# Library mathcomp.algebra.finalg

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B.                                  *)

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.

The algebraic part of the algebraic hierarchy for finite types This file clones the entire ssralg hierarchy for finite types; this allows type inference to function properly on expressions that mix combinatorial and algebraic operators, e.g., [set x + y | x in A, y in A]. finNmodType, finZmodType, finSemiRingType, finRingType, finComRingType, finComSemiRingType, finUnitRingType, finComUnitRingType, finIdomType, finFieldType, finLmodType, finLalgType, finAlgType, finUnitAlgType == the finite counterparts of nmodType, etc Note that a finFieldType is canonically decidable. This file also provides direct tie-ins with finite group theory: [finGroupMixin of R for +%R] == structures for R {unit R} == the type of units of R, which has a canonical group structure FinRing.unit R Ux == the element of {unit R} corresponding to x, where Ux : x \in GRing.unit 'U%act == the action by right multiplication of {unit R} on R, via FinRing.unit_act (This is also a group action.)

Local Open Scope ring_scope.

Set Implicit Arguments.

Module FinRing.

Import GRing.Theory.

#[short(type="finNmodType")]
HB.structure Definition Nmodule := {M of GRing.Nmodule M & Finite M}.

Module NmoduleExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finNmodType' 'of' T ]" := (Nmodule.clone T _)
(at level 0, format "[ 'finNmodType' 'of' T ]") : form_scope.
End NmoduleExports.

#[short(type="finZmodType")]
HB.structure Definition Zmodule := {M of GRing.Zmodule M & Finite M}.

Module ZmoduleExports.
Notation "[ 'finZmodType' 'of' T ]" := (Zmodule.clone T%type _)
(at level 0, format "[ 'finZmodType' 'of' T ]") : form_scope.
Notation "[ 'finGroupMixin' 'of' R 'for' +%R ]" :=
(at level 0, format "[ 'finGroupMixin' 'of' R 'for' +%R ]") : form_scope.
End ZmoduleExports.

#[short(type="finSemiRingType")]
HB.structure Definition SemiRing := {R of GRing.SemiRing R & Finite R}.

Module SemiRingExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finSemiRingType' 'of' T ]" := (SemiRing.clone T _)
(at level 0, format "[ 'finSemiRingType' 'of' T ]") : form_scope.
End SemiRingExports.

#[short(type="finRingType")]
HB.structure Definition Ring := {R of GRing.Ring R & Finite R}.

Module RingExports.
Notation "[ 'finRingType' 'of' T ]" := (Ring.clone T%type _)
(at level 0, format "[ 'finRingType' 'of' T ]") : form_scope.
End RingExports.

#[short(type="finComSemiRingType")]
HB.structure Definition ComSemiRing := {R of GRing.ComSemiRing R & Finite R}.

Module ComSemiRingExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finComSemiRingType' 'of' T ]" := (ComSemiRing.clone T _)
(at level 0, format "[ 'finComSemiRingType' 'of' T ]") : form_scope.
End ComSemiRingExports.

#[short(type="finComRingType")]
HB.structure Definition ComRing := {R of GRing.ComRing R & Finite R}.

Module ComRingExports.
Notation "[ 'finComRingType' 'of' T ]" := (ComRing.clone T%type _)
(at level 0, format "[ 'finComRingType' 'of' T ]") : form_scope.
End ComRingExports.

#[short(type="finUnitRingType")]
HB.structure Definition UnitRing := {R of GRing.UnitRing R & Finite R}.

Module UnitRingExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finUnitRingType' 'of' T ]" := (UnitRing.clone T%type _)
(at level 0, format "[ 'finUnitRingType' 'of' T ]") : form_scope.
End UnitRingExports.

#[short(type="finComUnitRingType")]
HB.structure Definition ComUnitRing := {R of GRing.ComUnitRing R & Finite R}.

Module ComUnitRingExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finComUnitRingType' 'of' T ]" := (ComUnitRing.clone T%type _)
(at level 0, format "[ 'finComUnitRingType' 'of' T ]") : form_scope.
End ComUnitRingExports.

#[short(type="finIdomainType")]
HB.structure Definition IntegralDomain :=
{R of GRing.IntegralDomain R & Finite R}.
#[deprecated(since="mathcomp 2.1.0",
note="Use finIdomainType (not available in mathcomp 2.0.0).")]
Notation finIntegralDomainType := finIdomainType.

Module IntegralDomainExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finIntegralDomainType' 'of' T ]" := (IntegralDomain.clone T%type _)
(at level 0, format "[ 'finIntegralDomainType' 'of' T ]") : form_scope.
End IntegralDomainExports.

#[short(type="finFieldType")]
HB.structure Definition Field := {R of GRing.Field R & Finite R}.

Module FieldExports.
Notation "[ 'finFieldType' 'of' T ]" := (Field.clone T%type _)
(at level 0, format "[ 'finFieldType' 'of' T ]") : form_scope.
End FieldExports.

