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14
15 include "basic_2/computation/cprs.ma".
16 include "basic_2/equivalence/cpcs.ma".
17
18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
19
20 (* Properties about context sensitive computation on terms ******************)
21
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
25 /3 width=3 by cpcs_cpr_strap1, cpcs_strap1, cpc_cpcs/
26 qed.
27
28 (* Basic_1: was: pc3_pr3_x *)
29 lemma cpcs_cprs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡* T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
30 #G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2
31 /3 width=3 by cpcs_cpr_div, cpcs_strap1, cpcs_cprs_dx/
32 qed.
33
34 lemma cpcs_cprs_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
35 #G #L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /2 width=3 by cpcs_cpr_strap1/
36 qed-.
37
38 lemma cpcs_cprs_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
39 #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /2 width=3 by cpcs_cpr_strap2/
40 qed-.
41
42 lemma cpcs_cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
43 #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=3 by cpcs_cpr_div/
44 qed-.
45
46 (* Basic_1: was: pc3_pr3_conf *)
47 lemma cpcs_cprs_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
48 #G #L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /2 width=3 by cpcs_cpr_conf/
49 qed-.
50
51 (* Basic_1: was: pc3_pr3_t *)
52 (* Basic_1: note: pc3_pr3_t should be renamed *)
53 lemma cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
54 #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2
55 /2 width=3 by cpcs_cpr_div, cpcs_cprs_dx/
56 qed.
57
58 lemma cprs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
59 /3 width=5 by cpr_cprs, cprs_div/ qed-.
60
61 lemma cpr_cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
62 /3 width=3 by cpr_cprs, cprs_div/ qed-.