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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 set "baseuri" "cic:/matita/RELATIONAL/NLE/fwd".
17 include "logic/connectives.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.
24 intros. inversion H; clear H; intros;
25 [ apply (eq_gen_succ_zero ? ? H)
26 | lapply linear eq_gen_succ_succ to H2 as H0.
28 apply ex_intro; [|auto] (**)
32 theorem nle_gen_succ_succ: \forall x,y. x < succ y \to x <= y.
33 intros; inversion H; clear H; intros;
34 [ apply (eq_gen_succ_zero ? ? H)
35 | lapply linear eq_gen_succ_succ to H2 as H0.
36 lapply linear eq_gen_succ_succ to H3 as H2.
41 theorem nle_gen_succ_zero: \forall (P:Prop). \forall x. x < zero \to P.
43 lapply linear nle_gen_succ_1 to H. decompose.
44 apply (eq_gen_zero_succ ? ? H1).
47 theorem nle_gen_zero_2: \forall x. x <= zero \to x = zero.
48 intros 1. elim x; clear x; intros;
50 | apply (nle_gen_succ_zero ? ? H1)
54 theorem nle_gen_succ_2: \forall y,x. x <= succ y \to
55 x = zero \lor \exists z. x = succ z \land z <= y.
56 intros 2; elim x; clear x; intros;
58 | lapply linear nle_gen_succ_succ to H1.
59 right. apply ex_intro; [|auto] (**)