Answer:
This lemma can be proved constructively. If you think about what can be done at each step to make progress the lemma proves itself:
Lemma PeirceContra :
forall P Q, ~P -> ~((P -> Q) -> P).
Proof.
intros P Q np.
unfold "~".
intros pq_p.
apply np. (* this is pretty much the only thing we can do at this point *)
apply pq_p. (* this is almost inevitable too *)
(* the rest should be easy *)
(* Qed. *)
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Answers & Comments
Answer:
This lemma can be proved constructively. If you think about what can be done at each step to make progress the lemma proves itself:
Lemma PeirceContra :
forall P Q, ~P -> ~((P -> Q) -> P).
Proof.
intros P Q np.
unfold "~".
intros pq_p.
apply np. (* this is pretty much the only thing we can do at this point *)
apply pq_p. (* this is almost inevitable too *)
(* the rest should be easy *)
(* Qed. *)