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Decidable setoid equality theory.

Author: Matthieu Sozeau Institution: LRI, CNRS UMR 8623 - University Paris Sud 
Set Implicit Arguments.
Unset Strict Implicit.

Generalizable Variables A B .
Export notations. 
Require Export Coq.Classes.SetoidClass.
The DecidableSetoid class asserts decidability of a Setoid. It can be useful in proofs to reason more classically. 
Require Import Coq.Logic.Decidable.

Class DecidableSetoid `(S : Setoid A) :=
  setoid_decidable : forall x y : A, decidable (x == y).
The EqDec class gives a decision procedure for a particular setoid equality. 
Class EqDec `(S : Setoid A) :=
  equiv_dec : forall x y : A, { x == y } + { x =/= y }.
We define the == overloaded notation for deciding equality. It does not take precedence of == defined in the type scope, hence we can have both at the same time. 
Notation " x == y " := (equiv_dec (x :>) (y :>)) (no associativity, at level 70).

Definition swap_sumbool {A B} (x : { A } + { B }) : { B } + { A } :=
  match x with
    | left H => @right _ _ H
    | right H => @left _ _ H
  end.

Require Import Coq.Program.Program.

Local Open Scope program_scope.
Invert the branches. 
Definition nequiv_dec `{EqDec A} (x y : A) : { x =/= y } + { x == y } := swap_sumbool (x == y).
Overloaded notation for inequality. 
Infix "=/=" := nequiv_dec (no associativity, at level 70).
Define boolean versions, losing the logical information. 
Definition equiv_decb `{EqDec A} (x y : A) : bool :=
  if x == y then true else false.

Definition nequiv_decb `{EqDec A} (x y : A) : bool :=
  negb (equiv_decb x y).

Infix "==b" := equiv_decb (no associativity, at level 70).
Infix "<>b" := nequiv_decb (no associativity, at level 70).
Decidable leibniz equality instances. 
Require Import Coq.Arith.Arith.
The equiv is buried inside the setoid, but we can recover it by specifying which setoid we're talking about. 
Instance eq_setoid A : Setoid A | 10 :=
  { equiv := eq ; setoid_equiv := eq_equivalence }.

Instance nat_eq_eqdec : EqDec (eq_setoid nat) :=
  eq_nat_dec.

Require Import Coq.Bool.Bool.

Instance bool_eqdec : EqDec (eq_setoid bool) :=
  bool_dec.

Instance unit_eqdec : EqDec (eq_setoid unit) :=
  fun x y => in_left.

  
x, y:()
x = y

  
x, y:()
x = y

    
() = ()

    reflexivity.
  Qed.

Instance prod_eqdec `(! EqDec (eq_setoid A), ! EqDec (eq_setoid B))
 : EqDec (eq_setoid (prod A B)) :=
  fun x y =>
    let '(x1, x2) := x in
    let '(y1, y2) := y in
    if x1 == y1 then
      if x2 == y2 then in_left
      else in_right
    else in_right.

  Solve Obligations with unfold complement ; program_simpl.
Objects of function spaces with countable domains like bool have decidable equality. 
Instance bool_function_eqdec `(! EqDec (eq_setoid A))
 : EqDec (eq_setoid (bool -> A)) :=
  fun f g =>
    if f true == g true then
      if f false == g false then in_left
      else in_right
    else in_right.

  Solve Obligations with try red ; unfold complement ; program_simpl.

  
A:Type
EqDec0:EqDec (eq_setoid A)
f, g:bool -> A
H:f true = g true
H0:f false = g false
f = g

  
A:Type
EqDec0:EqDec (eq_setoid A)
f, g:bool -> A
H:f true = g true
H0:f false = g false
f = g

    
A:Type
EqDec0:EqDec (eq_setoid A)
f, g:bool -> A
H:f true = g true
H0:f false = g false
x:bool
f x = g x

    destruct x ; auto.
  Qed.