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 |
|---|---|---|---|---|---|---|
val_unitVx : val (x^-1 : unit_of)%g = (val x)^-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_unitV
| |
val_unitXn x : val (x ^+ n : unit_of)%g = val x ^+ n.
Proof. by case: n; last by elim=> //= n ->. 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_unitX
| |
unit_actx u := x * val 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_act
| |
unit_actEx u : unit_act x u = x * val 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
|
unit_actE
| |
unit_action:=
@TotalAction _ _ unit_act (@mulr1 _) (fun _ _ _ => mulrA _ _ _).
|
Canonical
|
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_action
| |
unit_is_groupAction: @is_groupAction _ R setT setT unit_action.
Proof.
move=> u _ /[1!inE]; apply/andP; split; first by apply/subsetP=> x /[1!inE].
by apply/morphicP=> x y _ _; rewrite !actpermE /= [_ u]mulrDl.
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_is_groupAction
| |
unit_groupAction:= GroupAction unit_is_groupAction.
|
Canonical
|
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_groupAction
| |
unit_action.
|
Canonical
|
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_action
| |
unit_groupAction.
|
Canonical
|
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_groupAction
| |
unitR Ux := (@Unit R%type _ Ux).
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ComUnitRing.type) := [finGroupMixin of R for +%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
|
unit
| |
ComUnitRing_to_baseFinGroup(R : ComUnitRing.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
|
ComUnitRing_to_baseFinGroup
| |
ComUnitRing_to_finGroup(R : ComUnitRing.type) :=
FinGroup.clone R _.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : IntegralDomain.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
|
ComUnitRing_to_finGroup
| |
IntegralDomain_to_baseFinGroup(R : IntegralDomain.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
|
IntegralDomain_to_baseFinGroup
| |
IntegralDomain_to_finGroup(R : IntegralDomain.type) :=
FinGroup.clone R _.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : Field.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
|
IntegralDomain_to_finGroup
| |
Field_to_baseFinGroup(R : Field.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
|
Field_to_baseFinGroup
| |
Field_to_finGroup(R : Field.type) := FinGroup.clone R _.
HB.factory Record isField F of Field F := {}.
HB.builders Context F of isField F.
|
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
|
Field_to_finGroup
| |
sate f :=
match f with
| GRing.Bool b => b
| t1 == t2 => (GRing.eval e t1 == GRing.eval e t2)%bool
| GRing.Unit t => GRing.eval e t \is a GRing.unit
| f1 /\ f2 => sat e f1 && sat e f2
| f1 \/ f2 => sat e f1 || sat e f2
| f1 ==> f2 => (sat e f1 ==> sat e f2)%bool
| ~ f1 => ~~ sat e f1
| ('exists 'X_k, f1) => [exists x : F, sat (set_nth 0%R e k x) f1]
| ('forall 'X_k, f1) => [forall x : F, sat (set_nth 0%R e k x) f1]
end%T.
|
Fixpoint
|
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
|
sat
| |
decidable: GRing.decidable_field_axiom sat.
Proof.
move=> e f; elim: f e;
try by move=> f1 IH1 f2 IH2 e /=; case IH1; case IH2; constructor; tauto.
- by move=> b e; apply: idP.
- by move=> t1 t2 e; apply: eqP.
- by move=> t e; apply: idP.
- by move=> f IH e /=; case: IH; constructor.
- by move=> i f IH e; apply: (iffP existsP) => [] [x fx]; exists x; apply/IH.
by move=> i f IH e; apply: (iffP forallP) => f_ x; apply/IH.
Qed.
HB.instance Definition _ := GRing.Field_isDecField.Build F decidable.
HB.end.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : nzRingType) (M : Lmodule.type R) :=
[finGroupMixin of M 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
|
decidable
| |
Lmodule_to_baseFinGroup(R : nzRingType) (M : Lmodule.type R) :=
FinStarMonoid.clone M _.
|
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
|
Lmodule_to_baseFinGroup
| |
Lmodule_to_finGroup(R : nzRingType) (M : Lmodule.type R)
: finGroupType :=
FinGroup.clone (M : Type) _.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : nzRingType) (M : Lalgebra.type R) :=
[finGroupMixin of M 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
|
Lmodule_to_finGroup
| |
Lalgebra_to_baseFinGroup(R : nzRingType) (M : Lalgebra.type R) :=
FinStarMonoid.clone M _.
|
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
|
Lalgebra_to_baseFinGroup
| |
Lalgebra_to_finGroup(R : nzRingType) (M : Lalgebra.type R) :=
FinGroup.clone M _.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : nzRingType) (M : Algebra.type R) :=
[finGroupMixin of M 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
|
Lalgebra_to_finGroup
| |
Algebra_to_baseFinGroup(R : nzRingType) (M : Algebra.type R) :=
FinStarMonoid.clone M _.
|
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
|
Algebra_to_baseFinGroup
| |
Algebra_to_finGroup(R : nzRingType) (M : Algebra.type R) :=
FinGroup.clone M _.
