(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) 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. (*****************************************************************************) (* This file clones the entire ssralg hierachy 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]). *) (* finZmodType, finRingType, finComRingType, finUnitRingType, *) (* finComUnitRingType, finIdomType, finFieldType finLmodType, *) (* finLalgType finAlgType finUnitAlgType *) (* == the finite counterparts of zmodType, etc. *) (* Note that a finFieldType is canonically decidable. All these structures *) (* can be derived using [xxxType of T] forms, e.g., if R has both canonical *) (* finType and ringType structures, then *) (* Canonical R_finRingType := Eval hnf in [finRingType of R]. *) (* declares the derived finRingType structure for R. As the implementation *) (* of the derivation is somewhat involved, the Eval hnf normalization is *) (* strongly recommended. *) (* This file also provides direct tie-ins with finite group theory: *) (* [baseFinGroupType of R for +%R] == the (canonical) additive group *) (* [finGroupType 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. Unset Strict Implicit. Unset Printing Implicit Defensive. Module FinRing. Local Notation mixin_of T b := (Finite.mixin_of (EqType T b)). Section Generic. (* Implicits *) Variables (type base_type : Type) (class_of base_of : Type -> Type). Variable to_choice : forall T, base_of T -> Choice.class_of T. Variable base_sort : base_type -> Type. (* Explicits *) Variable Pack : forall T, class_of T -> type. Variable Class : forall T b, mixin_of T (to_choice b) -> class_of T. Variable base_class : forall bT, base_of (base_sort bT). Definition gen_pack T := fun bT b & phant_id (base_class bT) b => fun fT m & phant_id (Finite.class fT) (Finite.Class m) => Pack (@Class T b m). End Generic. Arguments gen_pack [type base_type class_of base_of to_choice base_sort]. Local Notation fin_ c := (@Finite.Class _ c c). Local Notation do_pack pack T := (pack T _ _ id _ _ id). Import GRing.Theory. Definition groupMixin V := FinGroup.Mixin (@addrA V) (@add0r V) (@addNr V). Local Notation base_group T vT fT := (@FinGroup.PackBase T (groupMixin vT) (Finite.class fT)). Local Notation fin_group B V := (@FinGroup.Pack B (@addNr V)). Module Zmodule. Section ClassDef. Set Primitive Projections. Record class_of M := Class { base : GRing.Zmodule.class_of M; mixin : mixin_of M base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.Zmodule.class_of. Local Coercion base2 R (c : class_of R) : CountRing.Zmodule.class_of R := CountRing.Zmodule.Class c (mixin c). Local Coercion mixin : class_of >-> mixin_of. Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.Zmodule.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition zmod_finType := @Finite.Pack zmodType (fin_ class). Definition zmod_baseFinGroupType := base_group zmodType zmodType finType. Definition zmod_finGroupType := fin_group zmod_baseFinGroupType zmodType. Definition countZmod_finType := @Finite.Pack countZmodType (fin_ class). Definition countZmod_baseFinGroupType := base_group countZmodType zmodType finType. Definition countZmod_finGroupType := fin_group countZmod_baseFinGroupType zmodType. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.Zmodule.class_of. Coercion base2 : class_of >-> CountRing.Zmodule.class_of. Coercion mixin : class_of >-> mixin_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Canonical zmod_finType. Canonical zmod_baseFinGroupType. Canonical zmod_finGroupType. Canonical countZmod_finType. Canonical countZmod_baseFinGroupType. Canonical countZmod_finGroupType. Notation finZmodType := type. Notation "[ 'finZmodType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finZmodType' 'of' T ]") : form_scope. Notation "[ 'baseFinGroupType' 'of' R 'for' +%R ]" := (BaseFinGroupType R (groupMixin _)) (at level 0, format "[ 'baseFinGroupType' 'of' R 'for' +%R ]") : form_scope. Notation "[ 'finGroupType' 'of' R 'for' +%R ]" := (@FinGroup.clone R _ (finGroupType _) id _ id) (at level 0, format "[ 'finGroupType' 'of' R 'for' +%R ]") : form_scope. End Exports. End Zmodule. Import Zmodule.Exports. Section AdditiveGroup. Variable U : finZmodType. Implicit Types x y : U. Lemma zmod1gE : 1%g = 0 :> U. Proof. by []. Qed. Lemma zmodVgE x : x^-1%g = - x. Proof. by []. Qed. Lemma zmodMgE x y : (x * y)%g = x + y. Proof. by []. Qed. Lemma zmodXgE n x : (x ^+ n)%g = x *+ n. Proof. by []. Qed. Lemma zmod_mulgC x y : commute x y. Proof. exact: addrC. Qed. Lemma zmod_abelian (A : {set U}) : abelian A. Proof. by apply/centsP=> x _ y _; apply: zmod_mulgC. Qed. End AdditiveGroup. Module Ring. Section ClassDef. Set Primitive Projections. Record class_of R := Class { base : GRing.Ring.class_of R; mixin : mixin_of R base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.Ring.class_of. Local Coercion base2 R (c : class_of R) : CountRing.Ring.class_of R := CountRing.Ring.Class c (mixin c). Local Coercion base3 R (c : class_of R) : Zmodule.class_of R := Zmodule.Class (mixin c). Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.Ring.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition ring_finType := @Finite.Pack ringType (fin_ class). Definition ring_baseFinGroupType := base_group ringType zmodType finType. Definition ring_finGroupType := fin_group ring_baseFinGroupType zmodType. Definition ring_finZmodType := @Zmodule.Pack ringType class. Definition countRing_finType := @Finite.Pack countRingType (fin_ class). Definition countRing_baseFinGroupType := base_group countRingType zmodType finType. Definition countRing_finGroupType := fin_group countRing_baseFinGroupType zmodType. Definition countRing_finZmodType := @Zmodule.Pack countRingType class. End ClassDef. Module Import Exports. Coercion base : class_of >-> GRing.Ring.class_of. Coercion base2 : class_of >-> CountRing.Ring.class_of. Coercion base3 : class_of >-> Zmodule.