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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 *)
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15 include "ground/arith/nat_le.ma".
16 include "ground/arith/ynat_nat.ma".
18 (* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
21 inductive yle: relation ynat ≝
22 | yle_inj: ∀m,n. m ≤ n → yle (yinj_nat m) (yinj_nat n)
23 | yle_inf: ∀x. yle x (∞)
27 "less equal (non-negative integers with infinity)"
30 (* Basic inversions *********************************************************)
33 lemma yle_inv_inj_dx (x) (n):
35 ∃∃m. m ≤ n & x = yinj_nat m.
36 #x #n0 @(insert_eq_1 … (yinj_nat n0))
39 lapply (eq_inv_yinj_nat_bi … H) -H #H destruct
40 /2 width=3 by ex2_intro/
42 elim (eq_inv_yinj_nat_inf … H)
47 lemma yle_inv_inj_bi (m) (n):
48 yinj_nat m ≤ yinj_nat n → m ≤ n.
50 elim (yle_inv_inj_dx … H) -H #x #Hxn #H
51 lapply (eq_inv_yinj_nat_bi … H) -H #H destruct //
55 lemma yle_inv_zero_dx (x):
58 elim (yle_inv_inj_dx ? (𝟎) H) -H #m #Hm #H destruct
59 <(nle_inv_zero_dx … Hm) -m //
63 lemma yle_inv_inf_sn (y): ∞ ≤ y → ∞ = y.
64 #y @(insert_eq_1 … (∞))
67 elim (eq_inv_inf_yinj_nat … H)
71 lemma yle_antisym (x) (y):
72 x ≤ y → y ≤ x → x = y.
75 <(nle_antisym … Hmn) -Hmn /2 width=1 by yle_inv_inj_bi/
76 | /2 width=1 by yle_inv_inf_sn/
80 (* Basic constructions ******************************************************)
83 lemma yle_zero_sn (y): 𝟎 ≤ y.
84 #y @(ynat_split_nat_inf … y) -y
85 /2 width=1 by yle_inj/
89 lemma yle_refl: reflexive … yle.
90 #x @(ynat_split_nat_inf … x) -x
91 /2 width=1 by yle_inj, yle_inf, nle_refl/
95 lemma ynat_split_le_ge (x) (y):
97 #x @(ynat_split_nat_inf … x) -x
98 [| /2 width=1 by or_intror/ ]
99 #m #y @(ynat_split_nat_inf … y) -y
100 [| /3 width=1 by yle_inf, or_introl/ ]
101 #n elim (nat_split_le_ge m n)
102 /3 width=1 by yle_inj, or_introl, or_intror/
105 (* Main constructions *******************************************************)
108 theorem yle_trans: Transitive … yle.
110 [ #m #n #Hxy #z @(ynat_split_nat_inf … z) -z //
111 /4 width=3 by yle_inv_inj_bi, nle_trans, yle_inj/
112 | #x #z #H <(yle_inv_inf_sn … H) -H //