#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : unitRingType) (M : UnitAlgebra.type R) :=
[finGroupMixin of M 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
|
Algebra_to_finGroup
| |
UnitAlgebra_to_baseFinGroup(R : unitRingType) (M : UnitAlgebra.type R) := FinStarMonoid.clone M _.
|
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
|
UnitAlgebra_to_baseFinGroup
| |
UnitAlgebra_to_finGroup(R : unitRingType) (M : UnitAlgebra.type R) :=
FinGroup.clone M _.
|
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
|
UnitAlgebra_to_finGroup
| |
Definition_ (R : finType) := Finite.on R^o.
HB.instance Definition _ (R : finNmodType) := Nmodule.on R^o.
HB.instance Definition _ (R : finZmodType) := Zmodule.on R^o.
HB.instance Definition _ (R : finPzSemiRingType) := PzSemiRing.on R^o.
HB.instance Definition _ (R : finPzRingType) := PzRing.on R^o.
HB.instance Definition _ (R : finNzSemiRingType) := NzSemiRing.on R^o.
HB.instance Definition _ (R : finNzRingType) := NzRing.on R^o.
HB.instance Definition _ (R : finComPzSemiRingType) := PzSemiRing.on R^o.
HB.instance Definition _ (R : finComPzRingType) := PzRing.on R^o.
HB.instance Definition _ (R : finComNzSemiRingType) := NzSemiRing.on R^o.
HB.instance Definition _ (R : finComNzRingType) := NzRing.on R^o.
HB.instance Definition _ (R : finUnitRingType) := NzRing.on R^o.
HB.instance Definition _ (R : finComUnitRingType) := NzRing.on R^o.
HB.instance Definition _ (R : finIdomainType) := NzRing.on R^o.
HB.instance Definition _ (R : finFieldType) := NzRing.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 finset fingroup morphism perm action",
"From mathcomp Require Import ssralg countalg"
] |
algebra/finalg.v
|
Definition
| |
zmod1gE:= zmod1gE.
|
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
|
zmod1gE
| |
zmodVgE:= zmodVgE.
|
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
|
zmodVgE
| |
zmodMgE:= zmodMgE.
|
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
|
zmodMgE
| |
zmodXgE:= zmodXgE.
|
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
|
zmodXgE
| |
zmod_mulgC:= zmod_mulgC.
|
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
|
zmod_mulgC
| |
zmod_abelian:= zmod_abelian.
|
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
|
zmod_abelian
| |
val_unit1:= val_unit1.
|
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
|
val_unit1
| |
val_unitM:= val_unitM.
|
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
|
val_unitM
| |
val_unitX:= val_unitX.
|
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
|
val_unitX
| |
val_unitV:= val_unitV.
|
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
|
val_unitV
| |
unit_actE:= unit_actE.
|
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_actE
| |
finSemiRingType:= (finNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use finNzRingType 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
|
finSemiRingType
| |
finRingType:= (finNzRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use finComNzSemiRingType 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
|
finRingType
| |
finComSemiRingType:= (finComNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use finComNzRingType 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
|
finComSemiRingType
| |
finComRingType:= (finComNzRingType) (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
|
finComRingType
| |
card_finNzRing_gt1(R : finNzRingType) : 1 < #|R|.
Proof. by rewrite (cardD1 0) (cardD1 1) !inE GRing.oner_neq0. Qed.
#[deprecated(since="mathcomp 2.4.0",
note="Use card_finNzRing_gt1 instead.")]
|
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
|
card_finNzRing_gt1
| |
card_finRing_gt1:= (card_finNzRing_gt1) (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
|
card_finRing_gt1
| |
card_finField_unit(F : finFieldType) : #|[set: {unit F}]| = #|F|.-1.
Proof.
rewrite -(cardC1 0) cardsT card_sub; apply: eq_card => x.
by rewrite GRing.unitfE.
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
|
card_finField_unit
| |
ratio:= mkRatio { frac :> R * R; _ : frac.2 != 0 }.
HB.instance Definition _ := [isSub for frac].
HB.instance Definition _ := [Choice of ratio by <:].
|
Inductive
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
ratio
| |
denom_ratioP: forall f : ratio, f.2 != 0. Proof. by case. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
denom_ratioP
| |
ratio0:= (@mkRatio (0, 1) (oner_neq0 _)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
ratio0
| |
Ratiox y : ratio := insubd ratio0 (x, y).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Ratio
| |
numer_Ratiox y : y != 0 -> (Ratio x y).1 = x.
Proof. by move=> ny0; rewrite /Ratio /insubd insubT. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
numer_Ratio
| |
denom_Ratiox y : y != 0 -> (Ratio x y).2 = y.