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Canonical ring_finType. Canonical ring_baseFinGroupType. Canonical ring_finGroupType. Canonical ring_finZmodType. Canonical countRing_finType. Canonical countRing_baseFinGroupType. Canonical countRing_finGroupType. Canonical countRing_finZmodType. Notation finRingType := type. Notation "[ 'finRingType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finRingType' 'of' T ]") : form_scope. End Exports. Section Unit. Variable R : finRingType. Definition is_inv (x y : R) := (x * y == 1) && (y * x == 1). Definition unit := [qualify a x : R | [exists y, is_inv x y]]. Definition inv x := odflt x (pick (is_inv x)). Lemma 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 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 intro_unit x 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 invr_out : {in [predC unit], inv =1 id}. Proof. rewrite /inv => x nUx; case: pickP => // y invxy. by case/existsP: nUx; exists y. Qed. Definition UnitMixin := GRing.UnitRing.Mixin mulVr mulrV intro_unit invr_out. End Unit. End Ring. Import Ring.Exports. Module ComRing. Section ClassDef. Set Primitive Projections. Record class_of R := Class { base : GRing.ComRing.class_of R; mixin : mixin_of R base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.ComRing.class_of. Local Coercion base2 R (c : class_of R) : CountRing.ComRing.class_of R := CountRing.ComRing.Class c (mixin c). Local Coercion base3 R (c : class_of R) : Ring.class_of R := Ring.Class (mixin c). Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.ComRing.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition comRingType := @GRing.ComRing.Pack cT class. Definition countComRingType := @CountRing.ComRing.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition comRing_finType := @Finite.Pack comRingType (fin_ class). Definition comRing_baseFinGroupType := base_group comRingType zmodType finType. Definition comRing_finGroupType := fin_group comRing_baseFinGroupType zmodType. Definition comRing_finZmodType := @Zmodule.Pack comRingType class. Definition comRing_finRingType := @Ring.Pack comRingType class. Definition countComRing_finType := @Finite.Pack countComRingType (fin_ class). Definition countComRing_baseFinGroupType := base_group countComRingType zmodType finType. Definition countComRing_finGroupType := fin_group countComRing_baseFinGroupType zmodType. Definition countComRing_finZmodType := @Zmodule.Pack countComRingType class. Definition countComRing_finRingType := @Ring.Pack countComRingType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.ComRing.class_of. Coercion base2 : class_of >-> CountRing.ComRing.class_of. Coercion base3 : class_of >-> Ring.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion comRingType : type >-> GRing.ComRing.type. Canonical comRingType. Coercion countComRingType : type >-> CountRing.ComRing.type. Canonical countComRingType. Canonical comRing_finType. Canonical comRing_baseFinGroupType. Canonical comRing_finGroupType. Canonical comRing_finZmodType. Canonical comRing_finRingType. Canonical countComRing_finType. Canonical countComRing_baseFinGroupType. Canonical countComRing_finGroupType. Canonical countComRing_finZmodType. Canonical countComRing_finRingType. Notation finComRingType := FinRing.ComRing.type. Notation "[ 'finComRingType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finComRingType' 'of' T ]") : form_scope. End Exports. End ComRing. Import ComRing.Exports. Module UnitRing. Section ClassDef. Set Primitive Projections. Record class_of R := Class { base : GRing.UnitRing.class_of R; mixin : mixin_of R base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.UnitRing.class_of. Local Coercion base2 R (c : class_of R) : CountRing.UnitRing.class_of R := CountRing.UnitRing.Class c (mixin c). Local Coercion base3 R (c : class_of R) : Ring.class_of R := Ring.Class (mixin c). Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.UnitRing.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition unitRingType := @GRing.UnitRing.Pack cT class. Definition countUnitRingType := @CountRing.UnitRing.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition unitRing_finType := @Finite.Pack unitRingType (fin_ class). Definition unitRing_baseFinGroupType := base_group unitRingType zmodType finType. Definition unitRing_finGroupType := fin_group unitRing_baseFinGroupType zmodType. Definition unitRing_finZmodType := @Zmodule.Pack unitRingType class. Definition unitRing_finRingType := @Ring.Pack unitRingType class. Definition countUnitRing_finType := @Finite.Pack countUnitRingType (fin_ class). Definition countUnitRing_baseFinGroupType := base_group countUnitRingType zmodType finType. Definition countUnitRing_finGroupType := fin_group countUnitRing_baseFinGroupType zmodType. Definition countUnitRing_finZmodType := @Zmodule.Pack countUnitRingType class. Definition countUnitRing_finRingType := @Ring.Pack countUnitRingType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.UnitRing.class_of. Coercion base2 : class_of >-> CountRing.UnitRing.class_of. Coercion base3 : class_of >-> Ring.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion unitRingType : type >-> GRing.UnitRing.type. Canonical unitRingType. Coercion countUnitRingType : type >-> CountRing.UnitRing.type. Canonical countUnitRingType. Canonical unitRing_finType. Canonical unitRing_baseFinGroupType. Canonical unitRing_finGroupType. Canonical unitRing_finZmodType. Canonical unitRing_finRingType. Canonical countUnitRing_finType. Canonical countUnitRing_baseFinGroupType. Canonical countUnitRing_finGroupType. Canonical countUnitRing_finZmodType. Canonical countUnitRing_finRingType. Notation finUnitRingType := FinRing.UnitRing.type. Notation "[ 'finUnitRingType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finUnitRingType' 'of' T ]") : form_scope. End Exports. End UnitRing. Import UnitRing.Exports. 