Library Coq.Classes.EquivDec


Decidable equivalences.

Author: Matthieu Sozeau Institution: LRI, CNRS UMR 8623 - University Paris Sud
Export notations.

Require Export Coq.Classes.Equivalence.

The DecidableSetoid class asserts decidability of a Setoid. It can be useful in proofs to reason more classically.

Require Import Coq.Logic.Decidable.
Require Import Coq.Bool.Bool.
Require Import Coq.Arith.Peano_dec.
Require Import Coq.Program.Program.


Open Scope equiv_scope.

Class DecidableEquivalence `(equiv : Equivalence A) :=
  setoid_decidable : forall x y : A, decidable (x === y).

The EqDec class gives a decision procedure for a particular setoid equality.

Class EqDec A R {equiv : Equivalence R} :=
  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) : equiv_scope.

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

Local Open Scope program_scope.

Invert the branches.

Program 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) : equiv_scope.

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.
The equiv is buried inside the setoid, but we can recover it by specifying which setoid we're talking about.

#[global]
Program Instance nat_eq_eqdec : EqDec nat eq := eq_nat_dec.

#[global]
Program Instance bool_eqdec : EqDec bool eq := bool_dec.

#[global]
Program Instance unit_eqdec : EqDec unit eq := fun x y => in_left.


Obligation Tactic := unfold complement, equiv ; program_simpl.

#[global]
Program Instance prod_eqdec `(EqDec A eq, EqDec B eq) :
  EqDec (prod A B) eq :=
  { equiv_dec 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 }.

#[global]
Program Instance sum_eqdec `(EqDec A eq, EqDec B eq) :
  EqDec (sum A B) eq := {
  equiv_dec x y :=
    match x, y with
      | inl a, inl b => if a == b then in_left else in_right
      | inr a, inr b => if a == b then in_left else in_right
      | inl _, inr _ | inr _, inl _ => in_right
    end }.

Objects of function spaces with countable domains like bool have decidable equality. Proving the reflection requires functional extensionality though.

#[global]
Program Instance bool_function_eqdec `(EqDec A eq) : EqDec (bool -> A) eq :=
  { equiv_dec f g :=
    if f true == g true then
      if f false == g false then in_left
      else in_right
    else in_right }.


Require Import List.

#[global]
Program Instance list_eqdec `(eqa : EqDec A eq) : EqDec (list A) eq :=
  { equiv_dec :=
    fix aux (x y : list A) :=
    match x, y with
      | nil, nil => in_left
      | cons hd tl, cons hd' tl' =>
        if hd == hd' then
          if aux tl tl' then in_left else in_right
          else in_right
      | _, _ => in_right
    end }.


  Solve Obligations with unfold equiv, complement in * ;
    program_simpl ; intuition (discriminate || eauto).