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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/ynat/ynat_le.ma".
17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
19 (* strict order relation *)
20 inductive ylt: relation ynat ≝
21 | ylt_inj: ∀m,n. m < n → ylt m n
22 | ylt_Y : ∀m:nat. ylt m (∞)
25 interpretation "ynat 'less than'" 'lt x y = (ylt x y).
27 (* Basic inversion lemmas ***************************************************)
29 fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
30 ∃∃m. m < n & x = yinj m.
32 [ #x #y #Hxy #n #Hy elim (le_inv_S1 … Hxy) -Hxy
33 #m #Hm #H destruct /3 width=3 by le_S_S, ex2_intro/
38 lemma ylt_inv_inj2: ∀x,n. x < yinj n →
39 ∃∃m. m < n & x = yinj m.
40 /2 width=3 by ylt_inv_inj2_aux/ qed-.
42 lemma ylt_inv_inj: ∀m,n. yinj m < yinj n → m < n.
43 #m #n #H elim (ylt_inv_inj2 … H) -H
47 fact ylt_inv_Y2_aux: ∀x,y. x < y → y = ∞ → ∃m. x = yinj m.
48 #x #y * -x -y /2 width=2 by ex_intro/
51 lemma ylt_inv_Y2: ∀x. x < ∞ → ∃m. x = yinj m.
52 /2 width=3 by ylt_inv_Y2_aux/ qed-.
54 lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
55 * // #n #H lapply (ylt_inv_inj … H) -H normalize
56 /3 width=1 by S_pred, eq_f/
59 (* Inversion lemmas on successor ********************************************)
61 fact ylt_inv_succ1_aux: ∀x,y. x < y → ∀m. x = ⫯m → ∃∃n. m < n & y = ⫯n.
63 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
64 #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
66 @(ex2_intro … m) /2 width=1 by ylt_inj/ (**) (* explicit constructor *)
67 | #x #y #H elim (ysucc_inv_inj_sn … H) -H
69 @(ex2_intro … (∞)) /2 width=1 by/ (**) (* explicit constructor *)
73 lemma ylt_inv_succ1: ∀m,y. ⫯m < y → ∃∃n. m < n & y = ⫯n.
74 /2 width=3 by ylt_inv_succ1_aux/ qed-.
76 lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
77 #m #n #H elim (ylt_inv_succ1 … H) -H
81 fact ylt_inv_succ2_aux: ∀x,y. x < y → ∀n. y = ⫯n → x ≤ n.
83 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
84 #n #H1 #H2 destruct /3 width=1 by yle_inj, le_S_S_to_le/
85 | #x #n #H lapply (ysucc_inv_Y_sn … H) -H //
89 (* Forward lemmas on successor **********************************************)
91 lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
92 /2 width=3 by ylt_inv_succ2_aux/ qed-.
94 (* inversion and forward lemmas on yle **************************************)
96 lemma lt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
97 #m #n * -m -n /3 width=1 by yle_pred_sn, yle_inj, yle_Y/
100 lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
102 [ #m #n #Hmn #H lapply (yle_inv_inj … H) -H
103 #H elim (lt_refl_false n) /2 width=3 by le_to_lt_to_lt/
104 | #m #H lapply (yle_inv_Y1 … H) -H
109 (* Properties on successor **************************************************)
111 lemma ylt_O_succ: ∀n. 0 < ⫯n.
112 * /2 width=1 by ylt_inj/
115 (* Properties on yle ********************************************************)
117 lemma yle_to_ylt_or_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.
119 [ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn
120 /3 width=1 by or_introl, ylt_inj/
121 | * /2 width=1 by or_introl, ylt_Y/
125 lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z.
127 [ #y #z #Hyz #H elim (ylt_inv_inj2 … H) -H
128 #m #Hm #H destruct /3 width=3 by ylt_inj, lt_to_le_to_lt/
133 lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x < z.
135 [ #y #z #Hyz #H elim (yle_inv_inj2 … H) -H
136 #m #Hm #H destruct /3 width=3 by ylt_inj, le_to_lt_to_lt/
137 | #y #H elim (yle_inv_inj2 … H) -H //
141 (* Main properties **********************************************************)
143 theorem ylt_trans: Transitive … ylt.
146 #z #H lapply (ylt_inv_inj … H) -H
147 /3 width=3 by transitive_lt, ylt_inj/ (**) (* full auto too slow *)
148 | #x #z #H elim (ylt_yle_false … H) //