matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.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/multiple/llpx_sn_drop.ma".
  16 include "basic_2/multiple/lleq.ma".
  17
  18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
  23                         ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W.
  24 /2 width=7 by llpx_sn_bind_repl_O/ qed-.
  25
  26 lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
  27                           ∀L1,L2,T,l. L1 ≡[T, l] L2 →
  28                           ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L.
  29 #R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l
  30 [1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
  31 |4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
  32 | #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2
  33   /3 width=11 by llpx_sn_lref/
  34 | #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H
  35   /3 width=1 by llpx_sn_bind/
  36 | #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H
  37   /3 width=1 by llpx_sn_flat/
  38 ]
  39 qed-.
  40
  41 lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
  42                          ∀L1,L2,T,l. L1 ≡[T, l] L2 →
  43                          ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L.
  44 /3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
  45
  46 (* Advanced inversion lemmas ************************************************)
  47
  48 lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
  49                            ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V →
  50                            ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2.
  51 #L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2
  52 /2 width=3 by ex2_intro/
  53 qed-.
  54
  55 lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
  56                            ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V →
  57                            ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2.
  58 #L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1
  59 /2 width=3 by ex2_intro/
  60 qed-.
  61
  62 lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
  63                            ∀I1,I2,K1,K2,V1,V2.
  64                            ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
  65                            ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2.
  66 /2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
  67
  68 lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
  69                         ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
  70                         K1 ≡[V, 0] K2.
  71 #L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2
  72 elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) //
  73 qed-.
  74
  75 lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 →
  76                   ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V →
  77                   K1 ≡[V, 0] K2 → L1 ≡[T, l] L2.
  78 /2 width=9 by llpx_sn_inv_S/ qed-.
  79
  80 lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
  81                        L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
  82 /2 width=2 by llpx_sn_inv_bind_O/ qed-.
  83
  84 (* Advanced forward lemmas **************************************************)
  85
  86 lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
  87                         ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V →
  88                         i < l ∨
  89                         ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i.
  90 #L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
  91 [ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
  92 qed-.
  93
  94 lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
  95                         ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V →
  96                         i < l ∨
  97                         ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i.
  98 #L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
  99 [ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
 100 qed-.
 101
 102 lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
 103                           L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
 104 /2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
 105
 106 (* Inversion lemmas on negated lazy quivalence for local environments *******)
 107
 108 lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) →
 109                       (L1 ≡[V, l] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → ⊥).
 110 /3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
 111
 112 lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) →
 113                       (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥).
 114 /3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
 115
 116 lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) →
 117                         (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥).
 118 /3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.