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15 include "basic_2/conversion/lfpc.ma".
17 (* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
19 definition lfpcs: relation lenv ≝ TC … lfpc.
21 interpretation "focalized parallel equivalence (local environment)"
22 'FocalizedPConvStar L1 L2 = (lfpcs L1 L2).
24 (* Basic eliminators ********************************************************)
26 lemma lfpcs_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 lfpcs_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 lfpcs_refl: reflexive … lfpcs.
44 lemma lfpcs_sym: symmetric … lfpcs.
47 lemma lfpc_lfpcs: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
50 lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
53 lemma lfpcs_strap2: ∀L1,L,L2. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
56 lemma lfpcs_lfpr_dx: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
59 lemma lfpcs_lfpr_sn: ∀L1,L2. ⦃L2⦄ ➡ ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
62 lemma lfpcs_lfpr_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
65 lemma lfpcs_lfpr_strap2: ∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
68 lemma lfpcs_lfpr_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡ ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
71 lemma lfpcs_lfpr_conf: ∀L1,L. ⦃L⦄ ➡ ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.