(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
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(* This file defines two "base" combinatorial interfaces:                     *)
(*    eqType == the structure for types with a decidable equality.            *)
(* subType P == the structure for types isomorphic to {x : T | P x} with      *)
(*              P : pred T for some type T.                                   *)
(* The following are used to construct eqType instances:                      *)
(*         EqType T m == the packed eqType class for type T and mixin m.      *)
(* --> As eqType is a root class, equality mixins and classes coincide.       *)
(*   Equality.axiom e <-> e : rel T is a valid comparison decision procedure  *)
(*                       for type T: reflect (x = y) (e x y) for all x y : T. *)
(*         EqMixin eP == the equality mixin for eP : Equality.axiom e.        *)
(* --> Such manifest equality mixins should be declared Canonical to allow    *)
(* for generic folding of equality predicates (see lemma eqE below).          *)
(*  [eqType of T for eT] == clone for T of eT, where eT is an eqType for a    *)
(*                      type convertible, but usually not identical, to T.    *)
(*      [eqType of T] == clone for T of the eqType inferred for T, possibly   *)
(*                       after unfolding some definitions.                    *)
(*     [eqMixin of T] == mixin of the eqType inferred for T.                  *)
(*       comparable T <-> equality on T is decidable.                         *)
(*                    := forall x y : T, decidable (x = y)                    *)
(*  comparableMixin compT == equality mixin for compT : comparable T.         *)
(*   InjEqMixin injf == an Equality mixin for T, using an f : T -> U  where   *)
(*                      U has an eqType structure and injf : injective f.     *)
(*    PcanEqMixin fK == an Equality mixin similarly derived from f and a left *)
(*                      inverse partial function g and fK : pcancel f g.      *)
(*     CanEqMixin fK == an Equality mixin similarly derived from f and a left *)
(*                      inverse function g and fK : cancel f g.               *)
(* --> Equality mixins derived by the above should never be made Canonical as *)
(* they provide only comparisons with a generic head constant.                *)
(*   The eqType interface supports the following operations:                  *)
(*              x == y <=> x compares equal to y (this is a boolean test).    *)
(*         x == y :> T <=> x == y at type T.                                  *)
(*              x != y <=> x and y compare unequal.                           *)
(*         x != y :> T <=> x and y compare unequal at type T.                 *)
(*             x =P y  :: a proof of reflect (x = y) (x == y); x =P y coerces *)
(*                     to x == y -> x = y.                                    *)
(*               eq_op == the boolean relation behing the == notation.        *)
(*             pred1 a == the singleton predicate [pred x | x == a].          *)
(* pred2, pred3, pred4 == pair, triple, quad predicates.                      *)
(*            predC1 a == [pred x | x != a].                                  *)
(*      [predU1 a & A] == [pred x | (x == a) || (x \in A)].                   *)
(*      [predD1 A & a] == [pred x | x != a & x \in A].                        *)
(*  predU1 a P, predD1 P a == applicative versions of the above.              *)
(*              frel f == the relation associated with f : T -> T.            *)
(*                     := [rel x y | f x == y].                               *)
(*       invariant k f == elements of T whose k-class is f-invariant.         *)
(*                     := [pred x | k (f x) == k x] with f : T -> T.          *)
(*  [fun x : T => e0 with a1 |-> e1, .., a_n |-> e_n]                         *)
(*  [eta f with a1 |-> e1, .., a_n |-> e_n] ==                                *)
(*    the auto-expanding function that maps x = a_i to e_i, and other values  *)
(*    of x to e0 (resp. f x). In the first form the `: T' is optional and x   *)
(*    can occur in a_i or e_i.                                                *)
(* Equality on an eqType is proof-irrelevant (lemma eq_irrelevance).          *)
(*   The eqType interface is implemented for most standard datatypes:         *)
(*  bool, unit, void, option, prod (denoted A * B), sum (denoted A + B),      *)
(*  sig (denoted {x | P}), sigT (denoted {i : I & T}). We also define         *)
(*   tagged_as u v == v cast as T_(tag u) if tag v == tag u, else u.          *)
(*  -> We have u == v <=> (tag u == tag v) && (tagged u == tagged_as u v).    *)
(* The subType interface supports the following operations:                   *)
(*      val == the generic injection from a subType S of T into T.            *)
(*             For example, if u : {x : T | P}, then val u : T.               *)
