matita/matita/contribs/lambda_delta/basic_2/reducibility/crf.ma

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  14
  15 include "basic_2/grammar/term_simple.ma".
  16 include "basic_2/substitution/ldrop.ma".
  17
  18 (* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************)
  19
  20 (* reducible binary items *)
  21 definition ri2: item2 → Prop ≝
  22                 λI. I = Bind2 true Abbr ∨ I = Flat2 Cast.
  23
  24 (* irreducible binary binders *)
  25 definition ib2: bool → bind2 → Prop ≝
  26                 λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr.
  27
  28 (* reducible terms *)
  29 inductive crf: lenv → predicate term ≝
  30 | crf_delta  : ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → crf L (#i)
  31 | crf_appl_sn: ∀L,V,T. crf L V → crf L (ⓐV. T)
  32 | crf_appl_dx: ∀L,V,T. crf L T → crf L (ⓐV. T)
  33 | crf_ri2    : ∀I,L,V,T. ri2 I → crf L (②{I}V. T)
  34 | crf_ib2_sn : ∀a,I,L,V,T. ib2 a I → crf L V → crf L (ⓑ{a,I}V. T)
  35 | crf_ib2_dx : ∀a,I,L,V,T. ib2 a I → crf (L.ⓑ{I}V) T → crf L (ⓑ{a,I}V. T)
  36 | crf_beta   : ∀a,L,V,W,T. crf L (ⓐV. ⓛ{a}W. T)
  37 | crf_theta  : ∀a,L,V,W,T. crf L (ⓐV. ⓓ{a}W. T)
  38 .
  39
  40 interpretation
  41    "context-sensitive reducibility (term)"
  42    'Reducible L T = (crf L T).
  43
  44 (* Basic inversion lemmas ***************************************************)
  45
  46 fact trf_inv_atom_aux: ∀I,L,T. L ⊢ 𝐑⦃T⦄ → L = ⋆ → T =  ⓪{I} → ⊥.
  47 #I #L #T * -L -T
  48 [ #L #K #V #i #HLK #H1 #H2 destruct
  49   lapply (ldrop_inv_atom1 … HLK) -HLK #H destruct
  50 | #L #V #T #_ #_ #H destruct
  51 | #L #V #T #_ #_ #H destruct
  52 | #J #L #V #T #_ #_ #H destruct
  53 | #a #J #L #V #T #_ #_ #_ #H destruct
  54 | #a #J #L #V #T #_ #_ #_ #H destruct
  55 | #a #L #V #W #T #_ #H destruct
  56 | #a #L #V #W #T #_ #H destruct
  57 ]
  58 qed.
  59
  60 lemma trf_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥.
  61 /2 width=6/ qed-.
  62
  63 fact crf_inv_abst_aux: ∀a,L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓛ{a}W. U →
  64                        L ⊢ 𝐑⦃W⦄ ∨ L.ⓛW ⊢ 𝐑⦃U⦄.
  65 #a #L #W #U #T * -T
  66 [ #L #K #V #i #_ #H destruct
  67 | #L #V #T #_ #H destruct
  68 | #L #V #T #_ #H destruct
  69 | #J #L #V #T #H1 #H2 destruct
  70   elim H1 -H1 #H destruct
  71 | #b #J #L #V #T #_ #HV #H destruct /2 width=1/
  72 | #b #J #L #V #T #_ #HT #H destruct /2 width=1/
  73 | #b #L #V #W #T #H destruct
  74 | #b #L #V #W #T #H destruct
  75 ]
  76 qed.
  77
  78 lemma crf_inv_abst: ∀a,L,W,T. L ⊢ 𝐑⦃ⓛ{a}W.T⦄ → L ⊢ 𝐑⦃W⦄ ∨ L.ⓛW ⊢ 𝐑⦃T⦄.
  79 /2 width=4/ qed-.
  80
  81 fact crf_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW. U →
  82                        ∨∨ L ⊢ 𝐑⦃W⦄ | L ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
  83 #L #W #U #T * -T
  84 [ #L #K #V #i #_ #H destruct
  85 | #L #V #T #HV #H destruct /2 width=1/
  86 | #L #V #T #HT #H destruct /2 width=1/
  87 | #J #L #V #T #H1 #H2 destruct
  88   elim H1 -H1 #H destruct
  89 | #a #I #L #V #T #_ #_ #H destruct
  90 | #a #I #L #V #T #_ #_ #H destruct
  91 | #a #L #V #W0 #T #H destruct
  92   @or3_intro2 #H elim (simple_inv_bind … H)
  93 | #a #L #V #W0 #T #H destruct
  94   @or3_intro2 #H elim (simple_inv_bind … H)
  95 ]
  96 qed.
  97
  98 lemma crf_inv_appl: ∀L,V,T. L ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ L ⊢ 𝐑⦃V⦄ | L ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
  99 /2 width=3/ qed-.
 100