ExtLib.Data.Prop

From Coq Require Import Setoid.

NOTE: These should fit into a larger picture, e.g. lattices or monoids And/Conjunction

Lemma and_True_iff : forall P, (P /\ True ) <-> P.
Proof. intuition. Qed.

Lemma and_and_iff : forall P, (P /\ P ) <-> P.
Proof. intuition. Qed.

Lemma and_assoc : forall P Q R, (P /\ Q /\ R ) <-> ((P /\ Q ) /\ R ).
Proof. intuition. Qed.

Lemma and_comm : forall P Q, (P /\ Q ) <-> (Q /\ P ).
Proof. intuition. Qed.

Lemma and_False_iff : forall P, (P /\ False ) <-> False.
Proof. intuition. Qed.

Lemma and_cancel
: forall P Q R : Prop, (P -> (Q <-> R )) -> ((P /\ Q ) <-> (P /\ R )).
Proof. intuition. Qed.

Lemma and_iff
: forall P Q R S : Prop,
    (P <-> R ) ->
    (P -> (Q <-> S )) ->
    ((P /\ Q ) <-> (R /\ S )).
Proof. clear; intuition. Qed.

Or/Disjunction

Lemma or_False_iff : forall P, (P \/ False ) <-> P.
Proof. intuition. Qed.

Lemma or_or_iff : forall P, (P \/ P ) <-> P.
Proof. intuition. Qed.

Lemma or_assoc : forall P Q R, (P \/ Q \/ R ) <-> ((P \/ Q ) \/ R ).
Proof. intuition. Qed.

Lemma or_comm : forall P Q, (P \/ Q ) <-> (Q \/ P ).
Proof. intuition. Qed.

Lemma or_True_iff : forall P, (P \/ True ) <-> True.
Proof. intuition. Qed.

Implication

Lemma impl_True_iff : forall (P : Prop), (True -> P ) <-> P.
Proof.
  clear; intros; tauto.
Qed.

Lemma impl_iff
: forall P Q R S : Prop,
    (P <-> R ) ->
    (P -> (Q <-> S )) ->
    ((P -> Q ) <-> (R -> S )).
Proof. clear. intuition. Qed.

Lemma impl_eq : forall (P Q : Prop), P = Q -> (P -> Q ).
Proof. clear. intros; subst; auto. Qed.

Lemma uncurry : forall (P Q R : Prop),
    (P /\ Q -> R ) <-> (P -> Q -> R ).
Proof. clear. tauto. Qed.

Forall

Lemma forall_iff : forall T P Q,
                     (forall x,
                        P x <-> Q x ) ->
                     ((forall x : T, P x ) <-> (forall x : T, Q x )).
Proof.
   intros. setoid_rewrite H. reflexivity.
Qed.

Lemma forall_impl : forall {T} (P Q : T -> Prop),
                      (forall x, P x -> Q x ) ->
                      (forall x, P x ) -> (forall x, Q x ).
Proof.
  clear. intuition.
Qed.

Exists

Lemma exists_iff : forall T P Q,
                     (forall x,
                        P x <-> Q x ) ->
                     ((exists x : T , P x ) <-> (exists x : T , Q x )).
Proof.
   intros. setoid_rewrite H. reflexivity.
Qed.

Lemma exists_impl : forall {T} (P Q : T -> Prop),
                      (forall x, P x -> Q x ) ->
                      (exists x , P x ) -> (exists x , Q x ).
Proof.
  clear. intuition.
  destruct H0; eauto.
Qed.

Lemma iff_eq : forall (P Q : Prop), P = Q -> (P <-> Q ).
Proof. clear. intros; subst; reflexivity. Qed.