matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma

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  14
  15 include "basic_2/notation/relations/notreducible_4.ma".
  16 include "basic_2/reduction/cir.ma".
  17 include "basic_2/reduction/crx.ma".
  18
  19 (* CONTEXT-SENSITIVE EXTENDED IRREDUCIBLE TERMS *****************************)
  20
  21 definition cix: ∀h. sd h → lenv → predicate term ≝ λh,g,L,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → ⊥.
  22
  23 interpretation "context-sensitive extended irreducibility (term)"
  24    'NotReducible h g L T = (cix h g L T).
  25
  26 (* Basic inversion lemmas ***************************************************)
  27
  28 lemma cix_inv_sort: ∀h,g,L,k,l. deg h g k (l+1) → ⦃h, L⦄ ⊢ 𝐈[g]⦃⋆k⦄ → ⊥.
  29 /3 width=2/ qed-.
  30
  31 lemma cix_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃h, L⦄ ⊢ 𝐈[g]⦃#i⦄ → ⊥.
  32 /3 width=4/ qed-.
  33
  34 lemma cix_inv_ri2: ∀h,g,I,L,V,T. ri2 I → ⦃h, L⦄ ⊢ 𝐈[g]⦃②{I}V.T⦄ → ⊥.
  35 /3 width=1/ qed-.
  36
  37 lemma cix_inv_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄ →
  38                    ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄.
  39 /4 width=1/ qed-.
  40
  41 lemma cix_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄ →
  42                     ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄ & ib2 a I.
  43 #h #g #a * [ elim a -a ]
  44 [ #L #V #T #H elim H -H /3 width=1/
  45 |*: #L #V #T #H elim (cix_inv_ib2 … H) -H /2 width=1/ /3 width=1/
  46 ]
  47 qed-.
  48
  49 lemma cix_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓐV.T⦄ →
  50                     ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ & 𝐒⦃T⦄.
  51 #h #g #L #V #T #HVT @and3_intro /3 width=1/
  52 generalize in match HVT; -HVT elim T -T //
  53 * // #a * #U #T #_ #_ #H elim H -H /2 width=1/
  54 qed-.
  55
  56 lemma cix_inv_flat: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓕ{I}V.T⦄ →
  57                     ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
  58 #h #g * #L #V #T #H
  59 [ elim (cix_inv_appl … H) -H /2 width=1/
  60 | elim (cix_inv_ri2 … H) -H /2 width=1/
  61 ]
  62 qed-.
  63
  64 (* Basic forward lemmas *****************************************************)
  65
  66 lemma cix_inv_cir: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ → L ⊢ 𝐈⦃T⦄.
  67 /3 width=1/ qed-.
  68
  69 (* Basic properties *********************************************************)
  70
  71 lemma cix_sort: ∀h,g,L,k. deg h g k 0 → ⦃h, L⦄ ⊢ 𝐈[g]⦃⋆k⦄.
  72 #h #g #L #k #Hk #H elim (crx_inv_sort … H) -L #l #Hkl
  73 lapply (deg_mono … Hk Hkl) -h -k <plus_n_Sm #H destruct
  74 qed.
  75
  76 lemma tix_lref: ∀h,g,i. ⦃h, ⋆⦄ ⊢ 𝐈[g]⦃#i⦄.
  77 #h #g #i #H elim (trx_inv_atom … H) -H #k #l #_ #H destruct
  78 qed.
  79
  80 lemma cix_gref: ∀h,g,L,p. ⦃h, L⦄ ⊢ 𝐈[g]⦃§p⦄.
  81 #h #g #L #p #H elim (crx_inv_gref … H)
  82 qed.
  83
  84 lemma cix_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ → ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄ →
  85                                ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄.
  86 #h #g #a #I #L #V #T #HI #HV #HT #H
  87 elim (crx_inv_ib2 … HI H) -HI -H /2 width=1/
  88 qed.
  89
  90 lemma cix_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ →  𝐒⦃T⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓐV.T⦄.
  91 #h #g #L #V #T #HV #HT #H1 #H2
  92 elim (crx_inv_appl … H2) -H2 /2 width=1/
  93 qed.