(**
15-817 Spring 2012
HW1
Due: 7 March 2012, 15:00
Prove the following lemmae and send the file to soonhok+15817@cs.cmu.edu
**)
(** This file is based on Prof. Anita Wasilewska's Lecture Note **)
(** http://www.cs.sunysb.edu/~cse371/chapter10.pdf **)
(** List of classical tautologies
which are also provable in intuitionistic tautologies **)
Lemma c1: forall a:Prop, a -> a.
admit.
Qed.
Lemma c2: forall a b:Prop, a -> (b -> a).
admit.
Qed.
Lemma c3: forall a b:Prop, a -> (b -> (a /\ a)).
admit.
Qed.
Lemma c4: forall a b c:Prop, (a -> (b -> c)) -> ((a -> b) -> (a -> c)).
admit.
Qed.
Lemma c5: forall a:Prop, a -> ~~a.
admit.
Qed.
Lemma c6: forall a:Prop, ~(a /\ ~a).
admit.
Qed.
Lemma c7: forall a b:Prop, (~a \/ b) -> (a -> b).
admit.
Qed.
Lemma c8: forall a b:Prop, ~(a \/ b) -> (a -> b).
admit.
Qed.
Lemma c9: forall a b:Prop, (~a /\ ~b) -> (~(a \/ b)).
admit.
Qed.
Lemma c10: forall a b c:Prop, (c -> a) -> (c -> (a -> b)) -> (c -> b).
admit.
Qed.
Lemma c11: forall a b:Prop, (a -> b) -> (~b -> ~a).
admit.
Qed.
Lemma c12: forall a b:Prop, (a -> ~b) -> (b -> ~a).
admit.
Qed.
Lemma c13: forall a b:Prop, (~~~a -> ~a).
admit.
Qed.
Lemma c14: forall a b:Prop, (~a -> ~~~a).
admit.
Qed.
Lemma c15: forall a b:Prop, (~~(a -> b)) -> (a -> ~~b).
admit.
Qed.
(** The following classical tautologies are not intuitionistic tautologies. **)
Require Import Classical.
Lemma i1: forall a:Prop, (a \/ ~a).
Proof.
admit.
Qed.
Lemma i2: forall a b:Prop, (~~a -> a).
Qed.
Lemma i3: forall a b:Prop, (a -> b) -> (~a \/ b).
admit.
Qed.
Lemma i4: forall a b:Prop, ~(a \/ b) -> (~a \/ ~b).
admit.
Qed.
Lemma i5: forall a b:Prop, (~a -> b) -> (~b -> a).
admit.
Qed.
Lemma i6: forall a b:Prop, (~a -> ~b) -> (b -> a).
admit.
Qed.
Lemma i7: forall a b:Prop, ((a -> b) -> a) -> a.
admit.
Qed.