How do I prove the following using Coq?
(q V p) ∧ (¬p -> q) <-> (p V q).
My Attempt
Lemma work: (forall p q: Prop, (q \/ p)/\(~p -> q) <-> (p \/ q)).
Proof.
intros p q.
split.
intros q_or_p_and_not_p_implies_q.
intros p_or_q.
split.
Here is a proof of a very similar statement. It requires a little more work to swap the first p \\/ q
to q \\/ p
in order to match the statement you gave.
Theorem work : (forall p q : Prop, (p \/ q) /\ (~p -> q) <-> (p \/ q)).
Proof.
intros p q.
split.
(* Prove the "->" direction *)
intros given.
destruct given as [p_or_q _].
exact p_or_q.
(* Prove the "<-" direction *)
intros p_or_q.
refine (conj p_or_q _).
case p_or_q.
(* We're given that p is true, so ~p implies anything *)
intros p_true p_false.
case (p_false p_true).
(* We're given that q is true *)
intros q_true p_false.
exact q_true.
Qed.
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