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
.