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. Canonical unit_subType := [subType for uval]. Definition unit_eqMixin := Eval hnf in [eqMixin of uT by <:]. Canonical unit_eqType := Eval hnf in EqType uT unit_eqMixin. Definition unit_choiceMixin := [choiceMixin of uT by <:]. Canonical unit_choiceType := Eval hnf in ChoiceType uT unit_choiceMixin. Definition unit_countMixin := [countMixin of uT by <:]. Canonical unit_countType := Eval hnf in CountType uT unit_countMixin. Canonical unit_subCountType := Eval hnf in [subCountType of uT]. Definition unit_finMixin := [finMixin of uT by <:]. Canonical unit_finType := Eval hnf in FinType uT unit_finMixin. Canonical unit_subFinType := Eval hnf in [subFinType of uT]. Definition unit1 := Unit phR (@GRing.unitr1 _). Lemma unit_inv_proof u : (val u)^-1 \is a GRing.unit. Proof. by rewrite unitrV ?(valP u). Qed. Definition unit_inv u := Unit phR (unit_inv_proof u). Lemma unit_mul_proof u v : val u * val v \is a GRing.unit. Proof. by rewrite (unitrMr _ (valP u)) ?(valP v). Qed. Definition unit_mul u v := Unit phR (unit_mul_proof u v). Lemma unit_muluA : associative unit_mul. Proof. by move=> u v w; apply/val_inj/mulrA. Qed. Lemma unit_mul1u : left_id unit1 unit_mul. Proof. by move=> u; apply/val_inj/mul1r. Qed. Lemma unit_mulVu : left_inverse unit1 unit_inv unit_mul. Proof. by move=> u; apply/val_inj/(mulVr (valP u)). Qed. Definition unit_GroupMixin := FinGroup.Mixin unit_muluA unit_mul1u unit_mulVu. Canonical unit_baseFinGroupType := Eval hnf in BaseFinGroupType uT unit_GroupMixin. Canonical unit_finGroupType := Eval hnf in FinGroupType unit_mulVu. Lemma val_unit1 : val (1%g : uT) = 1. Proof. by []. Qed. Lemma val_unitM x y : val (x * y : uT)%g = val x * val y. Proof. by []. Qed. Lemma val_unitV x : val (x^-1 : uT)%g = (val x)^-1. Proof. by []. Qed. Lemma val_unitX n x : val (x ^+ n : uT)%g = val x ^+ n. Proof. by case: n; last by elim=> //= n ->. Qed. Definition unit_act x u := x * val u. Lemma unit_actE x u : unit_act x u = x * val u. Proof. by []. Qed. Canonical unit_action := @TotalAction _ _ unit_act (@mulr1 _) (fun _ _ _ => mulrA _ _ _). Lemma 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. Canonical unit_groupAction := GroupAction unit_is_groupAction. End UnitsGroup. Module Import UnitsGroupExports. Bind Scope group_scope with unit_of. Canonical unit_subType. Canonical unit_eqType. Canonical unit_choiceType. Canonical unit_countType. Canonical unit_subCountType. Canonical unit_finType. Canonical unit_subFinType. Canonical unit_baseFinGroupType. Canonical unit_finGroupType. Canonical unit_action. Canonical unit_groupAction. End UnitsGroupExports. Notation unit R Ux := (Unit (Phant R) Ux). Module ComUnitRing. Section ClassDef. Set Primitive Projections. Record class_of R := Class { base : GRing.ComUnitRing.class_of R; mixin : mixin_of R base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.ComUnitRing.class_of. Local Coercion base2 R (c : class_of R) : CountRing.ComUnitRing.class_of R := CountRing.ComUnitRing.Class c (mixin c). Local Coercion base3 R (c : class_of R) : ComRing.class_of R := ComRing.Class (mixin c). Local Coercion base4 R (c : class_of R) : UnitRing.class_of R := @UnitRing.Class R (base c) (mixin c). Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.ComUnitRing.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition comRingType := @GRing.ComRing.Pack cT class. Definition countComRingType := @CountRing.ComRing.Pack cT class. Definition finComRingType := @ComRing.Pack cT class. Definition unitRingType := @GRing.UnitRing.Pack cT class. Definition countUnitRingType := @CountRing.UnitRing.Pack cT class. Definition finUnitRingType := @UnitRing.Pack cT class. Definition comUnitRingType := @GRing.ComUnitRing.Pack cT class. Definition countComUnitRingType := @CountRing.ComUnitRing.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition comUnitRing_finType := @Finite.Pack comUnitRingType (fin_ class). Definition comUnitRing_baseFinGroupType := base_group comUnitRingType zmodType finType. Definition comUnitRing_finGroupType := fin_group comUnitRing_baseFinGroupType zmodType. Definition comUnitRing_finZmodType := @Zmodule.Pack comUnitRingType class. Definition comUnitRing_finRingType := @Ring.Pack comUnitRingType class. Definition comUnitRing_finComRingType := @ComRing.Pack comUnitRingType class. Definition comUnitRing_finUnitRingType := @UnitRing.Pack comUnitRingType class. Definition countComUnitRing_finType := @Finite.Pack countComUnitRingType (fin_ class). Definition countComUnitRing_baseFinGroupType := base_group countComUnitRingType zmodType finType. Definition countComUnitRing_finGroupType := fin_group countComUnitRing_baseFinGroupType zmodType. Definition countComUnitRing_finZmodType := @Zmodule.Pack countComUnitRingType class. Definition countComUnitRing_finRingType := @Ring.Pack countComUnitRingType class. Definition countComUnitRing_finComRingType := @ComRing.Pack countComUnitRingType class. Definition countComUnitRing_finUnitRingType := @UnitRing.Pack countComUnitRingType class. Definition unitRing_finComRingType := @ComRing.Pack unitRingType class. Definition countUnitRing_finComRingType := @ComRing.Pack countUnitRingType class. Definition comRing_finUnitRingType := @UnitRing.Pack comRingType class. Definition countComRing_finUnitRingType := @UnitRing.Pack countComRingType class. Definition finComRing_finUnitRingType := @UnitRing.Pack finComRingType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.ComUnitRing.class_of. Coercion base2 : class_of >-> CountRing.ComUnitRing.class_of. Coercion base3 : class_of >-> ComRing.class_of. Coercion base4 : class_of >-> UnitRing.