matita/matita/contribs/lambda_delta/basic_2/static/lsuba.ma

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  14
  15 include "Basic_2/static/aaa.ma".
  16
  17 (* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
  18
  19 inductive lsuba: relation lenv ≝
  20 | lsuba_atom: lsuba (⋆) (⋆)
  21 | lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V)
  22 | lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ÷ A → L2 ⊢ W ÷ A →
  23               lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW)
  24 .
  25
  26 interpretation
  27   "local environment refinement (atomic arity assigment)"
  28   'CrSubEqA L1 L2 = (lsuba L1 L2).
  29
  30 (* Basic inversion lemmas ***************************************************)
  31
  32 fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ÷⊑ L2 → L1 = ⋆ → L2 = ⋆.
  33 #L1 #L2 * -L1 -L2
  34 [ //
  35 | #I #L1 #L2 #V #_ #H destruct
  36 | #L1 #L2 #V #W #A #_ #_ #_ #H destruct
  37 ]
  38 qed.
  39
  40 lemma lsuba_inv_atom1: ∀L2. ⋆ ÷⊑ L2 → L2 = ⋆.
  41 /2 width=3/ qed-.
  42
  43 fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ÷⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
  44                           (∃∃K2. K1 ÷⊑ K2 & L2 = K2. ⓑ{I} V) ∨
  45                           ∃∃K2,W,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
  46                                     L2 = K2. ⓛW & I = Abbr.
  47 #L1 #L2 * -L1 -L2
  48 [ #I #K1 #V #H destruct
  49 | #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
  50 | #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
  51 ]
  52 qed.
  53
  54 lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ÷⊑ L2 →
  55                        (∃∃K2. K1 ÷⊑ K2 & L2 = K2. ⓑ{I} V) ∨
  56                        ∃∃K2,W,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
  57                                  L2 = K2. ⓛW & I = Abbr.
  58 /2 width=3/ qed-.
  59
  60 fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ÷⊑ L2 → L2 = ⋆ → L1 = ⋆.
  61 #L1 #L2 * -L1 -L2
  62 [ //
  63 | #I #L1 #L2 #V #_ #H destruct
  64 | #L1 #L2 #V #W #A #_ #_ #_ #H destruct
  65 ]
  66 qed.
  67
  68 lemma lsubc_inv_atom2: ∀L1. L1 ÷⊑ ⋆ → L1 = ⋆.
  69 /2 width=3/ qed-.
  70
  71 fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ÷⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
  72                           (∃∃K1. K1 ÷⊑ K2 & L1 = K1. ⓑ{I} W) ∨
  73                           ∃∃K1,V,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
  74                                     L1 = K1. ⓓV & I = Abst.
  75 #L1 #L2 * -L1 -L2
  76 [ #I #K2 #W #H destruct
  77 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
  78 | #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
  79 ]
  80 qed.
  81
  82 lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ÷⊑ K2. ⓑ{I} W →
  83                        (∃∃K1. K1 ÷⊑ K2 & L1 = K1. ⓑ{I} W) ∨
  84                        ∃∃K1,V,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
  85                                  L1 = K1. ⓓV & I = Abst.
  86 /2 width=3/ qed-.
  87
  88 (* Basic properties *********************************************************)
  89
  90 lemma lsuba_refl: ∀L. L ÷⊑ L.
  91 #L elim L -L // /2 width=1/
  92 qed.