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 |
|---|---|---|---|---|---|---|
nat:= nat_num.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
nat
| |
int:= int_num.
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealDomain instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
int
| |
ArchiDomainT := (ArchiRealDomain T).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
ArchiDomain
| |
type:= ArchiRealDomain.type.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
type
| |
copyT C := (ArchiRealDomain.copy T C).
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
copy
| |
onT := (ArchiRealDomain.on T).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
on
| |
ArchiFieldT := (ArchiRealField T).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
ArchiField
| |
type:= ArchiRealField.type.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealField.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
type
| |
copyT C := (ArchiRealField.copy T C).
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealField.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
copy
| |
onT := (ArchiRealField.on T).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
on
| |
floorD:= real_floorDzr.
#[deprecated(since="mathcomp 2.3.0", note="Use real_ceilDzr instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
floorD
| |
ceilD:= real_ceilDzr.
#[deprecated(since="mathcomp 2.3.0", note="Use real_ceilDzr instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
ceilD
| |
real_ceilD:= real_ceilDzr.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
real_ceilD
| |
archiDomainType:= archiRealDomainType (only parsing).
#[deprecated(since="mathcomp 2.3.0", note="Use archiRealFieldType instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
archiDomainType
| |
archiFieldType:= archiRealFieldType (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint"
] |
algebra/archimedean.v
|
archiFieldType
| |
DefinitionNmodule := {M of GRing.Nmodule M & Countable M}.
#[short(type="countZmodType")]
HB.structure Definition Zmodule := {M of GRing.Zmodule M & Countable M}.
#[short(type="countPzSemiRingType")]
HB.structure Definition PzSemiRing := {R of GRing.PzSemiRing R & Countable R}.
#[short(type="countNzSemiRingType")]
HB.structure Definition NzSemiRing := {R of GRing.NzSemiRing R & Countable R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.NzSemiRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Definition
| |
SemiRingR := (NzSemiRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
SemiRing
| |
sort:= (NzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.NzSemiRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
sort
| |
onR := (NzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.NzSemiRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
on
| |
copyT U := (NzSemiRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
copy
| |
DefinitionPzRing := {R of GRing.PzRing R & Countable R}.
#[short(type="countNzRingType")]
HB.structure Definition NzRing := {R of GRing.NzRing R & Countable R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.NzRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Definition
| |
RingR := (NzRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Ring
| |
sort:= (NzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.NzRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
sort
| |
onR := (NzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.NzRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
on
| |
copyT U := (NzRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
copy
| |
DefinitionComPzSemiRing :=
{R of GRing.ComPzSemiRing R & Countable R}.
#[short(type="countComNzSemiRingType")]
HB.structure Definition ComNzSemiRing :=
{R of GRing.ComNzSemiRing R & Countable R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.ComNzSemiRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Definition
| |
ComSemiRingR := (ComNzSemiRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
ComSemiRing
| |
sort:= (ComNzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.ComNzSemiRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
sort
| |
onR := (ComNzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.ComNzSemiRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.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 seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
copy
| |
DefinitionComPzRing := {R of GRing.ComPzRing R & Countable R}.
#[short(type="countComNzRingType")]
HB.structure Definition ComNzRing := {R of GRing.ComNzRing R & Countable R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.ComNzRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Definition
| |
ComRingR := (ComNzRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
ComRing
| |
sort:= (ComNzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.ComNzRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
sort
| |
onR := (ComNzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use CountRing.ComNzRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.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 seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
copy
| |
DefinitionUnitRing := {R of GRing.UnitRing R & Countable R}.
#[short(type="countComUnitRingType")]
HB.structure Definition ComUnitRing := {R of GRing.ComUnitRing R & Countable R}.
#[short(type="countIdomainType")]
HB.structure Definition IntegralDomain :=
{R of GRing.IntegralDomain R & Countable R}.
#[short(type="countFieldType")]
HB.structure Definition Field := {R of GRing.Field R & Countable R}.
#[short(type="countDecFieldType")]
HB.structure Definition DecidableField :=
{R of GRing.DecidableField R & Countable R}.
#[short(type="countClosedFieldType")]
HB.structure Definition ClosedField := {R of GRing.ClosedField R & Countable R}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Definition
| |
Definition_ (R : countType) := Countable.on R^o.
HB.instance Definition _ (R : countNmodType) := Nmodule.on R^o.
HB.instance Definition _ (R : countZmodType) := Zmodule.on R^o.
HB.instance Definition _ (R : countPzSemiRingType) := PzSemiRing.on R^o.
HB.instance Definition _ (R : countNzSemiRingType) := NzSemiRing.on R^o.
HB.instance Definition _ (R : countPzRingType) := PzRing.on R^o.
HB.instance Definition _ (R : countNzRingType) := NzRing.on R^o.