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion comRingType : type >-> GRing.ComRing.type. Canonical comRingType. Coercion countComRingType : type >-> CountRing.ComRing.type. Canonical countComRingType. Coercion finComRingType : type >-> ComRing.type. Canonical finComRingType. Coercion unitRingType : type >-> GRing.UnitRing.type. Canonical unitRingType. Coercion countUnitRingType : type >-> CountRing.UnitRing.type. Canonical countUnitRingType. Coercion finUnitRingType : type >-> UnitRing.type. Canonical finUnitRingType. Coercion comUnitRingType : type >-> GRing.ComUnitRing.type. Canonical comUnitRingType. Coercion countComUnitRingType : type >-> CountRing.ComUnitRing.type. Canonical countComUnitRingType. Canonical comUnitRing_finType. Canonical comUnitRing_baseFinGroupType. Canonical comUnitRing_finGroupType. Canonical comUnitRing_finZmodType. Canonical comUnitRing_finRingType. Canonical comUnitRing_finComRingType. Canonical comUnitRing_finUnitRingType. Canonical countComUnitRing_finType. Canonical countComUnitRing_baseFinGroupType. Canonical countComUnitRing_finGroupType. Canonical countComUnitRing_finZmodType. Canonical countComUnitRing_finRingType. Canonical countComUnitRing_finComRingType. Canonical countComUnitRing_finUnitRingType. Canonical unitRing_finComRingType. Canonical countUnitRing_finComRingType. Canonical comRing_finUnitRingType. Canonical countComRing_finUnitRingType. Canonical finComRing_finUnitRingType. Notation finComUnitRingType := FinRing.ComUnitRing.type. Notation "[ 'finComUnitRingType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finComUnitRingType' 'of' T ]") : form_scope. End Exports. End ComUnitRing. Import ComUnitRing.Exports. Module IntegralDomain. Section ClassDef. Set Primitive Projections. Record class_of R := Class { base : GRing.IntegralDomain.class_of R; mixin : mixin_of R base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.IntegralDomain.class_of. Local Coercion base2 R (c : class_of R) : CountRing.IntegralDomain.class_of R := CountRing.IntegralDomain.Class c (mixin c). Local Coercion base3 R (c : class_of R) : ComUnitRing.class_of R := ComUnitRing.Class (mixin c). Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.IntegralDomain.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition comRingType := @GRing.ComRing.Pack cT class. Definition countComRingType := @CountRing.ComRing.Pack cT class. Definition finComRingType := @ComRing.Pack cT class. Definition unitRingType := @GRing.UnitRing.Pack cT class. Definition countUnitRingType := @CountRing.UnitRing.Pack cT class. Definition finUnitRingType := @UnitRing.Pack cT class. Definition comUnitRingType := @GRing.ComUnitRing.Pack cT class. Definition countComUnitRingType := @CountRing.ComUnitRing.Pack cT class. Definition finComUnitRingType := @ComUnitRing.Pack cT class. Definition idomainType := @GRing.IntegralDomain.Pack cT class. Definition countIdomainType := @CountRing.IntegralDomain.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition idomain_finType := @Finite.Pack idomainType (fin_ class). Definition idomain_baseFinGroupType := base_group idomainType zmodType finType. Definition idomain_finGroupType := fin_group idomain_baseFinGroupType zmodType. Definition idomain_finZmodType := @Zmodule.Pack idomainType class. Definition idomain_finRingType := @Ring.Pack idomainType class. Definition idomain_finUnitRingType := @UnitRing.Pack idomainType class. Definition idomain_finComRingType := @ComRing.Pack idomainType class. Definition idomain_finComUnitRingType := @ComUnitRing.Pack idomainType class. Definition countIdomain_finType := @Finite.Pack countIdomainType (fin_ class). Definition countIdomain_baseFinGroupType := base_group countIdomainType zmodType finType. Definition countIdomain_finGroupType := fin_group countIdomain_baseFinGroupType zmodType. Definition countIdomain_finZmodType := @Zmodule.Pack countIdomainType class. Definition countIdomain_finRingType := @Ring.Pack countIdomainType class. Definition countIdomain_finUnitRingType := @UnitRing.Pack countIdomainType class. Definition countIdomain_finComRingType := @ComRing.Pack countIdomainType class. Definition countIdomain_finComUnitRingType := @ComUnitRing.Pack countIdomainType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.IntegralDomain.class_of. Coercion base2 : class_of >-> CountRing.IntegralDomain.class_of. Coercion base3 : class_of >-> ComUnitRing.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion comRingType : type >-> GRing.ComRing.type. Canonical comRingType. Coercion countComRingType : type >-> CountRing.ComRing.type. Canonical countComRingType. Coercion finComRingType : type >-> ComRing.type. Canonical finComRingType. Coercion unitRingType : type >-> GRing.UnitRing.type. Canonical unitRingType. Coercion countUnitRingType : type >-> CountRing.UnitRing.type. Canonical countUnitRingType. Coercion finUnitRingType : type >-> UnitRing.type. Canonical finUnitRingType. Coercion comUnitRingType : type >-> GRing.ComUnitRing.type. Canonical comUnitRingType. Coercion countComUnitRingType : type >-> CountRing.ComUnitRing.type. Canonical countComUnitRingType. Coercion finComUnitRingType : type >-> ComUnitRing.type. Canonical finComUnitRingType. Coercion idomainType : type >-> GRing.IntegralDomain.type. Canonical idomainType. Coercion countIdomainType : type >-> CountRing.IntegralDomain.type. Canonical countIdomainType. Canonical idomain_finType. Canonical idomain_baseFinGroupType. Canonical idomain_finGroupType. Canonical idomain_finZmodType. Canonical idomain_finRingType. Canonical idomain_finUnitRingType. Canonical idomain_finComRingType. Canonical idomain_finComUnitRingType. Canonical countIdomain_finType. Canonical countIdomain_baseFinGroupType. Canonical countIdomain_finGroupType. Canonical countIdomain_finZmodType. Canonical countIdomain_finRingType. Canonical countIdomain_finUnitRingType. Canonical countIdomain_finComRingType. Canonical countIdomain_finComUnitRingType. Notation finIdomainType := FinRing.IntegralDomain.type. Notation "[ 'finIdomainType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finIdomainType' 'of' T ]") : form_scope. End Exports. End IntegralDomain. Import IntegralDomain.Exports. Module Field. Section ClassDef. Set Primitive Projections. Record class_of R := Class { base : GRing.Field.class_of R; mixin : mixin_of R base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.Field.class_of. Local Coercion base2 R (c : class_of R) : CountRing.Field.class_of R := CountRing.Field.Class c (mixin c). Local Coercion base3 R (c : class_of R) : IntegralDomain.class_of R := IntegralDomain.Class (mixin c). Structure type := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Definition pack := gen_pack Pack Class GRing.Field.class. Variable cT : type. Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition comRingType := @GRing.ComRing.Pack cT class. Definition countComRingType := @CountRing.ComRing.Pack cT class. Definition finComRingType := @ComRing.Pack cT class. Definition unitRingType := @GRing.UnitRing.Pack cT class. Definition countUnitRingType := @CountRing.UnitRing.Pack cT class. Definition finUnitRingType := @UnitRing.Pack cT class. Definition comUnitRingType := @GRing.ComUnitRing.Pack cT class. Definition countComUnitRingType := @CountRing.ComUnitRing.Pack cT class. Definition finComUnitRingType := @ComUnitRing.Pack cT class. Definition idomainType := @GRing.IntegralDomain.Pack cT class. Definition countIdomainType := @CountRing.IntegralDomain.Pack cT class. Definition finIdomainType := @IntegralDomain.Pack cT class. Definition fieldType := @GRing.Field.Pack cT class. Definition countFieldType := @CountRing.Field.Pack cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition field_finType := @Finite.Pack fieldType (fin_ class). Definition field_baseFinGroupType := base_group fieldType zmodType finType. Definition field_finGroupType := fin_group field_baseFinGroupType zmodType. Definition field_finZmodType := @Zmodule.Pack fieldType class. Definition field_finRingType := @Ring.Pack fieldType class. Definition field_finUnitRingType := @UnitRing.Pack fieldType class. Definition field_finComRingType := @ComRing.Pack fieldType class. Definition field_finComUnitRingType := @ComUnitRing.Pack fieldType class. Definition field_finIdomainType := @IntegralDomain.Pack fieldType class. Definition countField_finType := @Finite.Pack countFieldType (fin_ class). Definition countField_baseFinGroupType := base_group countFieldType zmodType finType. Definition countField_finGroupType := fin_group countField_baseFinGroupType zmodType. Definition countField_finZmodType := @Zmodule.Pack countFieldType class. Definition countField_finRingType := @Ring.Pack countFieldType class. Definition countField_finUnitRingType := @UnitRing.Pack countFieldType class. Definition countField_finComRingType := @ComRing.Pack countFieldType class. Definition countField_finComUnitRingType := @ComUnitRing.Pack countFieldType class. Definition countField_finIdomainType := @IntegralDomain.Pack countFieldType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.Field.class_of. Coercion base2 : class_of >-> CountRing.Field.class_of. Coercion base3 : class_of >-> IntegralDomain.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion comRingType : type >-> GRing.ComRing.type. Canonical comRingType. Coercion countComRingType : type >-> CountRing.ComRing.type. Canonical countComRingType. Coercion finComRingType : type >-> ComRing.type. Canonical finComRingType. Coercion unitRingType : type >-> GRing.UnitRing.type. Canonical unitRingType. Coercion countUnitRingType : type >-> CountRing.UnitRing.type. Canonical countUnitRingType. Coercion finUnitRingType : type >-> UnitRing.type. Canonical finUnitRingType. Coercion comUnitRingType : type >-> GRing.ComUnitRing.type. Canonical comUnitRingType. Coercion countComUnitRingType : type >-> CountRing.ComUnitRing.type. Canonical countComUnitRingType. Coercion finComUnitRingType : type >-> ComUnitRing.type. Canonical finComUnitRingType. Coercion idomainType : type >-> GRing.IntegralDomain.type. Canonical idomainType. Coercion countIdomainType : type >-> CountRing.IntegralDomain.type. Canonical countIdomainType. Coercion finIdomainType : type >-> IntegralDomain.type. Canonical finIdomainType. Coercion fieldType : type >-> GRing.Field.type. Canonical fieldType. Coercion countFieldType : type >-> CountRing.Field.type. Canonical countFieldType. Canonical field_finType. Canonical field_baseFinGroupType. Canonical field_finGroupType. Canonical field_finZmodType. Canonical field_finRingType. Canonical field_finUnitRingType. Canonical field_finComRingType. Canonical field_finComUnitRingType. Canonical field_finIdomainType. Canonical countField_finType. Canonical countField_baseFinGroupType. Canonical countField_finGroupType. Canonical countField_finZmodType. Canonical countField_finRingType. Canonical countField_finUnitRingType. Canonical countField_finComRingType. Canonical countField_finComUnitRingType. Canonical countField_finIdomainType. Notation finFieldType := FinRing.Field.type. Notation "[ 'finFieldType' 'of' T ]" := (do_pack pack T) (at level 0, format "[ 'finFieldType' 'of' T ]") : form_scope. End Exports. End Field. Import Field.Exports. Section DecideField. Variable F : Field.type. Fixpoint sat e 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. Lemma decidable : GRing.DecidableField.