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15 include "basic_2/computation/lfprs.ma".
16 include "basic_2/equivalence/lfpcs.ma".
18 (* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
20 (* Properties on focalized computation for local environments ***************)
22 lemma lfpcs_lfprs_dx: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
23 #L1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /3 width=3/
26 lemma lfpcs_lfprs_sn: ∀L1,L2. ⦃L2⦄ ➡* ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
27 #L1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /3 width=3/
30 lemma lfpcs_lfprs_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
31 #L1 #L #HL1 #L2 #H @(lfprs_ind … H) -L2 /width=1/ /2 width=3/
34 lemma lfpcs_lfprs_strap2: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
35 #L1 #L #H #L2 #HL2 @(lfprs_ind_dx … H) -L1 /width=1/ /2 width=3/
38 lemma lfpcs_lfprs_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
39 #L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /width=1/ /2 width=3/
42 lemma lfpcs_lfprs_conf: ∀L1,L. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
43 #L1 #L #H #L2 #HL2 @(lfprs_ind … H) -L1 /width=1/ /2 width=3/
46 lemma lfprs_div: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
47 #L1 #L #HL1 #L2 #H @(lfprs_ind_dx … H) -L2 /2 width=1/ /2 width=3/