#[short(type="finLmodType")]
HB.structure Definition Lmodule (R : ringType) :=
{M of GRing.Lmodule R M & Finite M}.

Module LmoduleExports.
Notation "[ 'finLmodType' R 'of' T ]" := (Lmodule.clone R T%type _)
(at level 0, format "[ 'finLmodType' R 'of' T ]") : form_scope.
End LmoduleExports.

#[short(type="finLalgType")]
HB.structure Definition Lalgebra (R : ringType) :=
{M of GRing.Lalgebra R M & Finite M}.

Module LalgebraExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finLalgType' R 'of' T ]" := (Lalgebra.clone R T%type _)
(at level 0, format "[ 'finLalgType' R 'of' T ]") : form_scope.
End LalgebraExports.

#[short(type="finAlgType")]
HB.structure Definition Algebra (R : ringType) :=
{M of GRing.Algebra R M & Finite M}.

Module AlgebraExports.
Notation "[ 'finAlgType' R 'of' T ]" := (Algebra.clone R T%type _)
(at level 0, format "[ 'finAlgType' R 'of' T ]") : form_scope.
End AlgebraExports.

#[short(type="finUnitAlgType")]
HB.structure Definition UnitAlgebra (R : unitRingType) :=
{M of GRing.UnitAlgebra R M & Finite M}.

Module UnitAlgebraExports.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'finUnitAlgType' R 'of' T ]" := (UnitAlgebra.clone R T%type _)
(at level 0, format "[ 'finUnitAlgType' R 'of' T ]") : form_scope.
End UnitAlgebraExports.

Group structures

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : Zmodule.type) := [finGroupMixin of R for +%R].
Coercion Zmodule_to_baseFinGroup (R : Zmodule.type) := BaseFinGroup.clone R _.
Coercion Zmodule_to_finGroup (R : Zmodule.type) := FinGroup.clone R _.

Variable U : finZmodType.
Implicit Types x y : U.

Lemma zmod1gE : 1%g = 0 :> U.
Lemma zmodVgE x : x^-1%g = - x.
Lemma zmodMgE x y : (x × y)%g = x + y.
Lemma zmodXgE n x : (x ^+ n)%g = x *+ n.
Lemma zmod_mulgC x y : commute x y.
Lemma zmod_abelian (A : {set U}) : abelian A.

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : Ring.type) := [finGroupMixin of R for +%R].
Coercion Ring_to_baseFinGroup (R : Ring.type) := BaseFinGroup.clone R _.
Coercion Ring_to_finGroup (R : Ring.type) := FinGroup.clone R _.

Definition is_inv (x y : R) := (x × y == 1) && (y × x == 1).
Definition unit := [qualify a x : R | [ y, is_inv x y]].
Definition inv x := odflt x (pick (is_inv x)).

Lemma mulVr : {in unit, left_inverse 1 inv *%R}.

Lemma mulrV : {in unit, right_inverse 1 inv *%R}.

Lemma intro_unit x y : y × x = 1 x × y = 1 x \is a unit.

Lemma invr_out : {in [predC unit], inv =1 id}.

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ComRing.type) := [finGroupMixin of R for +%R].
Coercion ComRing_to_baseFinGroup (R : ComRing.type) := BaseFinGroup.clone R _.
Coercion ComRing_to_finGroup (R : ComRing.type) := FinGroup.clone R _.

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : UnitRing.type) := [finGroupMixin of R for +%R].
Coercion UnitRing_to_baseFinGroup (R : UnitRing.type) := BaseFinGroup.clone R _.
Coercion UnitRing_to_finGroup (R : UnitRing.type) := FinGroup.clone R _.

Section UnitsGroup.

Variable R : finUnitRingType.

Inductive unit_of (phR : phant R) := Unit (x : R) of x \is a GRing.unit.
Bind Scope group_scope with unit_of.

Let phR := Phant R.
Local Notation uT := (unit_of phR).
Implicit Types u v : uT.
Definition uval u := let: Unit x _ := u in x.

#[export] HB.instance Definition _ := [isSub for uval].
#[export] HB.instance Definition _ := [Finite of uT by <:].

Definition unit1 := Unit phR (@GRing.unitr1 _).
Lemma unit_inv_proof u : (val u)^-1 \is a GRing.unit.
Definition unit_inv u := Unit phR (unit_inv_proof u).
Lemma unit_mul_proof u v : val u × val v \is a GRing.unit.
Definition unit_mul u v := Unit phR (unit_mul_proof u v).
Lemma unit_muluA : associative unit_mul.
Lemma unit_mul1u : left_id unit1 unit_mul.
Lemma unit_mulVu : left_inverse unit1 unit_inv unit_mul.

#[export] HB.instance Definition _ := isMulGroup.Build uT
unit_muluA unit_mul1u unit_mulVu.

