proposition: Theorem div2_le : forall n:nat, div2 n <= n.
proof: Proof. induction n as [ | n IHn]. - simpl; auto. - induction n. + simpl; auto. + Abort. End bad_proof_for_div2_le. Theorem div2_rect (P: nat -> Type) : P 0 -> P 1 -> (forall n, P n -> P (S n) -> P(S (S n)))-> forall n:nat, P n. Proof. intros H0 H1 H n; assert (X: ( P n * P (S n))%type). - induction n; intuition. - now destruct X. Defined.
imports: ZArith List Arith Lia Zwf Program