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

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  14
  15 include "basic_2/notation/relations/lrsubeqa_3.ma".
  16 include "basic_2/static/aaa.ma".
  17
  18 (* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
  19
  20 inductive lsuba (G:genv): relation lenv ≝
  21 | lsuba_atom: lsuba G (⋆) (⋆)
  22 | lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
  23 | lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
  24               lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
  25 .
  26
  27 interpretation
  28   "local environment refinement (atomic arity assignment)"
  29   'LRSubEqA G L1 L2 = (lsuba G L1 L2).
  30
  31 (* Basic inversion lemmas ***************************************************)
  32
  33 fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
  34 #G #L1 #L2 * -L1 -L2
  35 [ //
  36 | #I #L1 #L2 #V #_ #H destruct
  37 | #L1 #L2 #W #V #A #_ #_ #_ #H destruct
  38 ]
  39 qed-.
  40
  41 lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
  42 /2 width=4 by lsuba_inv_atom1_aux/ qed-.
  43
  44 fact lsuba_inv_pair1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
  45                           (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
  46                           ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
  47                                       G ⊢ K1 ⫃⁝ K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
  48 #G #L1 #L2 * -L1 -L2
  49 [ #J #K1 #X #H destruct
  50 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
  51 | #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by or_intror, ex6_4_intro/
  52 ]
  53 qed-.
  54
  55 lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃⁝ L2 →
  56                        (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
  57                        ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
  58                                    I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
  59 /2 width=3 by lsuba_inv_pair1_aux/ qed-.
  60
  61 fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
  62 #G #L1 #L2 * -L1 -L2
  63 [ //
  64 | #I #L1 #L2 #V #_ #H destruct
  65 | #L1 #L2 #W #V #A #_ #_ #_ #H destruct
  66 ]
  67 qed-.
  68
  69 lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
  70 /2 width=4 by lsuba_inv_atom2_aux/ qed-.
  71
  72 fact lsuba_inv_pair2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
  73                           (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
  74                           ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
  75                                     G ⊢ K1 ⫃⁝ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
  76 #G #L1 #L2 * -L1 -L2
  77 [ #J #K2 #U #H destruct
  78 | #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
  79 | #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by or_intror, ex5_3_intro/
  80 ]
  81 qed-.
  82
  83 lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⫃⁝ K2.ⓑ{I}W →
  84                        (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
  85                        ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
  86                                  I = Abst & L1 = K1.ⓓⓝW.V.
  87 /2 width=3 by lsuba_inv_pair2_aux/ qed-.
  88
  89 (* Basic properties *********************************************************)
  90
  91 lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
  92 #G #L elim L -L /2 width=1 by lsuba_atom, lsuba_pair/
  93 qed.