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14
15 include "static_2/syntax/ac.ma".
16
17 (* APPLICABILITY CONDITION PREORDER *****************************************)
18
19 definition acle: relation ac ≝
20            λa1,a2. ∀m. ad a1 m → ∃∃n. ad a2 n & m ≤ n.
21
22 interpretation "preorder (applicability domain)"
23   'subseteq a1 a2 = (acle a1 a2).
24
25 (* Basic properties *********************************************************)
26
27 lemma acle_refl: reflexive … acle.
28 /2 width=3 by ex2_intro/ qed.
29
30 lemma acle_omega (a): a ⊆ 𝛚.
31 /2 width=1 by acle_refl/
32 qed.
33
34 lemma acle_one (a): ∀n. ad a n → 𝟏 ⊆ a.
35 #a #n #Ha #m #Hm destruct
36 /2 width=3 by ex2_intro/
37 qed.
38
39 lemma acle_eq_monotonic_le (k1) (k2):
40       k1 ≤ k2 → (ac_eq k1) ⊆ (ac_eq k2).
41 #k1 #k2 #Hk #m #Hm destruct
42 /2 width=3 by ex2_intro/
43 qed.
44
45 lemma acle_le_monotonic_le (k1) (k2):
46       k1 ≤ k2 → (ac_le k1) ⊆ (ac_le k2).
47 #k1 #k2 #Hk #m #Hm
48 /3 width=3 by acle_refl, nle_trans/
49 qed.
50
51 lemma acle_eq_le (k): (ac_eq k) ⊆ (ac_le k).
52 #k #m #Hm destruct
53 /2 width=1 by nle_refl, acle_refl/
54 qed.
55
56 lemma acle_le_eq (k): (ac_le k) ⊆ (ac_eq k).
57 #k #m #Hm /2 width=3 by ex2_intro/
58 qed.