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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 "basic_2/conversion/cpc.ma".
17 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
19 definition cpcs: lenv → relation term ≝
22 interpretation "context-sensitive parallel equivalence (term)"
23 'PConvStar L T1 T2 = (cpcs L T1 T2).
25 (* Basic eliminators ********************************************************)
27 lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 →
28 (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) →
29 ∀T2. L ⊢ T1 ⬌* T2 → R T2.
30 #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
33 lemma cpcs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
34 (∀T1,T. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → R T → R T1) →
35 ∀T1. L ⊢ T1 ⬌* T2 → R T1.
36 #L #T2 #R #HT2 #IHT2 #T1 #HT12
37 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
40 (* Basic properties *********************************************************)
42 (* Basic_1: was: pc3_refl *)
43 lemma cpcs_refl: ∀L,T. L ⊢ T ⬌* T.
46 lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
49 lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
52 (* Basic_1: was: pc3_pr2_r *)
53 lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2.
56 (* Basic_1: was: pc3_pr2_x *)
57 lemma cpcs_cpr_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2.
60 lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
63 (* Basic_1: was: pc3_pr2_u *)
64 lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
67 lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
70 (* Basic_1: was: pc3_pr2_u2 *)
71 lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
74 (* Basic_1: was: pc3_s *)
75 lemma cprs_comm: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → L ⊢ T2 ⬌* T1.
76 #L #T1 #T2 #H @(cpcs_ind … H) -T2 // /3 width=3/
79 (* Basic_1: removed theorems 6:
80 clear_pc3_trans pc3_ind_left
81 pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
82 Basic_1: removed local theorems 5:
83 pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left