(** 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.