matita/matita/contribs/lambdadelta/basic_2/etc/delift/delift.etc

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   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
  16    non associative with precedence 45
  17    for @{ 'TSubst $L $T1 $d $e $T2 }.
  18
  19 include "basic_2/unfold/tpss.ma".
  20
  21 (* INVERSE BASIC TERM RELOCATION  *******************************************)
  22
  23 definition delift: nat → nat → lenv → relation term ≝
  24                    λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T.
  25
  26 interpretation "inverse basic relocation (term)"
  27    'TSubst L T1 d e T2 = (delift d e L T1 T2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 →
  32                    ∀L. L ⊢ ▼*[d, e] T2 ≡ T1.
  33 /2 width=3/ qed.
  34
  35 lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T.
  36 /2 width=3/ qed.
  37
  38 lemma delift_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 →
  39                           ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2.
  40 #L1 #T1 #T2 #d #e * /3 width=3/
  41 qed.
  42
  43 lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k.
  44 /2 width=3/ qed.
  45
  46 lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i.
  47 /3 width=3/ qed.
  48
  49 lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e).
  50 /3 width=3/ qed.
  51
  52 lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p.
  53 /2 width=3/ qed.
  54
  55 lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e.
  56                    L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 →
  57                    L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
  58 #a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
  59 lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
  60 qed.
  61
  62 lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
  63                    L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 →
  64                    L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
  65 #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
  66 qed.
  67
  68 (* Basic inversion lemmas ***************************************************)
  69
  70 lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k.
  71 #L #U2 #d #e #k * #U #HU
  72 >(tpss_inv_sort1 … HU) -HU #HU2
  73 >(lift_inv_sort2 … HU2) -HU2 //
  74 qed-.
  75
  76 lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p.
  77 #L #U #d #e #p * #U #HU
  78 >(tpss_inv_gref1 … HU) -HU #HU2
  79 >(lift_inv_gref2 … HU2) -HU2 //
  80 qed-.
  81
  82 lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 →
  83                         ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
  84                                  L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 &
  85                                  U2 = ⓑ{a,I} V2. T2.
  86 #a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
  87 elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
  88 elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
  89 lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
  90 qed-.
  91
  92 lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 →
  93                         ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
  94                                  L ⊢ ▼*[d, e] T1 ≡ T2 &
  95                                  U2 = ⓕ{I} V2. T2.
  96 #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
  97 elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
  98 elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
  99 qed-.
 100
 101 lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2.
 102 #L #T1 #T2 #d * #T #HT1
 103 >(tpss_inv_refl_O2 … HT1) -HT1 #HT2
 104 >(lift_inv_refl_O2 … HT2) -HT2 //
 105 qed-.
 106
 107 (* Basic forward lemmas *****************************************************)
 108
 109 lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → ♯{T1} ≤ ♯{T2}.
 110 #L #T1 #T2 #d #e * #T #HT1 #HT2
 111 >(lift_fwd_tw … HT2) -T2 /2 width=4 by tpss_fwd_tw/
 112 qed-.