComRing
| |
sort:= (SubComNzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubComNzRing.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 := (SubComNzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use SubComNzRing.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 := (SubComNzRing.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
| |
RecordSubNzRing_isSubComNzRing (R : comNzRingType) S U
of SubNzRing R S U := {}.
|
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 S U := (SubNzRing_isSubComNzRing.Build R S 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
|
Build
| |
SubRing_isSubComRingR S U :=
(SubNzRing_isSubComNzRing R S U) (only parsing).
HB.builders Context R S U of SubNzRing_isSubComNzRing R S U.
HB.instance Definition _ := SubPzRing_isSubComPzRing.Build R S U.
HB.end.
HB.mixin Record isSubLSemiModule (R : pzSemiRingType) (V : lSemiModType R)
(S : pred V) W of SubNmodule V S W & LSemiModule R W := {
valZ : scalable (val : W -> V);
}.
#[short(type="subLSemiModType")]
HB.structure Definition SubLSemiModule (R : pzSemiRingType) (V : lSemiModType R)
(S : pred V) :=
{ W of SubNmodule V S W &
Nmodule_isLSemiModule R W & isSubLSemiModule R V S W}.
#[short(type="subLmodType")]
HB.structure Definition SubLmodule (R : pzRingType) (V : lmodType R)
(S : pred V) :=
{ W of SubZmodule V S W &
Nmodule_isLSemiModule R W & isSubLSemiModule R V S W}.
|
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
|
SubRing_isSubComRing
| |
val:= (val : W -> V).
#[export]
HB.instance Definition _ := isScalable.Build R W V *:%R val valZ.
|
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
|
val
| |
RecordisSubLmodule (R : pzRingType) (V : lmodType R) (S : pred V)
W of SubZmodule V S W & Lmodule R W := {
valZ : scalable (val : W -> V);
}.
HB.builders Context R V S W of isSubLmodule R V S W.
HB.instance Definition _ := isSubLSemiModule.Build R V S W valZ.
HB.end.
HB.factory Record SubNmodule_isSubLSemiModule
(R : pzSemiRingType) (V : lSemiModType R) S W of SubNmodule V S W := {
subsemimod_closed_subproof : subsemimod_closed S
}.
HB.builders Context R V S W of SubNmodule_isSubLSemiModule R V S W.
HB.instance Definition _ :=
isSubSemiModClosed.Build R V S subsemimod_closed_subproof.
Let inW v Sv : W := Sub v Sv.
Let scaleW a (w : W) := inW (rpredZ a _ (valP w)).
|
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
| |
scalerA'a b v : scaleW a (scaleW b v) = scaleW (a * b) v.
Proof. by apply: val_inj; rewrite !SubK scalerA. 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
|
scalerA'
| |
scale0rv : scaleW 0 v = 0.
Proof. by apply: val_inj; rewrite SubK scale0r 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
|
scale0r
| |
scale1r: left_id 1 scaleW.
Proof. by move=> x; apply: val_inj; rewrite SubK scale1r. 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
|
scale1r
| |
scalerDr: right_distributive scaleW +%R.
Proof. by move=> a u v; apply: val_inj; rewrite SubK !raddfD/= !SubK. 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
|
scalerDr
| |
scalerDlv : {morph scaleW^~ v : a b / a + b}.
Proof. by move=> a b; apply: val_inj; rewrite raddfD/= !SubK scalerDl. Qed.
HB.instance Definition _ := Nmodule_isLSemiModule.Build R W
scalerA' scale0r scale1r scalerDr scalerDl.
Fact valZ : scalable (val : W -> _). Proof. by move=> k w; rewrite SubK. Qed.
HB.instance Definition _ := isSubLSemiModule.Build R V S W valZ.
HB.end.
HB.factory Record SubZmodule_isSubLmodule (R : pzRingType) (V : lmodType R) S W
of SubZmodule V S W := {
subsemimod_closed_subproof : subsemimod_closed S
}.
HB.builders Context R V S W of SubZmodule_isSubLmodule R V S W.
HB.instance Definition _ := SubNmodule_isSubLSemiModule.Build R V S W
subsemimod_closed_subproof.
HB.end.
#[short(type="subLSemiAlgType")]
HB.structure Definition SubLSemiAlgebra
(R : pzSemiRingType) (V : lSemiAlgType R) S :=
{W of SubNzSemiRing V S W & @SubLSemiModule R V S W & LSemiAlgebra R W}.
