matita/matita/contribs/lambdadelta/static_2/syntax/acle.ma

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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.