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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 set "baseuri" "cic:/matita/RELATIONAL/NLE/fwd".
17 include "logic/connectives.ma".
19 include "NPlus/fwd.ma".
20 include "NLE/defs.ma".
22 theorem nle_gen_succ_1: \forall x,y. x < y \to
23 \exists z. y = succ z \land x <= z.
26 lapply linear nplus_gen_succ_2 to H1 as H.
28 apply ex_intro; auto. (**)
32 theorem nle_gen_succ_succ: \forall x,y. x < succ y \to x <= y.
34 lapply linear nle_gen_succ_1 to H as H0. decompose H0.
35 lapply linear eq_gen_succ_succ to H1 as H. subst.
39 theorem nle_gen_succ_zero: \forall x. x < zero \to False.
41 lapply linear nle_gen_succ_1 to H. decompose.
42 lapply linear eq_gen_zero_succ to H1. decompose.
45 theorem nle_gen_zero_2: \forall x. x <= zero \to x = zero.
46 intros 1. elim x; clear x; intros;
48 | lapply linear nle_gen_succ_zero to H1. decompose.
52 theorem nle_gen_succ_2: \forall y,x. x <= succ y \to
53 x = zero \lor \exists z. x = succ z \land z <= y.
54 intros 2; elim x; clear x; intros;
56 | lapply linear nle_gen_succ_succ to H1.
57 right. apply ex_intro; [|auto new timeout=30] (**)