matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/lsubr.etc

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  14
  15 notation "hvbox( L1 ⓝ ⊑ break term 46 L2 )"
  16    non associative with precedence 45
  17    for @{ 'LRSubEqT $L1 $L2 }.
  18
  19 include "basic_2/relocation/ldrop.ma".
  20
  21 (* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
  22
  23 inductive lsubr: relation lenv ≝
  24 | lsubr_sort: ∀L. lsubr L (⋆)
  25 | lsubr_abbr: ∀L1,L2,V. lsubr L1 L2 → lsubr (L1. ⓓV) (L2.ⓓV)
  26 | lsubr_abst: ∀I,L1,L2,V1,V2. lsubr L1 L2 → lsubr (L1. ⓑ{I}V1) (L2. ⓛV2)
  27 .
  28
  29 interpretation
  30   "local environment refinement (substitution)"
  31   'CrSubEq L1 L2 = (lsubr L1 L2).
  32
  33 (* Basic properties *********************************************************)
  34
  35 lemma lsubr_bind: ∀I,L1,L2,V. L1 ⊑ L2 → L1. ⓑ{I} V ⊑ L2.ⓑ{I} V.
  36 * /2 width=1/ qed.
  37
  38 lemma lsubr_abbr: ∀I,L1,L2,V. L1 ⊑ L2 → L1. ⓓV ⊑ L2. ⓑ{I}V.
  39 * /2 width=1/ qed.
  40
  41 lemma lsubr_refl: ∀L. L ⊑ L.
  42 #L elim L -L // /2 width=1/
  43 qed.
  44
  45 (* Basic inversion lemmas ***************************************************)
  46
  47 fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⊑ L2 → L1 = ⋆ → L2 = ⋆.
  48 #L1 #L2 * -L1 -L2 //
  49 [ #L1 #L2 #V #_ #H destruct
  50 | #I #L1 #L2 #V1 #V2 #_ #H destruct
  51 ]
  52 qed-.
  53
  54 lemma lsubr_inv_atom1: ∀L2. ⋆ ⊑ L2 → L2 = ⋆.
  55 /2 width=3 by lsubr_inv_atom1_aux/ qed-.
  56
  57 fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
  58                           ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
  59 #L1 #L2 * -L1 -L2
  60 [ #L #K2 #W #H destruct
  61 | #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3/
  62 | #I #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
  63 ]
  64 qed-.
  65
  66 lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⊑ K2.ⓓW →
  67                        ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
  68 /2 width=3 by lsubr_inv_abbr2_aux/ qed-.
  69
  70 fact lsubr_inv_abst2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W2. L2 = K2.ⓛW2 →
  71                           ∃∃I,K1,W1. K1 ⊑ K2 & L1 = K1.ⓑ{I}W1.
  72 #L1 #L2 * -L1 -L2
  73 [ #L #K2 #W2 #H destruct
  74 | #L1 #L2 #V #_ #K2 #W2 #H destruct
  75 | #I #L1 #L2 #V1 #V2 #HL12 #K2 #W2 #H destruct /2 width=5/
  76 ]
  77 qed-.
  78
  79 lemma lsubr_inv_abst2: ∀L1,K2,W2. L1 ⊑ K2.ⓛW2 →
  80                        ∃∃I,K1,W1. K1 ⊑ K2 & L1 = K1.ⓑ{I}W1.
  81 /2 width=4 by lsubr_inv_abst2_aux/ qed-.
  82
  83 (* Basic forward lemmas *****************************************************)
  84
  85 lemma lsubr_fwd_length: ∀L1,L2. L1 ⊑ L2 → |L2| ≤ |L1|.
  86 #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
  87 qed-.
  88
  89 lemma lsubr_fwd_ldrop2_abbr: ∀L1,L2. L1 ⊑ L2 →
  90                              ∀K2,W,i. ⇩[0, i] L2 ≡ K2. ⓓW →
  91                              ∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1. ⓓW.
  92 #L1 #L2 #H elim H -L1 -L2
  93 [ #L #K2 #W #i #H
  94   elim (ldrop_inv_atom1 … H) -H #H destruct
  95 | #L1 #L2 #V #HL12 #IHL12 #K2 #W #i #H
  96   elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
  97   [ /2 width=3/
  98   | elim (IHL12 … HLK2) -IHL12 -HLK2 /3 width=3/
  99   ]
 100 | #I #L1 #L2 #V1 #V2 #_ #IHL12 #K2 #W #i #H
 101   elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct
 102   elim (IHL12 … HLK2) -IHL12 -HLK2 /3 width=3/
 103 ]
 104 qed-.