#[short(type="subLalgType")]
HB.structure Definition SubLalgebra (R : pzRingType) (V : lalgType R) S :=
{W of SubNzRing V S W & @SubLmodule R V S W & Lalgebra R W}.
HB.factory Record SubNzSemiRing_SubLSemiModule_isSubLSemiAlgebra
(R : pzSemiRingType) (V : lSemiAlgType R) S W
of SubNzSemiRing V S W & @SubLSemiModule R V S W := {}.
HB.builders Context R V S W
of SubNzSemiRing_SubLSemiModule_isSubLSemiAlgebra R V S W.
|
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
|
scalerDl
| |
scalerAl(a : R) (u v : W) : a *: (u * v) = a *: u * v.
Proof. by apply: val_inj; rewrite !(linearZ, rmorphM) /= linearZ scalerAl. Qed.
HB.instance Definition _ := LSemiModule_isLSemiAlgebra.Build R W scalerAl.
HB.end.
HB.factory Record SubNzRing_SubLmodule_isSubLalgebra (R : pzRingType)
(V : lalgType R) S W of SubNzRing V S W & @SubLmodule R V S W := {}.
|
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
|
scalerAl
| |
BuildR V S U :=
(SubNzRing_SubLmodule_isSubLalgebra.Build R V S 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
|
Build
| |
SubRing_SubLmodule_isSubLalgebraR V S U :=
(SubNzRing_SubLmodule_isSubLalgebra R V S U) (only parsing).
HB.builders Context R V S W of SubNzRing_SubLmodule_isSubLalgebra R V S W.
HB.instance Definition _ :=
SubNzSemiRing_SubLSemiModule_isSubLSemiAlgebra.Build R V S W.
HB.end.
#[short(type="subSemiAlgType")]
HB.structure Definition SubSemiAlgebra (R : pzSemiRingType) (V : semiAlgType R)
S :=
{W of @SubLSemiAlgebra R V S W & SemiAlgebra R W}.
#[short(type="subAlgType")]
HB.structure Definition SubAlgebra (R : pzRingType) (V : algType R) S :=
{W of @SubLalgebra R V S W & Algebra R W}.
HB.factory Record SubLSemiAlgebra_isSubSemiAlgebra (R : pzSemiRingType)
(V : semiAlgType R) S W of @SubLSemiAlgebra R V S W := {}.
HB.builders Context R V S W of SubLSemiAlgebra_isSubSemiAlgebra R V S W.
|
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
|
SubRing_SubLmodule_isSubLalgebra
| |
scalerAr(k : R) (x y : W) : k *: (x * y) = x * (k *: y).
Proof. by apply: val_inj; rewrite !(linearZ, rmorphM)/= linearZ scalerAr. Qed.
HB.instance Definition _ := LSemiAlgebra_isSemiAlgebra.Build R W scalerAr.
HB.end.
HB.factory Record SubLalgebra_isSubAlgebra (R : pzRingType)
(V : algType R) S W of @SubLalgebra R V S W := {}.
HB.builders Context R V S W of SubLalgebra_isSubAlgebra R V S W.
HB.instance Definition _ := SubLSemiAlgebra_isSubSemiAlgebra.Build R V S W.
HB.end.
#[short(type="subUnitRingType")]
HB.structure Definition SubUnitRing (R : nzRingType) (S : pred R) :=
{U of SubNzRing R S U & UnitRing U}.
HB.factory Record SubNzRing_isSubUnitRing (R : unitRingType) S U
of SubNzRing R S U := {
divring_closed_subproof : divring_closed S
}.
HB.builders Context R S U of SubNzRing_isSubUnitRing R S U.
HB.instance Definition _ := isDivringClosed.Build R S divring_closed_subproof.
Let inU v Sv : U := Sub v Sv.
Let invU (u : U) := inU (rpredVr _ (valP u)).
|
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
|
scalerAr
| |
mulVr: {in [pred x | val x \is a unit], left_inverse 1 invU *%R}.
Proof.
by move=> x /[!inE] xu; apply: val_inj; rewrite rmorphM rmorph1 /= SubK mulVr.
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
|
mulVr
| |
divrr: {in [pred x | val x \is a unit], right_inverse 1 invU *%R}.
by move=> x /[!inE] xu; apply: val_inj; rewrite rmorphM rmorph1 /= SubK mulrV.
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
|
divrr
| |
unitrP(x y : U) : y * x = 1 /\ x * y = 1 -> val x \is a unit.
