include "NLE/defs.ma".
theorem nle_gen_succ_1: \forall x,y. x < y \to
- \exists z. y = succ z \land x <= z.
+ \exists z. y = succ z \land x <= z.
intros. inversion H; clear H; intros;
[ apply (eq_gen_succ_zero ? ? H)
| lapply linear eq_gen_succ_succ to H2 as H0.
- rewrite > H0. clear H0.
- apply ex_intro; [|auto] (**)
+ subst.
+ apply ex_intro; [|auto new] (**)
].
qed.
[ apply (eq_gen_succ_zero ? ? H)
| lapply linear eq_gen_succ_succ to H2 as H0.
lapply linear eq_gen_succ_succ to H3 as H2.
- rewrite > H0. rewrite > H2. clear H0 H2.
- auto
+ subst. auto new
].
qed.
theorem nle_gen_zero_2: \forall x. x <= zero \to x = zero.
intros 1. elim x; clear x; intros;
- [ auto
+ [ auto new
| apply (nle_gen_succ_zero ? ? H1)
].
qed.
theorem nle_gen_succ_2: \forall y,x. x <= succ y \to
- x = zero \lor \exists z. x = succ z \land z <= y.
+ x = zero \lor \exists z. x = succ z \land z <= y.
intros 2; elim x; clear x; intros;
- [ auto
+ [ auto new
| lapply linear nle_gen_succ_succ to H1.
- right. apply ex_intro; [|auto] (**)
+ right. apply ex_intro; [|auto new] (**)
].
qed.