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15 set "baseuri" "cic:/matita/RELATIONAL/NLE/props".
19 theorem nle_zero: \forall q. zero <= q.
21 intros. apply ex_intro; auto. (**)
24 theorem nle_succ: \forall p,q. p <= q \to succ p <= succ q.
27 apply ex_intro; auto. (**)
30 theorem nle_refl: \forall x. x <= x.
31 intros 1; elim x; clear x; intros; auto new timeout=100.
34 theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
35 intros 1. elim x; clear x; intros;
36 [ auto new timeout=100.
37 | lapply linear nle_gen_succ_1 to H1 as H0. decompose H0. subst.
42 theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.
43 intros 1. elim y; clear y; intros;
44 [ lapply linear nle_gen_zero_2 to H. auto new timeout=100
45 | lapply linear nle_gen_succ_2 to H1. decompose;
46 [ subst. auto new timeout=100
47 | lapply linear H to H3 as H0. decompose;
48 subst; auto new timeout=100