Proof.
move=> -[/(congr1 val) yx1 /(congr1 val) xy1].
by apply: rev_unitrP (val y) _; rewrite !rmorphM rmorph1 /= in yx1 xy1.
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
|
unitrP
| |
invr_out: {in [pred x | val x \isn't a unit], invU =1 id}.
Proof.
by move=> x /[!inE] xNU; apply: val_inj; rewrite SubK invr_out.
Qed.
HB.instance Definition _ := NzRing_hasMulInverse.Build U
mulVr divrr unitrP invr_out.
HB.end.
#[short(type="subComUnitRingType")]
HB.structure Definition SubComUnitRing (R : comUnitRingType) (S : pred R) :=
{U of SubComNzRing R S U & SubUnitRing R S U}.
#[short(type="subIdomainType")]
HB.structure Definition SubIntegralDomain (R : idomainType) (S : pred R) :=
{U of SubComNzRing R S U & IntegralDomain U}.
HB.factory Record SubComUnitRing_isSubIntegralDomain (R : idomainType) S U
of SubComUnitRing R S U := {}.
HB.builders Context R S U of SubComUnitRing_isSubIntegralDomain R S U.
|
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
|
invr_out
| |
id: IntegralDomain.axiom U.
Proof.
move=> x y /(congr1 val)/eqP; rewrite rmorphM /=.
by rewrite -!(inj_eq val_inj) rmorph0 -mulf_eq0.
Qed.
HB.instance Definition _ := ComUnitRing_isIntegral.Build U id.
HB.end.
#[short(type="subFieldType")]
HB.structure Definition SubField (F : fieldType) (S : pred F) :=
{U of SubIntegralDomain F S U & Field U}.
HB.factory Record SubIntegralDomain_isSubField (F : fieldType) S U
of SubIntegralDomain F S U := {
subfield_subproof : {mono (val : U -> F) : u / u \in unit}
}.
HB.builders Context F S U of SubIntegralDomain_isSubField F S U.
|
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
|
id
| |
fieldP: Field.axiom U.
Proof.
by move=> u; rewrite -(inj_eq val_inj) rmorph0 -unitfE subfield_subproof.
Qed.
HB.instance Definition _ := UnitRing_isField.Build U fieldP.
HB.end.
HB.factory Record SubChoice_isSubPzSemiRing (R : pzSemiRingType) S U
of SubChoice R S U := {
semiring_closed_subproof : semiring_closed S
}.
HB.builders Context R S U of SubChoice_isSubPzSemiRing R S U.
HB.instance Definition _ := SubChoice_isSubNmodule.Build R S U
(semiring_closedD semiring_closed_subproof).
HB.instance Definition _ := SubNmodule_isSubPzSemiRing.Build R S U
(semiring_closedM semiring_closed_subproof).
HB.end.
HB.factory Record SubChoice_isSubNzSemiRing (R : nzSemiRingType) S U
of SubChoice R S U := {
semiring_closed_subproof : semiring_closed S
}.
|
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
|
fieldP
| |
BuildR S U := (SubChoice_isSubNzSemiRing.Build R S 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
|
Build
| |
SubChoice_isSubSemiRingR S U :=
(SubChoice_isSubNzSemiRing R S U) (only parsing).
HB.builders Context R S U of SubChoice_isSubNzSemiRing R S U.
HB.instance Definition _ := SubChoice_isSubPzSemiRing.Build R S U
semiring_closed_subproof.
HB.instance Definition _ := SubPzSemiRing_isNonZero.Build R S U.
HB.end.
HB.factory Record SubChoice_isSubComPzSemiRing (R : comPzSemiRingType) S U
of SubChoice R S U := {
semiring_closed_subproof : semiring_closed S
}.
HB.builders Context R S U of SubChoice_isSubComPzSemiRing R S U.
HB.instance Definition _ := SubChoice_isSubPzSemiRing.Build R S U
semiring_closed_subproof.
HB.instance Definition _ := SubPzSemiRing_isSubComPzSemiRing.Build R S U.
HB.end.
HB.factory Record SubChoice_isSubComNzSemiRing (R : comNzSemiRingType) S U
of SubChoice R S U := {
semiring_closed_subproof : semiring_closed S
}.
|
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
|
SubChoice_isSubSemiRing
| |
BuildR S U :=
(SubChoice_isSubComNzSemiRing.Build R S 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
|
Build
| |
SubChoice_isSubComSemiRingR S U :=
(SubChoice_isSubComNzSemiRing R S U) (only parsing).