HB.instance Definition _ (R : countComPzSemiRingType) := ComPzSemiRing.on R^o.
HB.instance Definition _ (R : countComNzSemiRingType) := ComNzSemiRing.on R^o.
HB.instance Definition _ (R : countComPzRingType) := ComPzRing.on R^o.
HB.instance Definition _ (R : countComNzRingType) := ComNzRing.on R^o.
HB.instance Definition _ (R : countUnitRingType) := UnitRing.on R^o.
HB.instance Definition _ (R : countComUnitRingType) := ComUnitRing.on R^o.
HB.instance Definition _ (R : countIdomainType) := IntegralDomain.on R^o.
HB.instance Definition _ (R : countFieldType) := Field.on R^o.
HB.instance Definition _ (R : countDecFieldType) := DecidableField.on R^o.
HB.instance Definition _ (R : countClosedFieldType) := ClosedField.on R^o.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
Definition
| |
countSemiRingType:= (countNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use countNzRingType instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
countSemiRingType
| |
countRingType:= (countNzRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use countComNzSemiRingType instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
countRingType
| |
countComSemiRingType:= (countComNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use countComNzRingType instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
countComSemiRingType
| |
countComRingType:= (countComNzRingType) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg"
] |
algebra/countalg.v
|
countComRingType
| |
DefinitionNmodule := {M of GRing.Nmodule M & Finite M}.
#[short(type="finZmodType")]
HB.structure Definition Zmodule := {M of GRing.Zmodule M & Finite M}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
DefinitionPzSemiRing := {R of GRing.PzSemiRing R & Finite R}.
#[short(type="finNzSemiRingType")]
HB.structure Definition NzSemiRing := {R of GRing.NzSemiRing R & Finite R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.NzSemiRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
SemiRingR := (NzSemiRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
SemiRing
| |
sort:= (NzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.NzSemiRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
sort
| |
onR := (NzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.NzSemiRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
on
| |
copyT U := (NzSemiRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
copy
| |
DefinitionPzRing := {R of GRing.PzRing R & Finite R}.
#[short(type="finNzRingType")]
HB.structure Definition NzRing := {R of GRing.NzRing R & Finite R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.NzRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
RingR := (NzRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Ring
| |
sort:= (NzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.NzRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
sort
| |
onR := (NzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.NzRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
on
| |
copyT U := (NzRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
copy
| |
DefinitionComPzSemiRing :=
{R of GRing.ComPzSemiRing R & Finite R}.
#[short(type="finComNzSemiRingType")]
HB.structure Definition ComNzSemiRing :=
{R of GRing.ComNzSemiRing R & Finite R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.ComNzSemiRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
ComSemiRingR := (ComNzSemiRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
ComSemiRing
| |
sort:= (ComNzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.ComNzSemiRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
sort
| |
onR := (ComNzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.ComNzSemiRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.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 seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
copy
| |
DefinitionComPzRing := {R of GRing.ComPzRing R & Finite R}.
#[short(type="finComNzRingType")]
HB.structure Definition ComNzRing := {R of GRing.ComNzRing R & Finite R}.
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.ComNzRing instead.")]
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
ComRingR := (ComNzRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
ComRing
| |
sort:= (ComNzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.ComNzRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
sort
| |
onR := (ComNzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use FinRing.ComNzRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.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 seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
copy
| |
DefinitionUnitRing := {R of GRing.UnitRing R & Finite R}.
#[short(type="finComUnitRingType")]
HB.structure Definition ComUnitRing := {R of GRing.ComUnitRing R & Finite R}.
#[short(type="finIdomainType")]
HB.structure Definition IntegralDomain :=
{R of GRing.IntegralDomain R & Finite R}.
#[short(type="finFieldType")]
HB.structure Definition Field := {R of GRing.Field R & Finite R}.
#[short(type="finLmodType")]
HB.structure Definition Lmodule (R : nzRingType) :=
{M of GRing.Lmodule R M & Finite M}.
#[short(type="finLalgType")]
HB.structure Definition Lalgebra (R : nzRingType) :=
{M of GRing.Lalgebra R M & Finite M}.
#[short(type="finAlgType")]
HB.structure Definition Algebra (R : nzRingType) :=
{M of GRing.Algebra R M & Finite M}.
#[short(type="finUnitAlgType")]
HB.structure Definition UnitAlgebra (R : unitRingType) :=
{M of GRing.UnitAlgebra R M & Finite M}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
Definition_ (R : Zmodule.type) := [finGroupMixin of R for +%R].
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
Zmodule_to_baseFinGroup(R : Zmodule.type) := FinStarMonoid.clone R _.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Zmodule_to_baseFinGroup
| |
Zmodule_to_finGroup(R : Zmodule.type) := FinGroup.clone R _.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Zmodule_to_finGroup
| |
zmod1gE: 1%g = 0 :> U. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
zmod1gE
| |
zmodVgEx : x^-1%g = - x. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
zmodVgE
| |
zmodMgEx y : (x * y)%g = x + y. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
zmodMgE
| |
zmodXgEn x : (x ^+ n)%g = x *+ n. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
zmodXgE
| |
zmod_mulgCx y : commute x y.