Proof. by move=> ny0; rewrite /Ratio /insubd insubT. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
denom_Ratio
| |
numden_Ratio:= (numer_Ratio, denom_Ratio).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
numden_Ratio
| |
Ratio_spec(n d : R) : ratio -> R -> R -> Type :=
| RatioNull of d = 0 : Ratio_spec n d ratio0 n 0
| RatioNonNull (d_neq0 : d != 0) :
Ratio_spec n d (@mkRatio (n, d) d_neq0) n d.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Ratio_spec
| |
RatioPn d : Ratio_spec n d (Ratio n d) n d.
Proof.
rewrite /Ratio /insubd; case: insubP=> /= [x /= d_neq0 hx|].
have ->: x = @mkRatio (n, d) d_neq0 by apply: val_inj.
by constructor.
by rewrite negbK=> /eqP hx; rewrite {2}hx; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
RatioP
| |
Ratio0x : Ratio x 0 = ratio0.
Proof. by rewrite /Ratio /insubd; case: insubP; rewrite //= eqxx. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Ratio0
| |
equivfx y := equivf_notation x y.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf
| |
equivfEx y : equivf x y = equivf_notation x y.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivfE
| |
equivf_refl: reflexive equivf.
Proof. by move=> x; rewrite /equivf mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_refl
| |
equivf_sym: symmetric equivf.
Proof. by move=> x y; rewrite /equivf eq_sym; congr (_==_); rewrite mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_sym
| |
equivf_trans: transitive equivf.
Proof.
move=> [x Px] [y Py] [z Pz]; rewrite /equivf /= mulrC => /eqP xy /eqP yz.
by rewrite -(inj_eq (mulfI Px)) mulrA xy -mulrA yz mulrCA.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_trans
| |
equivf_equiv:= EquivRel equivf equivf_refl equivf_sym equivf_trans.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_equiv
| |
type:= {eq_quot equivf}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
type
| |
Definition_ : EqQuotient _ equivf type := EqQuotient.on type.
HB.instance Definition _ := Choice.on type.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Definition
| |
equivf_def(x y : ratio R) : x == y %[mod type]
= (\n_x * \d_y == \d_x * \n_y).
Proof. by rewrite eqmodE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_def
| |
equivf_rx : \n_x * \d_(repr (\pi_type x)) = \d_x * \n_(repr (\pi_type x)).
Proof. by apply/eqP; rewrite -equivf_def reprK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_r
| |
equivf_lx : \n_(repr (\pi_type x)) * \d_x = \d_(repr (\pi_type x)) * \n_x.
Proof. by apply/eqP; rewrite -equivf_def reprK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf_l
| |
numer0x : (\n_x == 0) = (x == (ratio0 R) %[mod_eq equivf]).
Proof. by rewrite eqmodE /= !equivfE // mulr1 mulr0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
numer0
| |
Ratio_numden: forall x, Ratio \n_x \d_x = x.
Proof.
case=> [[n d] /= nd]; rewrite /Ratio /insubd; apply: val_inj=> /=.
by case: insubP=> //=; rewrite nd.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Ratio_numden
| |
tofrac:= lift_embed type (fun x : R => Ratio x 1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac
| |
tofrac_pi_morph:= PiEmbed tofrac.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_pi_morph
| |
addfx y : dom := Ratio (\n_x * \d_y + \n_y * \d_x) (\d_x * \d_y).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
addf
| |
add:= lift_op2 type addf.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
add
| |
pi_add: {morph \pi : x y / addf x y >-> add x y}.
Proof.
move=> x y; unlock add; apply/eqmodP; rewrite /= equivfE /addf /=.
rewrite !numden_Ratio ?mulf_neq0 ?domP // mulrDr mulrDl; apply/eqP.
symmetry; rewrite (AC (2*2) (3*1*2*4)) (AC (2*2) (3*2*1*4))/=.
by rewrite !equivf_l (ACl ((2*3)*(1*4))) (ACl ((2*3)*(4*1)))/=.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_add
| |
pi_add_morph:= PiMorph2 pi_add.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_add_morph
| |
oppfx : dom := Ratio (- \n_x) \d_x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
oppf
| |
opp:= lift_op1 type oppf.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
opp
| |
pi_opp: {morph \pi : x / oppf x >-> opp x}.
Proof.
move=> x; unlock opp; apply/eqmodP; rewrite /= /equivf /oppf /=.
by rewrite !numden_Ratio ?(domP,mulf_neq0) // mulNr mulrN -equivf_r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_opp
| |
pi_opp_morph:= PiMorph1 pi_opp.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_opp_morph
| |
mulfx y : dom := Ratio (\n_x * \n_y) (\d_x * \d_y).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mulf
| |
mul:= lift_op2 type mulf.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mul
| |
pi_mul: {morph \pi : x y / mulf x y >-> mul x y}.
