matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llpxs_lleq.etc

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/substitution/lleq_leq.ma".
  16 include "basic_2/reduction/llpx_lleq.ma".
  17 include "basic_2/computation/cpxs_lleq.ma".
  18 include "basic_2/computation/llpxs_cpxs.ma".
  19
  20 (* LAZY SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS *************)
  21
  22 (* Properties on lazy equivalence for local environments ********************)
  23
  24 lemma llpxs_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
  25 /3 width=1 by llpx_lrefl, llpx_llpxs/ qed-.
  26
  27 lemma lleq_llpxs_trans: ∀h,g,G,L2,L,T,d. ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L →
  28                         ∀L1. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L.
  29 #h #g #G #L2 #L #T #d #H @(llpxs_ind … H) -L
  30 /3 width=3 by llpxs_strap1, llpxs_lrefl/
  31 qed-.
  32
  33 lemma lleq_llpxs_conf: ∀h,g,G,L1,L,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L →
  34                        ∀L2. L1 ⋕[T, d] L2 → ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L.
  35 /3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-.
  36 (*
  37 foct leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
  38                               ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
  39                               ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
  40                                    (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
  41 #h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
  42 [ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
  43   /3 width=5 by ex3_intro, conj/
  44 | #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
  45 | #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
  46   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
  47   lapply (ysucc_inv_Y_dx … He) -He #He
  48   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
  49   @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
  50   #T elim (IH T) #HL0dx #HL0sn
  51   @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
  52 | #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
  53   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
  54   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
  55   @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
  56   #T elim (IH T) #HL0dx #HL0sn
  57   @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
  58 ]
  59 qed-.
  60
  61 lamma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
  62                            ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
  63                            ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
  64                                 (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
  65 /2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
  66 *)