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15 include "basic_2/computation/lcprs.ma".
16 include "basic_2/equivalence/lcpcs.ma".
18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
20 (* Properties about context sensitive computation on local environments *****)
22 lemma lcpcs_lcprs_dx: ∀L1,L2. L1 ⊢ ➡* L2 → L1 ⊢ ⬌* L2.
23 #L1 #L2 #H @(lcprs_ind … H) -L2 /width=1/ /3 width=3/
26 lemma lcpcs_lcprs_sn: ∀L1,L2. L2 ⊢ ➡* L1 → L1 ⊢ ⬌* L2.
27 #L1 #L2 #H @(lcprs_ind_dx … H) -L2 /width=1/ /3 width=3/
30 lemma lcpcs_lcprs_strap1: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L ⊢ ➡* L2 → L1 ⊢ ⬌* L2.
31 #L1 #L #HL1 #L2 #H @(lcprs_ind … H) -L2 /width=1/ /2 width=3/
34 lemma lcpcs_lcprs_strap2: ∀L1,L. L1 ⊢ ➡* L → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
35 #L1 #L #H #L2 #HL2 @(lcprs_ind_dx … H) -L1 /width=1/ /2 width=3/
38 lemma lcpcs_lcprs_div: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L2 ⊢ ➡* L → L1 ⊢ ⬌* L2.
39 #L1 #L #HL1 #L2 #H @(lcprs_ind_dx … H) -L2 /width=1/ /2 width=3/
42 lemma lcpcs_lcprs_conf: ∀L1,L. L ⊢ ➡* L1 → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
43 #L1 #L #H #L2 #HL2 @(lcprs_ind … H) -L1 /width=1/ /2 width=3/
46 lemma lcprs_div: ∀L1,L. L1 ⊢ ➡* L → ∀L2. L2 ⊢ ➡* L → L1 ⊢ ⬌* L2.
47 #L1 #L #HL1 #L2 #H @(lcprs_ind_dx … H) -L2 /2 width=1/ /2 width=3/