matita/matita/contribs/lambdadelta/basic_2/substitution/lsubr_lsubr.ma

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  14
  15 include "basic_2/substitution/lsubr.ma".
  16
  17 (* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
  18
  19 fact lsubr_inv_abbr1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓓW →
  20                           ∨∨ L2 = ⋆
  21                            | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
  22                            | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
  23 #L1 #L2 * -L1 -L2
  24 [ #L #K1 #W #H destruct /2 width=1/
  25 | #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
  26 | #I #L1 #L2 #V1 #V2 #HL12 #K1 #W #H destruct /3 width=4/
  27 ]
  28 qed-.
  29
  30 lemma lsubr_inv_abbr1: ∀K1,L2,W. K1.ⓓW ⊑ L2 →
  31                        ∨∨ L2 = ⋆
  32                         | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
  33                         | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
  34 /2 width=3 by lsubr_inv_abbr1_aux/ qed-.
  35
  36 fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W1. L1 = K1.ⓛW1 →
  37                           L2 = ⋆ ∨
  38                           ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
  39 #L1 #L2 * -L1 -L2
  40 [ #L #K1 #W1 #H destruct /2 width=1/
  41 | #L1 #L2 #V #_ #K1 #W1 #H destruct
  42 | #I #L1 #L2 #V1 #V2 #HL12 #K1 #W1 #H destruct /3 width=4/
  43 ]
  44 qed-.
  45
  46 lemma lsubr_inv_abst1: ∀K1,L2,W1. K1.ⓛW1 ⊑ L2 →
  47                        L2 = ⋆ ∨
  48                        ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
  49 /2 width=4 by lsubr_inv_abst1_aux/ qed-.
  50
  51 (* Main properties **********************************************************)
  52
  53 theorem lsubr_trans: Transitive … lsubr.
  54 #L1 #L #H elim H -L1 -L
  55 [ #L1 #X #H
  56   lapply (lsubr_inv_atom1 … H) -H //
  57 | #L1 #L #V #_ #IHL1 #X #H
  58   elim (lsubr_inv_abbr1 … H) -H // *
  59   #L2 [2: #V2 ] #HL2 #H destruct /3 width=1/
  60 | #I #L1 #L #V1 #V #_ #IHL1 #X #H
  61   elim (lsubr_inv_abst1 … H) -H // *
  62   #L2 #V2 #HL2 #H destruct /3 width=1/
  63 ]
  64 qed-.