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+
+(* $Id$ *)
+
+(* Classical Propositional Logic *)
+
+Hints Unfold not : core.
+
+Axiom classic: (P:Prop)(P \/ ~(P)).
+
+Lemma NNPP : (p:Prop)~(~(p))->p.
+Proof.
+Unfold not; Intros; Elim (classic p); Auto.
+Intro NP; Elim (H NP).
+Qed.
+
+Lemma not_imply_elim : (P,Q:Prop)~(P->Q)->P.
+Proof.
+Intros; Apply NNPP; Red.
+Intro; Apply H; Intro; Absurd P; Trivial.
+Qed.
+
+Lemma not_imply_elim2 : (P,Q:Prop)~(P->Q) -> ~Q.
+Proof.
+Intros; Elim (classic Q); Auto.
+Qed.
+
+Lemma imply_to_or : (P,Q:Prop)(P->Q) -> ~P \/ Q.
+Proof.
+Intros; Elim (classic P); Auto.
+Qed.
+
+Lemma imply_to_and : (P,Q:Prop)~(P->Q) -> P /\ ~Q.
+Proof.
+Intros; Split.
+Apply not_imply_elim with Q;  Trivial.
+Apply not_imply_elim2 with P; Trivial.
+Qed.
+
+Lemma or_to_imply : (P,Q:Prop)(~P \/ Q) -> P->Q.
+Proof.
+Induction 1; Auto.
+Intros H1 H2; Elim (H1 H2).
+Qed.
+
+Lemma not_and_or : (P,Q:Prop)~(P/\Q)-> ~P \/ ~Q.
+Proof.
+Intros; Elim (classic P); Auto.
+Qed.
+
+Lemma or_not_and : (P,Q:Prop)(~P \/ ~Q) -> ~(P/\Q).
+Proof.
+Induction 1; Red; Induction 2; Auto.
+Qed.
+
+Lemma not_or_and : (P,Q:Prop)~(P\/Q)-> ~P /\ ~Q.
+Proof.
+Intros; Elim (classic P); Auto.
+Qed.
+
+Lemma and_not_or : (P,Q:Prop)(~P /\ ~Q) -> ~(P\/Q).
+Proof.
+Induction 1; Red; Induction 3; Trivial.
+Qed.
+
+Lemma imply_and_or: (P,Q:Prop)(P->Q) -> P \/ Q -> Q.
+Proof.
+Induction 2; Trivial.
+Qed.
+
+Lemma imply_and_or2: (P,Q,R:Prop)(P->Q) -> P \/ R -> Q \/ R.
+Proof.
+Induction 2; Auto.
+Qed.