matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx_ldrop.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/reduction/cpx_llpx_sn.ma".
  16 include "basic_2/reduction/cpx_lift.ma".
  17 include "basic_2/reduction/llpx.ma".
  18
  19 (* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma llpx_inv_lref_ge_dx: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
  24                            ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
  25                            ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
  26                                     ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
  27 /2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
  28
  29 lemma llpx_inv_lref_ge_sn: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
  30                            ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
  31                            ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
  32                                     ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
  33 /2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
  34
  35 lemma llpx_inv_lref_ge_bi: ∀h,g,G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2 → d ≤ i →
  36                            ∀I1,I2,K1,K2,V1,V2.
  37                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
  38                            ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2.
  39 /2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
  40
  41 lemma llpx_inv_bind_O: ∀h,g,a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [h, g, ⓑ{a,I}V.T, 0] L2 →
  42                        ⦃G, L1⦄ ⊢ ➡[h, g, V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V.
  43 /2 width=2 by llpx_sn_inv_bind_O/ qed-.
  44
  45 lemma llpx_bind_repl_O: ∀h,g,I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V2 →
  46                         ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[h, g, W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{J}W2.
  47 /2 width=4 by llpx_sn_bind_repl_O/ qed-.
  48
  49 (* Advanced forward lemmas **************************************************)
  50
  51 lemma llpx_fwd_bind_O_dx: ∀h,g,a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, 0] L2 →
  52                           ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, 0] L2.ⓑ{I}V.
  53 /2 width=3 by llpx_sn_fwd_bind_O_dx/ qed-.
  54
  55 (* Advanced properties ******************************************************)
  56
  57 lemma llpx_cpx_conf: ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx h g G 0).
  58 /3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/ qed-.
  59
  60 (* Inversion lemmas on relocation *******************************************)
  61
  62 lemma llpx_ldrop_trans_O: ∀h,g,G,L1,L2,U. ⦃G, L1⦄ ⊢ ➡[h, g, U, 0] L2 →
  63                           ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
  64                           ∃∃K1. ⇩[e] L1 ≡ K1 & ⦃G, K1⦄ ⊢ ➡[h, g, T, 0] K2.
  65 /2 width=5 by llpx_sn_ldrop_trans_O/ qed-.
  66
  67 (* Properties on supclosure *************************************************)
  68
  69 lemma llpx_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
  70                       ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
  71                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
  72 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  73 /3 width=7 by llpx_fwd_bind_O_dx, llpx_fwd_pair_sn,llpx_fwd_flat_dx, ex3_2_intro/
  74 [ #I #G1 #L1 #V1 #K1 #H elim (llpx_inv_lref_ge_dx … H) -H [5,6: // |2,3,4: skip ]
  75   #Y1 #W1 #HKL1 #HW1 #HWV1 elim (lift_total V1 0 1)
  76   /4 width=7 by llpx_cpx_conf, cpx_delta, fqu_drop, ldrop_fwd_drop2, ex3_2_intro/
  77 | #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HU1
  78   elim (llpx_ldrop_trans_O … HU1 … HLK1) -L1
  79   /3 width=5 by fqu_drop, ex3_2_intro/
  80 ]
  81 qed-.
  82
  83 lemma llpx_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
  84                        ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g, T1, 0] L1 →
  85                        ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g, T2, 0] L2.
  86 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
  87 [ #HT12 elim (llpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
  88 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
  89 ]
  90 qed-.