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. Definition DecidableFieldMixin := DecFieldMixin decidable. End DecideField. Module DecField. Section Joins. Variable cT : Field.type. Let class : Field.class_of cT := Field.class cT. Canonical decFieldType := Eval hnf in DecFieldType cT (DecidableFieldMixin cT). Definition countDecFieldType := [countDecFieldType of cT]. Definition decField_finType := @Finite.Pack decFieldType (fin_ class). Definition decField_finZmodType := @Zmodule.Pack decFieldType class. Definition decField_finRingType := @Ring.Pack decFieldType class. Definition decField_finUnitRingType := @UnitRing.Pack decFieldType class. Definition decField_finComRingType := @ComRing.Pack decFieldType class. Definition decField_finComUnitRingType := @ComUnitRing.Pack decFieldType class. Definition decField_finIdomainType := @IntegralDomain.Pack decFieldType class. Definition decField_baseFinGroupType := base_group decFieldType decField_finZmodType decField_finZmodType. Definition decField_finGroupType := fin_group decField_baseFinGroupType cT. Definition countDecField_finType := @Finite.Pack countDecFieldType (fin_ class). Definition countDecField_finZmodType := @Zmodule.Pack countDecFieldType class. Definition countDecField_finRingType := @Ring.Pack countDecFieldType class. Definition countDecField_finUnitRingType := @UnitRing.Pack countDecFieldType class. Definition countDecField_finComRingType := @ComRing.Pack countDecFieldType class. Definition countDecField_finComUnitRingType := @ComUnitRing.Pack countDecFieldType class. Definition countDecField_finIdomainType := @IntegralDomain.Pack countDecFieldType class. Definition countDecField_baseFinGroupType := base_group countDecFieldType countDecField_finZmodType countDecField_finZmodType. Definition countDecField_finGroupType := fin_group countDecField_baseFinGroupType cT. End Joins. Module Exports. Coercion decFieldType : Field.type >-> GRing.DecidableField.type. Coercion countDecFieldType : Field.type >-> CountRing.DecidableField.type. Canonical decFieldType. Canonical countDecFieldType. Canonical decField_finType. Canonical decField_finZmodType. Canonical decField_finRingType. Canonical decField_finUnitRingType. Canonical decField_finComRingType. Canonical decField_finComUnitRingType. Canonical decField_finIdomainType. Canonical decField_baseFinGroupType. Canonical decField_finGroupType. Canonical countDecField_finType. Canonical countDecField_finZmodType. Canonical countDecField_finRingType. Canonical countDecField_finUnitRingType. Canonical countDecField_finComRingType. Canonical countDecField_finComUnitRingType. Canonical countDecField_finIdomainType. Canonical countDecField_baseFinGroupType. Canonical countDecField_finGroupType. End Exports. End DecField. Module Lmodule. Section ClassDef. Variable R : ringType. Set Primitive Projections. Record class_of M := Class { base : GRing.Lmodule.class_of R M; mixin : mixin_of M base }. Unset Primitive Projections. Local Coercion base : class_of >-> GRing.Lmodule.class_of. Local Coercion base2 R (c : class_of R) : Zmodule.class_of R := Zmodule.Class (mixin c). Structure type (phR : phant R) := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Variables (phR : phant R) (cT : type phR). Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition pack := gen_pack (Pack phR) Class (@GRing.Lmodule.class R phR). Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition lmodType := @GRing.Lmodule.Pack R phR cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition lmod_countType := @Countable.Pack lmodType (fin_ class). Definition lmod_finType := @Finite.Pack lmodType (fin_ class). Definition lmod_baseFinGroupType := base_group lmodType zmodType finType. Definition lmod_finGroupType := fin_group lmod_baseFinGroupType zmodType. Definition lmod_countZmodType := @CountRing.Zmodule.Pack lmodType class. Definition lmod_finZmodType := @Zmodule.Pack lmodType class. End ClassDef. Module Import Exports. Coercion base : class_of >-> GRing.Lmodule.class_of. Coercion base2 : class_of >-> Zmodule.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion lmodType : type >-> GRing.Lmodule.type. Canonical lmodType. Canonical lmod_countType. Canonical lmod_finType. Canonical lmod_baseFinGroupType. Canonical lmod_finGroupType. Canonical lmod_countZmodType. Canonical lmod_finZmodType. Notation finLmodType R := (FinRing.Lmodule.type (Phant R)). Notation "[ 'finLmodType' R 'of' T ]" := (do_pack (@pack _ (Phant R)) T) (at level 0, format "[ 'finLmodType' R 'of' T ]") : form_scope. End Exports. End Lmodule. Import Lmodule.Exports. Module Lalgebra. Section ClassDef. Variable R : ringType. Set Primitive Projections. Record class_of M := Class { base : GRing.Lalgebra.class_of R M; mixin : mixin_of M base }. Unset Primitive Projections. Definition base2 M (c : class_of M) := Ring.Class (mixin c). Definition base3 M (c : class_of M) := @Lmodule.Class _ _ (base c) (mixin c). Local Coercion base : class_of >-> GRing.Lalgebra.class_of. Local Coercion base2 : class_of >-> Ring.class_of. Local Coercion base3 : class_of >-> Lmodule.class_of. Structure type (phR : phant R) := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Variables (phR : phant R) (cT : type phR). Definition pack := gen_pack (Pack phR) Class (@GRing.Lalgebra.class R phR). Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition lmodType := @GRing.Lmodule.Pack R phR cT class. Definition finLmodType := @Lmodule.Pack R phR cT class. Definition lalgType := @GRing.Lalgebra.Pack R phR cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition lalg_countType := @Countable.Pack lalgType (fin_ class). Definition lalg_finType := @Finite.Pack lalgType (fin_ class). Definition lalg_baseFinGroupType := base_group lalgType zmodType finType. Definition lalg_finGroupType := fin_group lalg_baseFinGroupType zmodType. Definition lalg_countZmodType := @CountRing.Zmodule.Pack lalgType class. Definition lalg_finZmodType := @Zmodule.Pack lalgType class. Definition lalg_finLmodType := @Lmodule.Pack R phR lalgType class. Definition lalg_countRingType := @CountRing.Ring.Pack lalgType class. Definition lalg_finRingType := @Ring.Pack lalgType class. Definition lmod_countRingType := @CountRing.Ring.Pack lmodType class. Definition lmod_finRingType := @Ring.Pack lmodType class. Definition finLmod_ringType := @GRing.Ring.Pack finLmodType class. Definition finLmod_countRingType := @CountRing.Ring.Pack finLmodType class. Definition finLmod_finRingType := @Ring.Pack finLmodType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.Lalgebra.class_of. Coercion base2 : class_of >-> Ring.class_of. Coercion base3 : class_of >-> Lmodule.