Proof. exact: addrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
zmod_mulgC
| |
zmod_abelian(A : {set U}) : abelian A.
Proof. by apply/centsP=> x _ y _; apply: zmod_mulgC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
zmod_abelian
| |
Definition_ (R : NzRing.type) := [finGroupMixin of R for +%R].
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
NzRing_to_baseFinGroup(R : NzRing.type) := FinStarMonoid.clone R _.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
NzRing_to_baseFinGroup
| |
NzRing_to_finGroup(R : NzRing.type) := FinGroup.clone R _.
HB.factory Record isNzRing R of NzRing R := {}.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
NzRing_to_finGroup
| |
BuildR := (isNzRing.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Build
| |
isRingR := (isNzRing R) (only parsing).
HB.builders Context R of isNzRing R.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
isRing
| |
is_inv(x y : R) := (x * y == 1) && (y * x == 1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
is_inv
| |
unit:= [qualify a x : R | [exists y, is_inv x y]].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit
| |
invx := odflt x (pick (is_inv x)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
inv
| |
mulVr: {in unit, left_inverse 1 inv *%R}.
Proof.
rewrite /inv => x Ux; case: pickP => [y | no_y]; last by case/pred0P: Ux.
by case/andP=> _; move/eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
mulVr
| |
mulrV: {in unit, right_inverse 1 inv *%R}.
Proof.
rewrite /inv => x Ux; case: pickP => [y | no_y]; last by case/pred0P: Ux.
by case/andP; move/eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
mulrV
| |
intro_unitx y : y * x = 1 /\ x * y = 1 -> x \is a unit.
Proof.
by case=> yx1 xy1; apply/existsP; exists y; rewrite /is_inv xy1 yx1 !eqxx.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
intro_unit
| |
invr_out: {in [predC unit], inv =1 id}.
Proof.
rewrite /inv => x nUx; case: pickP => // y invxy.
by case/existsP: nUx; exists y.
Qed.
HB.instance Definition _ :=
GRing.NzRing_hasMulInverse.Build R mulVr mulrV intro_unit invr_out.
HB.end.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ComNzRing.type) := [finGroupMixin of R for +%R].
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
invr_out
| |
ComNzRing_to_baseFinGroup(R : ComNzRing.type) :=
FinStarMonoid.clone R _.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
ComNzRing_to_baseFinGroup
| |
ComNzRing_to_finGroup(R : ComNzRing.type) := FinGroup.clone R _.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : UnitRing.type) := [finGroupMixin of R for +%R].
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
ComNzRing_to_finGroup
| |
UnitRing_to_baseFinGroup(R : UnitRing.type) := FinStarMonoid.clone R _.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
UnitRing_to_baseFinGroup
| |
UnitRing_to_finGroup(R : UnitRing.type) := FinGroup.clone R _.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
UnitRing_to_finGroup
| |
unit_of:= Unit (x : R) of x \is a GRing.unit.
Bind Scope group_scope with unit_of.
Implicit Types u v : unit_of.
|
Inductive
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_of
| |
uvalu := let: Unit x _ := u in x.
#[export] HB.instance Definition _ := [isSub for uval].
#[export] HB.instance Definition _ := [Finite of unit_of by <:].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
uval
| |
unit1:= Unit (@GRing.unitr1 _).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit1
| |
unit_inv_proofu : (val u)^-1 \is a GRing.unit.
Proof. by rewrite unitrV ?(valP u). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_inv_proof
| |
unit_invu := Unit (unit_inv_proof u).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_inv
| |
unit_mul_proofu v : val u * val v \is a GRing.unit.
Proof. by rewrite (unitrMr _ (valP u)) ?(valP v). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_mul_proof
| |
unit_mulu v := Unit (unit_mul_proof u v).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_mul
| |
unit_muluA: associative unit_mul.
Proof. by move=> u v w; apply/val_inj/mulrA. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_muluA
| |
unit_mul1u: left_id unit1 unit_mul.
Proof. by move=> u; apply/val_inj/mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_mul1u
| |
unit_mulVu: left_inverse unit1 unit_inv unit_mul.
Proof. by move=> u; apply/val_inj/(mulVr (valP u)). Qed.
#[export] HB.instance Definition _ := Finite_isGroup.Build unit_of
unit_muluA unit_mul1u unit_mulVu.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
unit_mulVu
| |
val_unit1: val (1%g : unit_of) = 1. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
val_unit1
| |
val_unitMx y : val (x * y : unit_of)%g = val x * val y.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
val_unitM
|
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