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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "ground_2/notation/functions/successor_1.ma".
16 include "ground_2/ynat/ynat_pred.ma".
18 (* NATURAL NUMBERS WITH INFINITY ********************************************)
20 (* the successor function *)
21 definition ysucc: ynat → ynat ≝ λm. match m with
26 interpretation "ynat successor" 'Successor m = (ysucc m).
28 (* Properties ***************************************************************)
30 lemma ypred_succ: ∀m. ⫰⫯m = m.
33 lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m. n = ⫯m.
35 [ * /2 width=1 by or_introl/
36 #n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
37 | @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
41 lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
43 #m #IHm whd in ⊢ (??%?); >IHm //
46 (* Inversion lemmas *********************************************************)
48 lemma ysucc_inj: ∀m,n. ⫯m = ⫯n → m = n.
49 #m #n #H <(ypred_succ m) <(ypred_succ n) //
52 lemma ysucc_inv_refl: ∀m. ⫯m = m → m = ∞.
54 #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
55 #H elim (lt_refl_false m) //
58 lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
59 ∃∃m1. n1 = yinj m1 & m2 = S m1.
61 [ #n1 #H destruct /2 width=3 by ex2_intro/
66 lemma ysucc_inv_inj_dx: ∀m2,n1. ⫯n1 = yinj m2 →
67 ∃∃m1. n1 = yinj m1 & m2 = S m1.
68 /2 width=1 by ysucc_inv_inj_sn/ qed-.
70 lemma ysucc_inv_Y_sn: ∀m. ∞ = ⫯m → m = ∞.
75 lemma ysucc_inv_Y_dx: ∀m. ⫯m = ∞ → m = ∞.
76 /2 width=1 by ysucc_inv_Y_sn/ qed-.
78 lemma ysucc_inv_O_sn: ∀m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
79 #m #H elim (ysucc_inv_inj_sn … H) -H
83 lemma ysucc_inv_O_dx: ∀m. ⫯m = 0 → ⊥.
84 /2 width=2 by ysucc_inv_O_sn/ qed-.