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion lmodType : type >-> GRing.Lmodule.type. Canonical lmodType. Coercion finLmodType : type >-> Lmodule.type. Canonical finLmodType. Coercion lalgType : type >-> GRing.Lalgebra.type. Canonical lalgType. Canonical lalg_countType. Canonical lalg_finType. Canonical lalg_baseFinGroupType. Canonical lalg_finGroupType. Canonical lalg_countZmodType. Canonical lalg_finZmodType. Canonical lalg_finLmodType. Canonical lalg_countRingType. Canonical lalg_finRingType. Canonical lmod_countRingType. Canonical lmod_finRingType. Canonical finLmod_ringType. Canonical finLmod_countRingType. Canonical finLmod_finRingType. Notation finLalgType R := (FinRing.Lalgebra.type (Phant R)). Notation "[ 'finLalgType' R 'of' T ]" := (do_pack (@pack _ (Phant R)) T) (at level 0, format "[ 'finLalgType' R 'of' T ]") : form_scope. End Exports. End Lalgebra. Import Lalgebra.Exports. Module Algebra. Section ClassDef. Variable R : ringType. Set Primitive Projections. Record class_of M := Class { base : GRing.Algebra.class_of R M; mixin : mixin_of M base }. Unset Primitive Projections. Definition base2 M (c : class_of M) := Lalgebra.Class (mixin c). Local Coercion base : class_of >-> GRing.Algebra.class_of. Local Coercion base2 : class_of >->Lalgebra.class_of. Structure type (phR : phant R) := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Variables (phR : phant R) (cT : type phR). Definition pack := gen_pack (Pack phR) Class (@GRing.Algebra.class R phR). Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition lmodType := @GRing.Lmodule.Pack R phR cT class. Definition finLmodType := @Lmodule.Pack R phR cT class. Definition lalgType := @GRing.Lalgebra.Pack R phR cT class. Definition finLalgType := @Lalgebra.Pack R phR cT class. Definition algType := @GRing.Algebra.Pack R phR cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition alg_countType := @Countable.Pack algType (fin_ class). Definition alg_finType := @Finite.Pack algType (fin_ class). Definition alg_baseFinGroupType := base_group algType zmodType finType. Definition alg_finGroupType := fin_group alg_baseFinGroupType zmodType. Definition alg_countZmodType := @CountRing.Zmodule.Pack algType class. Definition alg_finZmodType := @Zmodule.Pack algType class. Definition alg_countRingType := @CountRing.Ring.Pack algType class. Definition alg_finRingType := @Ring.Pack algType class. Definition alg_finLmodType := @Lmodule.Pack R phR algType class. Definition alg_finLalgType := @Lalgebra.Pack R phR algType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.Algebra.class_of. Coercion base2 : class_of >-> Lalgebra.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion lmodType : type >-> GRing.Lmodule.type. Canonical lmodType. Coercion finLmodType : type >-> Lmodule.type. Canonical finLmodType. Coercion lalgType : type >-> GRing.Lalgebra.type. Canonical lalgType. Coercion finLalgType : type >-> Lalgebra.type. Canonical finLalgType. Coercion algType : type >-> GRing.Algebra.type. Canonical algType. Canonical alg_countType. Canonical alg_finType. Canonical alg_baseFinGroupType. Canonical alg_finGroupType. Canonical alg_countZmodType. Canonical alg_finZmodType. Canonical alg_countRingType. Canonical alg_finRingType. Canonical alg_finLmodType. Canonical alg_finLalgType. Notation finAlgType R := (type (Phant R)). Notation "[ 'finAlgType' R 'of' T ]" := (do_pack (@pack _ (Phant R)) T) (at level 0, format "[ 'finAlgType' R 'of' T ]") : form_scope. End Exports. End Algebra. Import Algebra.Exports. Module UnitAlgebra. Section ClassDef. Variable R : unitRingType. Set Primitive Projections. Record class_of M := Class { base : GRing.UnitAlgebra.class_of R M; mixin : mixin_of M base }. Unset Primitive Projections. Definition base2 M (c : class_of M) := Algebra.Class (mixin c). Definition base3 M (c : class_of M) := @UnitRing.Class _ (base c) (mixin c). Local Coercion base : class_of >-> GRing.UnitAlgebra.class_of. Local Coercion base2 : class_of >-> Algebra.class_of. Local Coercion base3 : class_of >-> UnitRing.class_of. Structure type (phR : phant R) := Pack {sort; _ : class_of sort}. Local Coercion sort : type >-> Sortclass. Variables (phR : phant R) (cT : type phR). Definition pack := gen_pack (Pack phR) Class (@GRing.UnitAlgebra.class R phR). Definition class := let: Pack _ c as cT' := cT return class_of cT' in c. Definition eqType := @Equality.Pack cT class. Definition choiceType := @Choice.Pack cT class. Definition countType := @Countable.Pack cT (fin_ class). Definition finType := @Finite.Pack cT (fin_ class). Definition zmodType := @GRing.Zmodule.Pack cT class. Definition countZmodType := @CountRing.Zmodule.Pack cT class. Definition finZmodType := @Zmodule.Pack cT class. Definition ringType := @GRing.Ring.Pack cT class. Definition countRingType := @CountRing.Ring.Pack cT class. Definition finRingType := @Ring.Pack cT class. Definition unitRingType := @GRing.UnitRing.Pack cT class. Definition countUnitRingType := @CountRing.UnitRing.Pack cT class. Definition finUnitRingType := @UnitRing.Pack cT class. Definition lmodType := @GRing.Lmodule.Pack R phR cT class. Definition finLmodType := @Lmodule.Pack R phR cT class. Definition lalgType := @GRing.Lalgebra.Pack R phR cT class. Definition finLalgType := @Lalgebra.Pack R phR cT class. Definition algType := @GRing.Algebra.Pack R phR cT class. Definition finAlgType := @Algebra.Pack R phR cT class. Definition unitAlgType := @GRing.UnitAlgebra.Pack R phR cT class. Definition baseFinGroupType := base_group