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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_succ.ma".
17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
20 inductive yle: relation ynat ≝
21 | yle_inj: ∀m,n. m ≤ n → yle m n
22 | yle_Y : ∀m. yle m (∞)
25 interpretation "ynat 'less or equal to'" 'leq x y = (yle x y).
27 (* Basic inversion lemmas ***************************************************)
29 fact yle_inv_inj2_aux: ∀x,y. x ≤ y → ∀n. y = yinj n →
30 ∃∃m. m ≤ n & x = yinj m.
32 [ #x #y #Hxy #n #Hy destruct /2 width=3 by ex2_intro/
37 lemma yle_inv_inj2: ∀x,n. x ≤ yinj n → ∃∃m. m ≤ n & x = yinj m.
38 /2 width=3 by yle_inv_inj2_aux/ qed-.
40 lemma yle_inv_inj: ∀m,n. yinj m ≤ yinj n → m ≤ n.
41 #m #n #H elim (yle_inv_inj2 … H) -H
42 #x #Hxn #H destruct //
45 fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
47 [ #m #n #Hmn #H destruct /3 width=1 by le_n_O_to_eq, eq_f/
52 lemma yle_inv_O2: ∀m:ynat. m ≤ 0 → m = 0.
53 /2 width =3 by yle_inv_O2_aux/ qed-.
55 fact yle_inv_Y1_aux: ∀x,n. x ≤ n → x = ∞ → n = ∞.
60 lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
61 /2 width=3 by yle_inv_Y1_aux/ qed-.
63 lemma yle_antisym: ∀y,x. x ≤ y → y ≤ x → x = y.
65 /4 width=1 by yle_inv_Y1, yle_inv_inj, le_to_le_to_eq, eq_f/
68 (* Basic properties *********************************************************)
70 lemma le_O1: ∀n:ynat. 0 ≤ n.
71 * /2 width=1 by yle_inj/
74 lemma yle_refl: reflexive … yle.
75 * /2 width=1 by le_n, yle_inj/
78 lemma yle_split: ∀x,y:ynat. x ≤ y ∨ y ≤ x.
79 * /2 width=1 by or_intror/
80 #x * /2 width=1 by or_introl/
81 #y elim (le_or_ge x y) /3 width=1 by yle_inj, or_introl, or_intror/
84 (* Inversion lemmas on successor ********************************************)
86 fact yle_inv_succ1_aux: ∀x,y:ynat. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ ⫯⫰y = y.
88 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
89 #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
90 #m #Hnm #H destruct /3 width=1 by yle_inj, conj/
91 | #x #y #H destruct /2 width=1 by yle_Y, conj/
95 lemma yle_inv_succ1: ∀m,y:ynat. ⫯m ≤ y → m ≤ ⫰y ∧ ⫯⫰y = y.
96 /2 width=3 by yle_inv_succ1_aux/ qed-.
98 lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
99 #m #n #H elim (yle_inv_succ1 … H) -H //
102 lemma yle_inv_succ2: ∀x,y. x ≤ ⫯y → ⫰x ≤ y.
103 #x #y #Hxy elim (ynat_cases x)
105 | * #m #H destruct /2 width=1 by yle_inv_succ/
109 (* Properties on predecessor ************************************************)
111 lemma yle_pred_sn: ∀m,n. m ≤ n → ⫰m ≤ n.
112 #m #n * -m -n /3 width=3 by transitive_le, yle_inj/
115 lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
116 /2 width=1 by yle_refl, yle_pred_sn/ qed.
118 lemma yle_pred: ∀m,n. m ≤ n → ⫰m ≤ ⫰n.
119 #m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/
122 (* Properties on successor **************************************************)
124 lemma yle_succ: ∀m,n. m ≤ n → ⫯m ≤ ⫯n.
125 #m #n * -m -n /3 width=1 by yle_inj, le_S_S/
128 lemma yle_succ_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
129 #m #n * -m -n /3 width=1 by le_S, yle_inj/
132 lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
133 /2 width=1 by yle_succ_dx/ qed.
135 lemma yle_refl_SP_dx: ∀x. x ≤ ⫯⫰x.
139 lemma yle_succ2: ∀x,y. ⫰x ≤ y → x ≤ ⫯y.
140 #x #y #Hxy elim (ynat_cases x)
142 | * #m #H destruct /2 width=1 by yle_succ/
146 (* Main properties **********************************************************)
148 theorem yle_trans: Transitive … yle.
151 #z #H lapply (yle_inv_inj … H) -H
152 /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
153 | #x #z #H lapply (yle_inv_Y1 … H) //