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 *)
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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: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
24 #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /3 width=3/
27 (* Basic_1: was: pc3_pr3_x *)
28 lemma cpcs_cprs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡* T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
29 #G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /3 width=3/
32 lemma cpcs_cprs_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
33 #G #L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /2 width=3/
36 lemma cpcs_cprs_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
37 #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /width=1/ /2 width=3/
40 lemma cpcs_cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
41 #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /2 width=3/
44 (* Basic_1: was: pc3_pr3_conf *)
45 lemma cpcs_cprs_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
46 #G #L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /width=1/ /2 width=3/
49 (* Basic_1: was: pc3_pr3_t *)
50 (* Basic_1: note: pc3_pr3_t should be renamed *)
51 lemma cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
52 #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=1/ /2 width=3/
55 lemma cprs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
56 /3 width=5 by cpr_cprs, cprs_div/ qed-.
58 lemma cpr_cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
59 /3 width=3 by cpr_cprs, cprs_div/ qed-.