(** Peirce's law states [forall P Q:Prop, ((P -> Q) -> P) -> P]. Thanks to [forall P, False -> P], it is equivalent to the following form *)

(* Specifications: *)
Lemma Peirce : forall P:Prop, ((P -> False) -> P) -> P.
Lemma not_imply_elim : forall P Q:Prop, ~ (P -> Q) -> P.
Lemma not_imply_elim2 : forall P Q:Prop, ~ (P -> Q) -> ~ Q.
Lemma imply_to_or : forall P Q:Prop, (P -> Q) -> ~ P \/ Q.
Lemma imply_to_and : forall P Q:Prop, ~ (P -> Q) -> P /\ ~ Q.
Lemma or_to_imply : forall P Q:Prop, ~ P \/ Q -> P -> Q.
Lemma not_and_or : forall P Q:Prop, ~ (P /\ Q) -> ~ P \/ ~ Q.
Lemma or_not_and : forall P Q:Prop, ~ P \/ ~ Q -> ~ (P /\ Q).
Lemma not_or_and : forall P Q:Prop, ~ (P \/ Q) -> ~ P /\ ~ Q.
Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q).
Lemma imply_and_or : forall P Q:Prop, (P -> Q) -> P \/ Q -> Q.
Lemma imply_and_or2 : forall P Q R:Prop, (P -> Q) -> P \/ R -> Q \/ R.
Lemma proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.
Proof proof_irrelevance_cci classic.