+++ /dev/null
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-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lleq_leq.ma".
-include "basic_2/reduction/llpx_lleq.ma".
-include "basic_2/computation/cpxs_lleq.ma".
-include "basic_2/computation/llpxs_cpxs.ma".
-
-(* LAZY SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS *************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma llpxs_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-/3 width=1 by llpx_lrefl, llpx_llpxs/ qed-.
-
-lemma lleq_llpxs_trans: ∀h,g,G,L2,L,T,d. ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L →
- ∀L1. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L.
-#h #g #G #L2 #L #T #d #H @(llpxs_ind … H) -L
-/3 width=3 by llpxs_strap1, llpxs_lrefl/
-qed-.
-
-lemma lleq_llpxs_conf: ∀h,g,G,L1,L,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L →
- ∀L2. L1 ⋕[T, d] L2 → ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L.
-/3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-.
-(*
-foct leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
- ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
- ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
- (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
-#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
-[ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
- /3 width=5 by ex3_intro, conj/
-| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
-| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
- elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
- lapply (ysucc_inv_Y_dx … He) -He #He
- elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
- #T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
-| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
- elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
- elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
- #T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
-]
-qed-.
-
-lamma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
- ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
- ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
- (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
-/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
-*)