1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/computation/cprs.ma".
16 include "basic_2/equivalence/cpcs.ma".
18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
20 (* Properties about context sensitive computation on terms ******************)
22 (* Basic_1: was: pc3_pr3_r *)
23 lemma cpcs_cprs_dx: ∀L,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ T1 ⬌* T2.
24 #L #T1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /3 width=3/
27 lemma cpcs_tpr_dx: ∀L,T1,T2. T1 ➡ T2 → L ⊢ T1 ⬌* T2.
30 (* Basic_1: was: pc3_pr3_x *)
31 lemma cpcs_cprs_sn: ∀L,T1,T2. L ⊢ T2 ➡* T1 → L ⊢ T1 ⬌* T2.
32 #L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /3 width=3/
35 lemma cpcs_tpr_sn: ∀L,T1,T2. T2 ➡ T1 → L ⊢ T1 ⬌* T2.
38 lemma cpcs_cprs_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
39 #L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /2 width=3/
42 lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
45 lemma cpcs_cprs_strap2: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
46 #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /width=1/ /2 width=3/
49 lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
52 lemma cpcs_cprs_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
53 #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /2 width=3/
56 lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
59 (* Basic_1: was: pc3_pr3_conf *)
60 lemma cpcs_cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
61 #L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /width=1/ /2 width=3/
64 lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
67 (* Basic_1: was: pc3_pr3_t *)
68 (* Basic_1: note: pc3_pr3_t should be renamed *)
69 lemma cprs_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
70 #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=1/ /2 width=3/
73 lemma cprs_cpr_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
74 /3 width=5 by step, cprs_div/ qed-.
76 lemma cpr_cprs_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
77 /3 width=3 by step, cprs_div/ qed-.
79 lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
80 /3 width=5 by step, cprs_div/ qed-.