Proof.
move=> x y; unlock mul; apply/eqmodP=> /=.
rewrite equivfE /= /addf /= !numden_Ratio ?mulf_neq0 ?domP //.
by rewrite mulrACA !equivf_r mulrACA.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_mul
| |
pi_mul_morph:= PiMorph2 pi_mul.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_mul_morph
| |
invfx : dom := Ratio \d_x \n_x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
invf
| |
inv:= lift_op1 type invf.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
inv
| |
pi_inv: {morph \pi : x / invf x >-> inv x}.
Proof.
move=> x; unlock inv; apply/eqmodP=> /=; rewrite equivfE /invf eq_sym.
do 2?case: RatioP=> /= [/eqP|];
rewrite ?mul0r ?mul1r -?equivf_def ?numer0 ?reprK //.
by move=> hx /eqP hx'; rewrite hx' eqxx in hx.
by move=> /eqP ->; rewrite eqxx.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_inv
| |
pi_inv_morph:= PiMorph1 pi_inv.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
pi_inv_morph
| |
addA: associative add.
Proof.
elim/quotW=> x; elim/quotW=> y; elim/quotW=> z; rewrite !piE.
rewrite /addf /= !numden_Ratio ?mulf_neq0 ?domP // !mulrDl.
by rewrite !mulrA !addrA ![_ * _ * \d_x]mulrAC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
addA
| |
addC: commutative add.
Proof.
by elim/quotW=> x; elim/quotW=> y; rewrite !piE /addf addrC [\d__ * _]mulrC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
addC
| |
add0_l: left_id 0%:F add.
Proof.
elim/quotW=> x; rewrite !piE /addf !numden_Ratio ?oner_eq0 //.
by rewrite mul0r mul1r mulr1 add0r Ratio_numden.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
add0_l
| |
addN_l: left_inverse 0%:F opp add.
Proof.
elim/quotW=> x; apply/eqP; rewrite piE /equivf.
rewrite /addf /oppf !numden_Ratio ?(oner_eq0, mulf_neq0, domP) //.
by rewrite mulr1 mulr0 mulNr addNr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
addN_l
| |
Definition_ := GRing.isZmodule.Build type addA addC add0_l addN_l.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Definition
| |
mulA: associative mul.
Proof.
elim/quotW=> x; elim/quotW=> y; elim/quotW=> z; rewrite !piE.
by rewrite /mulf !numden_Ratio ?mulf_neq0 ?domP // !mulrA.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mulA
| |
mulC: commutative mul.
Proof.
elim/quotW=> x; elim/quotW=> y; rewrite !piE /mulf.
by rewrite [_ * (\d_x)]mulrC [_ * (\n_x)]mulrC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mulC
| |
mul1_l: left_id 1%:F mul.
Proof.
elim/quotW=> x; rewrite !piE /mulf.
by rewrite !numden_Ratio ?oner_eq0 // !mul1r Ratio_numden.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mul1_l
| |
mul_addl: left_distributive mul add.
Proof.
elim/quotW=> x; elim/quotW=> y; elim/quotW=> z; apply/eqP.
rewrite !piE /equivf /mulf /addf !numden_Ratio ?mulf_neq0 ?domP //; apply/eqP.
rewrite !(mulrDr, mulrDl) (AC (3*(2*2)) (4*2*7*((1*3)*(6*5))))/=.
by rewrite [X in _ + X](AC (3*(2*2)) (4*6*7*((1*3)*(2*5))))/=.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mul_addl
| |
nonzero1: 1%:F != 0%:F :> type.
Proof. by rewrite piE equivfE !numden_Ratio ?mul1r ?oner_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
nonzero1
| |
Definition_ :=
GRing.Zmodule_isComNzRing.Build type mulA mulC mul1_l mul_addl nonzero1.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Definition
| |
mulV_l: forall a, a != 0%:F -> mul (inv a) a = 1%:F.
Proof.
elim/quotW=> x /=; rewrite !piE.
rewrite /equivf !numden_Ratio ?oner_eq0 // mulr1 mulr0=> nx0.
apply/eqmodP; rewrite /= equivfE.
by rewrite !numden_Ratio ?(oner_eq0, mulf_neq0, domP) // !mulr1 mulrC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
mulV_l
| |
inv0: inv 0%:F = 0%:F.
Proof.
rewrite !piE /invf !numden_Ratio ?oner_eq0 // /Ratio /insubd.
do 2?case: insubP; rewrite //= ?eqxx ?oner_eq0 // => u _ hu _.
by congr \pi; apply: val_inj; rewrite /= hu.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
inv0
| |
Definition_ := GRing.ComNzRing_isField.Build type mulV_l inv0.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Definition
|
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