Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Classes.DecidableClass

# A typeclass to ease the handling of decidable properties.

A proposition is decidable whenever it is reflected by a boolean.
Class Decidable (P : Prop) := {
Decidable_witness : bool;
Decidable_spec : Decidable_witness = true <-> P
}.
Alternative ways of specifying the reflection property.
Lemma Decidable_sound : forall P (H : Decidable P),
Decidable_witness = true -> P.
Lemma Decidable_complete : forall P (H : Decidable P),
P -> Decidable_witness = true.
Lemma Decidable_sound_alt : forall P (H : Decidable P),
~ P -> Decidable_witness = false.
Lemma Decidable_complete_alt : forall P (H : Decidable P),
Decidable_witness = false -> ~ P.
The generic function that should be used to program, together with some useful tactics.
Definition decide P {H : Decidable P} := @Decidable_witness _ H.
Ltac _decide_ P H :=
let b := fresh "b" in
set (b := decide P) in *;
assert (H : decide P = b) by reflexivity;
clearbody b;
destruct b; [apply Decidable_sound in H|apply Decidable_complete_alt in H].
Tactic Notation "decide" constr(P) "as" ident(H) :=
_decide_ P H.
Tactic Notation "decide" constr(P) :=
let H := fresh "H" in _decide_ P H.
Some usual instances.
#[global,refine]
Instance Decidable_not {P} `{Decidable P} : Decidable (~ P) := {
Decidable_witness := negb Decidable_witness
}.