Library Coq.Structures.DecidableTypeEx
Require Import DecidableType OrderedType OrderedTypeEx.
Set Implicit Arguments.
Unset Strict Implicit.
NB: This file is here only for compatibility with earlier version of
FSets and FMap. Please use Structures/Equalities.v directly now.
A particular case of DecidableType where
the equality is the usual one of Coq.
Examples of Decidable Type structures.
a UsualDecidableType is in particular an DecidableType.
an shortcut for easily building a UsualDecidableType
Module Type MiniDecidableType := Equalities.MiniDecidableType.
Module Make_UDT (M:MiniDecidableType) <: UsualDecidableType
:= Equalities.Make_UDT M.
An OrderedType can now directly be seen as a DecidableType
(Usual) Decidable Type for nat, positive, N, Z
Module Nat_as_DT <: UsualDecidableType := Nat_as_OT.
Module Positive_as_DT <: UsualDecidableType := Positive_as_OT.
Module N_as_DT <: UsualDecidableType := N_as_OT.
Module Z_as_DT <: UsualDecidableType := Z_as_OT.
From two decidable types, we can build a new DecidableType
over their cartesian product.
Module PairDecidableType(D1 D2:DecidableType) <: DecidableType.
Definition t := prod D1.t D2.t.
Definition eq x y := D1.eq (fst x) (fst y) /\ D2.eq (snd x) (snd y).
Lemma eq_refl : forall x : t, eq x x.
Proof.
intros (x1,x2); red; simpl; auto.
Qed.
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Proof.
intros (x1,x2) (y1,y2); unfold eq; simpl; intuition.
Qed.
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Proof.
intros (x1,x2) (y1,y2) (z1,z2); unfold eq; simpl; intuition eauto.
Qed.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2); unfold eq; simpl.
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2); intuition.
Defined.
End PairDecidableType.
Similarly for pairs of UsualDecidableType
Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType.
Definition t := prod D1.t D2.t.
Definition eq := @eq t.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2);
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
unfold eq, D1.eq, D2.eq in *; simpl;
(left; f_equal; auto; fail) ||
(right; injection; auto).
Defined.
End PairUsualDecidableType.