(*             val is injective because P is proof-irrelevant (P is in bool,  *)
(*             and the is_true coercion expands to P = true).                 *)
(*     valP == the generic proof of P (val u) for u : subType P.              *)
(* Sub x Px == the generic constructor for a subType P; Px is a proof of P x  *)
(*             and P should be inferred from the expected return type.        *)
(*  insub x == the generic partial projection of T into a subType S of T.     *)
(*             This returns an option S; if S : subType P then                *)
(*                insub x = Some u with val u = x if P x,                     *)
(*                          None if ~~ P x                                    *)
(*             The insubP lemma encapsulates this dichotomy.                  *)
(*             P should be infered from the expected return type.             *)
(*  innew x == total (non-option) variant of insub when P = predT.            *)
(* {? x | P} == option {x | P} (syntax for casting insub x).                  *)
(* insubd u0 x == the generic projection with default value u0.               *)
(*             := odflt u0 (insub x).                                         *)
(* insigd A0 x == special case of insubd for S == {x | x \in A}, where A0 is  *)
(*                a proof of x0 \in A.                                        *)
(* insub_eq x == transparent version of insub x that expands to Some/None     *)
(*               when P x can evaluate.                                       *)
(* The subType P interface is most often implemented using one of:            *)
(*   [subType for S_val]                                                      *)
(*     where S_val : S -> T is the first projection of a type S isomorphic to *)
(*     {x : T | P}.                                                           *)
(*   [newType for S_val]                                                      *)
(*     where S_val : S -> T is the projection of a type S isomorphic to       *)
(*     wrapped T; in this case P must be predT.                               *)
(*   [subType for S_val by Srect], [newType for S_val by Srect]               *)
(*     variants of the above where the eliminator is explicitly provided.     *)
(*     Here S no longer needs to be syntactically identical to {x | P x} or   *)
(*     wrapped T, but it must have a derived constructor S_Sub statisfying an *)
(*     eliminator Srect identical to the one the Coq Inductive command would  *)
(*     have generated, and S_val (S_Sub x Px) (resp. S_val (S_sub x) for the  *)
(*     newType form) must be convertible to x.                                *)
(*     variant of the above when S is a wrapper type for T (so P = predT).    *)
(*   [subType of S], [subType of S for S_val]                                 *)
(*     clones the canonical subType structure for S; if S_val is specified,   *)
(*     then it replaces the inferred projector.                               *)
(* Subtypes inherit the eqType structure of their base types; the generic     *)
(* structure should be explicitly instantiated using the                      *)
(*   [eqMixin of S by <:]                                                     *)
(* construct to declare the equality mixin; this pattern is repeated for all  *)
(* the combinatorial interfaces (Choice, Countable, Finite). As noted above,  *)
(* such mixins should not be made Canonical.                                  *)
(*   We add the following to the standard suffixes documented in ssrbool.v:   *)
(*  1, 2, 3, 4 -- explicit enumeration predicate for 1 (singleton), 2, 3, or  *)
(*                4 values.                                                   *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module Equality.
Definition axiom T (e : rel T) := forall x y, reflect (x = y) (e x y).
Structure mixin_of T := Mixin {op : rel T; _ : axiom op}.
Notation class_of := mixin_of (only parsing).
Section ClassDef.
Structure type := Pack {sort; _ : class_of sort}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c := cT return class_of cT in c.
Definition clone := fun c & cT -> T & phant_id (@Pack T c) cT => Pack c.
End ClassDef.
Module Exports.
Coercion sort : type >-> Sortclass.
Notation eqType := type.
Notation EqMixin := Mixin.
Notation EqType T m := (@Pack T m).
Notation "[ 'eqMixin' 'of' T ]" := (class _ : mixin_of T)
(at level 0, format "[ 'eqMixin'  'of'  T ]") : form_scope.
Notation "[ 'eqType' 'of' T 'for' C ]" := (@clone T C _ idfun id)
(at level 0, format "[ 'eqType'  'of'  T  'for'  C ]") : form_scope.
Notation "[ 'eqType' 'of' T ]" := (@clone T _ _ id id)
(at level 0, format "[ 'eqType'  'of'  T ]") : form_scope.
End Exports.
End Equality.
Export Equality.Exports.
Definition eq_op T := Equality.op (Equality.class T).

Corollary eq_axiomK (T : eqType) (x : T) : all_equal_to (erefl x).