如何在 Coq 中证明逻辑等价？
How to prove logical equivalence in Coq?
如何使用 Coq 证明以下内容？
(q V p) ∧ (¬p -> q) <-> (p V q).
我的尝试
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.
这是一个非常相似的陈述的证明。将第一个 p \/ q 交换为 q \/ p 以匹配您给出的语句需要更多的工作。
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.
如何使用 Coq 证明以下内容？
(q V p) ∧ (¬p -> q) <-> (p V q).
我的尝试
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.
这是一个非常相似的陈述的证明。将第一个 p \/ q 交换为 q \/ p 以匹配您给出的语句需要更多的工作。
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.