Lemma val_unit1 : val (1%g : uT) = 1.
Lemma val_unitM x y : val (x × y : uT)%g = val x × val y.
Lemma val_unitV x : val (x^-1 : uT)%g = (val x)^-1.
Lemma val_unitX n x : val (x ^+ n : uT)%g = val x ^+ n.

Definition unit_act x u := x × val u.
Lemma unit_actE x u : unit_act x u = x × val u.

Canonical unit_action :=
@TotalAction _ _ unit_act (@mulr1 _) (fun _ _ _mulrA _ _ _).

Lemma unit_is_groupAction : @is_groupAction _ R setT setT unit_action.
Canonical unit_groupAction := GroupAction unit_is_groupAction.

End UnitsGroup.

Module Import UnitsGroupExports.
Bind Scope group_scope with unit_of.
Canonical unit_action.
Canonical unit_groupAction.
End UnitsGroupExports.

Notation unit R Ux := (Unit (Phant R) Ux).

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ComUnitRing.type) := [finGroupMixin of R for +%R].
Coercion ComUnitRing_to_baseFinGroup (R : ComUnitRing.type) :=
BaseFinGroup.clone R _.
Coercion 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 IntegralDomain_to_baseFinGroup (R : IntegralDomain.type) :=
BaseFinGroup.clone R _.
Coercion 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 Field_to_baseFinGroup (R : Field.type) := BaseFinGroup.clone R _.
Coercion Field_to_finGroup (R : Field.type) := FinGroup.clone R _.

Fixpoint sat e f :=
match f with
| GRing.Bool bb
| t1 == t2 ⇒ (GRing.eval e t1 == GRing.eval e t2)%bool
| GRing.Unit tGRing.eval e t \is a GRing.unit
| f1 f2sat e f1 && sat e f2
| f1 f2sat e f1 || sat e f2
| f1 ==> f2 ⇒ (sat e f1 ==> sat e f2)%bool
| ¬ f1~~ sat e f1
| (' 'X_k, f1) ⇒ [ x : F, sat (set_nth 0%R e k x) f1]
| (' 'X_k, f1) ⇒ [ x : F, sat (set_nth 0%R e k x) f1]
end%T.

Lemma decidable : GRing.decidable_field_axiom sat.

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ringType) (M : Lmodule.type R) :=
[finGroupMixin of M for +%R].

Coercion Lmodule_to_baseFinGroup (R : ringType) (M : Lmodule.type R) :=
BaseFinGroup.clone M _.
Coercion Lmodule_to_finGroup (R : ringType) (M : Lmodule.type R) : finGroupType :=
FinGroup.clone (M : Type) _.

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ringType) (M : Lalgebra.type R) :=
[finGroupMixin of M for +%R].
Coercion Lalgebra_to_baseFinGroup (R : ringType) (M : Lalgebra.type R) :=
BaseFinGroup.clone M _.
Coercion Lalgebra_to_finGroup (R : ringType) (M : Lalgebra.type R) :=
FinGroup.clone M _.

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ringType) (M : Algebra.type R) :=
[finGroupMixin of M for +%R].
Coercion Algebra_to_baseFinGroup (R : ringType) (M : Algebra.type R) :=
BaseFinGroup.clone M _.
Coercion Algebra_to_finGroup (R : ringType) (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 UnitAlgebra_to_baseFinGroup (R : unitRingType) (M : UnitAlgebra.type R) :=
BaseFinGroup.clone M _.
Coercion UnitAlgebra_to_finGroup (R : unitRingType) (M : UnitAlgebra.type R) :=
FinGroup.clone M _.

Module RegularExports.
End RegularExports.

Module Theory.

Definition zmod1gE := zmod1gE.
Definition zmodVgE := zmodVgE.
Definition zmodMgE := zmodMgE.
Definition zmodXgE := zmodXgE.
Definition zmod_mulgC := zmod_mulgC.
Definition zmod_abelian := zmod_abelian.
Definition val_unit1 := val_unit1.
Definition val_unitM := val_unitM.
Definition val_unitX := val_unitX.
Definition val_unitV := val_unitV.
Definition unit_actE := unit_actE.

End Theory.

End FinRing.

Import FinRing.

#[deprecated(since="mathcomp 2.1.0",
note="Use finIdomainType (not available in mathcomp 2.0.0).")]
Notation finIntegralDomainType := finIdomainType.

Lemma card_finRing_gt1 (R : finRingType) : 1 < #|R|.

Notation "{ 'unit' R }" := (unit_of (Phant R))
(at level 0, format "{ 'unit' R }") : type_scope.
Notation "''U'" := (unit_action _) (at level 8) : action_scope.
Notation "''U'" := (unit_groupAction _) (at level 8) : groupAction_scope.

Lemma card_finField_unit (F : finFieldType) : #|[set: {unit F}]| = #|F|.-1.

Finite Algebraic structure for bool