cT zmodType finType. Definition finGroupType := fin_group baseFinGroupType zmodType. Definition unitAlg_countType := @Countable.Pack unitAlgType (fin_ class). Definition unitAlg_finType := @Finite.Pack unitAlgType (fin_ class). Definition unitAlg_baseFinGroupType := base_group unitAlgType zmodType finType. Definition unitAlg_finGroupType := fin_group unitAlg_baseFinGroupType zmodType. Definition unitAlg_countZmodType := @CountRing.Zmodule.Pack unitAlgType class. Definition unitAlg_finZmodType := @Zmodule.Pack unitAlgType class. Definition unitAlg_countRingType := @CountRing.Ring.Pack unitAlgType class. Definition unitAlg_finRingType := @Ring.Pack unitAlgType class. Definition unitAlg_countUnitRingType := @CountRing.UnitRing.Pack unitAlgType class. Definition unitAlg_finUnitRingType := @UnitRing.Pack unitAlgType class. Definition unitAlg_finLmodType := @Lmodule.Pack R phR unitAlgType class. Definition unitAlg_finLalgType := @Lalgebra.Pack R phR unitAlgType class. Definition unitAlg_finAlgType := @Algebra.Pack R phR unitAlgType class. Definition unitRing_finLmodType := @Lmodule.Pack R phR unitRingType class. Definition unitRing_finLalgType := @Lalgebra.Pack R phR unitRingType class. Definition unitRing_finAlgType := @Algebra.Pack R phR unitRingType class. Definition countUnitRing_lmodType := @GRing.Lmodule.Pack R phR countUnitRingType class. Definition countUnitRing_finLmodType := @Lmodule.Pack R phR countUnitRingType class. Definition countUnitRing_lalgType := @GRing.Lalgebra.Pack R phR countUnitRingType class. Definition countUnitRing_finLalgType := @Lalgebra.Pack R phR countUnitRingType class. Definition countUnitRing_algType := @GRing.Algebra.Pack R phR countUnitRingType class. Definition countUnitRing_finAlgType := @Algebra.Pack R phR countUnitRingType class. Definition finUnitRing_lmodType := @GRing.Lmodule.Pack R phR finUnitRingType class. Definition finUnitRing_finLmodType := @Lmodule.Pack R phR finUnitRingType class. Definition finUnitRing_lalgType := @GRing.Lalgebra.Pack R phR finUnitRingType class. Definition finUnitRing_finLalgType := @Lalgebra.Pack R phR finUnitRingType class. Definition finUnitRing_algType := @GRing.Algebra.Pack R phR finUnitRingType class. Definition finUnitRing_finAlgType := @Algebra.Pack R phR finUnitRingType class. End ClassDef. Module Exports. Coercion base : class_of >-> GRing.UnitAlgebra.class_of. Coercion base2 : class_of >-> Algebra.class_of. Coercion base3 : class_of >-> UnitRing.class_of. Coercion sort : type >-> Sortclass. Bind Scope ring_scope with sort. Coercion eqType : type >-> Equality.type. Canonical eqType. Coercion choiceType : type >-> Choice.type. Canonical choiceType. Coercion countType : type >-> Countable.type. Canonical countType. Coercion finType : type >-> Finite.type. Canonical finType. Coercion baseFinGroupType : type >-> FinGroup.base_type. Canonical baseFinGroupType. Coercion finGroupType : type >-> FinGroup.type. Canonical finGroupType. Coercion zmodType : type >-> GRing.Zmodule.type. Canonical zmodType. Coercion countZmodType : type >-> CountRing.Zmodule.type. Canonical countZmodType. Coercion finZmodType : type >-> Zmodule.type. Canonical finZmodType. Coercion ringType : type >-> GRing.Ring.type. Canonical ringType. Coercion countRingType : type >-> CountRing.Ring.type. Canonical countRingType. Coercion finRingType : type >-> Ring.type. Canonical finRingType. Coercion unitRingType : type >-> GRing.UnitRing.type. Canonical unitRingType. Coercion countUnitRingType : type >-> CountRing.UnitRing.type. Canonical countUnitRingType. Coercion finUnitRingType : type >-> UnitRing.type. Canonical finUnitRingType. Coercion lmodType : type >-> GRing.Lmodule.type. Canonical lmodType. Coercion finLmodType : type >-> Lmodule.type. Canonical finLmodType. Coercion lalgType : type >-> GRing.Lalgebra.type. Canonical lalgType. Coercion finLalgType : type >-> Lalgebra.type. Canonical finLalgType. Coercion algType : type >-> GRing.Algebra.type. Canonical algType. Coercion finAlgType : type >-> Algebra.type. Canonical finAlgType. Coercion unitAlgType : type >-> GRing.UnitAlgebra.type. Canonical unitAlgType. Canonical unitAlg_countType. Canonical unitAlg_finType. Canonical unitAlg_baseFinGroupType. Canonical unitAlg_finGroupType. Canonical unitAlg_countZmodType. Canonical unitAlg_finZmodType. Canonical unitAlg_countRingType. Canonical unitAlg_finRingType. Canonical unitAlg_countUnitRingType. Canonical unitAlg_finUnitRingType. Canonical unitAlg_finLmodType. Canonical unitAlg_finLalgType. Canonical unitAlg_finAlgType. Canonical unitRing_finLmodType. Canonical unitRing_finLalgType. Canonical unitRing_finAlgType. Canonical countUnitRing_lmodType. Canonical countUnitRing_finLmodType. Canonical countUnitRing_lalgType. Canonical countUnitRing_finLalgType. Canonical countUnitRing_algType. Canonical countUnitRing_finAlgType. Canonical finUnitRing_lmodType. Canonical finUnitRing_finLmodType. Canonical finUnitRing_lalgType. Canonical finUnitRing_finLalgType. Canonical finUnitRing_algType. Canonical finUnitRing_finAlgType. Notation finUnitAlgType R := (type (Phant R)). Notation "[ 'finUnitAlgType' R 'of' T ]" := (do_pack (@pack _ (Phant R)) T) (at level 0, format "[ 'finUnitAlgType' R 'of' T ]") : form_scope. End Exports. End UnitAlgebra. Import UnitAlgebra.Exports. 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. Export Zmodule.Exports Ring.Exports ComRing.Exports. Export UnitRing.Exports UnitsGroupExports ComUnitRing.Exports. Export IntegralDomain.Exports Field.Exports DecField.Exports. Export Lmodule.Exports Lalgebra.Exports Algebra.Exports UnitAlgebra.Exports. Notation "{ 'unit' R }" := (unit_of (Phant R)) (at level 0, format "{ 'unit' R }") : type_scope. Prenex Implicits FinRing.uval. Notation "''U'" := (unit_action _) (at level 8) : action_scope. Notation "''U'" := (unit_groupAction _) (at level 8) : groupAction_scope.