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15 include "basic_2/conversion/lcpc.ma".
17 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL EBVIRONMENTS *************)
19 definition lcpcs: relation lenv ≝ TC … lcpc.
21 interpretation "context-sensitive parallel equivalence (local environment)"
22 'CPConvStar L1 L2 = (lcpcs L1 L2).
24 (* Basic eliminators ********************************************************)
26 lemma lcpcs_ind: ∀L1. ∀R:predicate lenv. R L1 →
27 (∀L,L2. L1 ⊢ ⬌* L → L ⊢ ⬌ L2 → R L → R L2) →
28 ∀L2. L1 ⊢ ⬌* L2 → R L2.
29 #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
32 lemma lcpcs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
33 (∀L1,L. L1 ⊢ ⬌ L → L ⊢ ⬌* L2 → R L → R L1) →
34 ∀L1. L1 ⊢ ⬌* L2 → R L1.
35 #L2 #R #HL2 #IHL2 #L1 #HL12
36 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
39 (* Basic properties *********************************************************)
41 lemma lcpcs_refl: ∀L. L ⊢ ⬌* L.
44 lemma lcpcs_strap1: ∀L1,L,L2. L1 ⊢ ⬌* L → L ⊢ ⬌ L2 → L1 ⊢ ⬌* L2.
47 lemma lcpcs_strap2: ∀L1,L,L2. L1 ⊢ ⬌ L → L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
50 lemma lcpcs_lcpr_dx: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ⬌* L2.
53 lemma lcpcs_lcpr_sn: ∀L1,L2. L2 ⊢ ➡ L1 → L1 ⊢ ⬌* L2.
56 lemma lcpcs_lcpr_strap1: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L ⊢ ➡ L2 → L1 ⊢ ⬌* L2.
59 lemma lcpcs_lcpr_strap2: ∀L1,L. L1 ⊢ ➡ L → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
62 lemma lcpcs_lcpr_div: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L2 ⊢ ➡ L → L1 ⊢ ⬌* L2.
65 lemma lcpcs_lcpr_conf: ∀L1,L. L ⊢ ➡ L1 → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
68 lemma lcprs_comm: ∀L1,L2. L1 ⊢ ⬌* L2 → L2 ⊢ ⬌* L1.
69 #L1 #L2 #H @(lcpcs_ind … H) -L2 // /3 width=3/