HB.builders Context R S U of SubChoice_isSubComNzSemiRing R S U.
HB.instance Definition _ := SubChoice_isSubComPzSemiRing.Build R S U
semiring_closed_subproof.
HB.instance Definition _ := SubPzSemiRing_isNonZero.Build R S U.
HB.end.
HB.factory Record SubChoice_isSubPzRing (R : pzRingType) S U
of SubChoice R S U := {
subring_closed_subproof : subring_closed S
}.
HB.builders Context R S U of SubChoice_isSubPzRing R S U.
HB.instance Definition _ := SubChoice_isSubZmodule.Build R S U
(subring_closedB subring_closed_subproof).
HB.instance Definition _ := SubZmodule_isSubPzRing.Build R S U
subring_closed_subproof.
HB.end.
HB.factory Record SubChoice_isSubNzRing (R : nzRingType) S U
of SubChoice R S U := {
subring_closed_subproof : subring_closed S
}.
|
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
|
SubChoice_isSubComSemiRing
| |
BuildR S U := (SubChoice_isSubNzRing.Build R S 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
|
Build
| |
SubChoice_isSubRingR S U :=
(SubChoice_isSubNzRing R S U) (only parsing).
HB.builders Context R S U of SubChoice_isSubNzRing R S U.
HB.instance Definition _ := SubChoice_isSubPzRing.Build R S U
subring_closed_subproof.
HB.instance Definition _ := SubPzSemiRing_isNonZero.Build R S U.
HB.end.
HB.factory Record SubChoice_isSubComPzRing (R : comPzRingType) S U
of SubChoice R S U := {
subring_closed_subproof : subring_closed S
}.
HB.builders Context R S U of SubChoice_isSubComPzRing R S U.
HB.instance Definition _ := SubChoice_isSubPzRing.Build R S U
subring_closed_subproof.
HB.instance Definition _ := SubPzRing_isSubComPzRing.Build R S U.
HB.end.
HB.factory Record SubChoice_isSubComNzRing (R : comNzRingType) S U
of SubChoice R S U := {
subring_closed_subproof : subring_closed S
}.
|
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
|
SubChoice_isSubRing
| |
BuildR S U := (SubChoice_isSubComNzRing.Build R S 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
|
Build
| |
SubChoice_isSubComRingR S U :=
(SubChoice_isSubComNzRing R S U) (only parsing).
HB.builders Context R S U of SubChoice_isSubComNzRing R S U.
HB.instance Definition _ := SubChoice_isSubComPzRing.Build R S U
subring_closed_subproof.
HB.instance Definition _ := SubPzSemiRing_isNonZero.Build R S U.
HB.end.
HB.factory Record SubChoice_isSubLSemiModule
(R : pzSemiRingType) (V : lSemiModType R) S W of SubChoice V S W := {
subsemimod_closed_subproof : subsemimod_closed S
}.
HB.builders Context R V S W of SubChoice_isSubLSemiModule R V S W.
HB.instance Definition _ := SubChoice_isSubNmodule.Build V S W
(subsemimod_closedD subsemimod_closed_subproof).
HB.instance Definition _ := SubNmodule_isSubLSemiModule.Build R V S W
subsemimod_closed_subproof.
HB.end.
HB.factory Record SubChoice_isSubLmodule (R : pzRingType) (V : lmodType R) S W
of SubChoice V S W := {
subsemimod_closed_subproof : subsemimod_closed S
}.
HB.builders Context R V S W of SubChoice_isSubLmodule R V S W.
HB.instance Definition _ := SubChoice_isSubZmodule.Build V S W
(subsemimod_closedB subsemimod_closed_subproof).
HB.instance Definition _ := SubZmodule_isSubLmodule.Build R V S W
subsemimod_closed_subproof.
HB.end.
HB.factory Record SubChoice_isSubLSemiAlgebra
(R : pzSemiRingType) (A : lSemiAlgType R) S W of SubChoice A S W := {
subsemialg_closed_subproof : subsemialg_closed S
}.
HB.builders Context R A S W of SubChoice_isSubLSemiAlgebra R A S W.
HB.instance Definition _ := SubChoice_isSubNzSemiRing.Build A S W
|
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
|
SubChoice_isSubComRing
| |
addrA:= @addrA.
|
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
|
addrA
| |
addrC:= @addrC.
|
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
|
addrC
| |
add0r:= @add0r.
|
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
|
add0r
| |
addNr:= @addNr.
|
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
|
addNr
| |
addr0:= addr0.
|
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
|
addr0
|
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