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

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  14
  15 include "basic_2/reducibility/trf.ma".
  16
  17 (* CONTEXT-FREE IRREDUCIBLE TERMS *******************************************)
  18
  19 definition tif: predicate term ≝ λT. 𝐑⦃T⦄ → ⊥.
  20
  21 interpretation "context-free irreducibility (term)"
  22    'NotReducible T = (tif T).
  23
  24 (* Basic inversion lemmas ***************************************************)
  25
  26 lemma tif_inv_abbr: ∀V,T. 𝐈⦃ⓓV.T⦄ → ⊥.
  27 /2 width=1/ qed-.
  28
  29 lemma tif_inv_abst: ∀V,T. 𝐈⦃ⓛV.T⦄ → 𝐈⦃V⦄ ∧ 𝐈⦃T⦄.
  30 /4 width=1/ qed-.
  31
  32 lemma tif_inv_appl: ∀V,T. 𝐈⦃ⓐV.T⦄ → ∧∧ 𝐈⦃V⦄ & 𝐈⦃T⦄ & 𝐒⦃T⦄.
  33 #V #T #HVT @and3_intro /3 width=1/
  34 generalize in match HVT; -HVT elim T -T //
  35 * // * #U #T #_ #_ #H elim (H ?) -H /2 width=1/
  36 qed-.
  37
  38 lemma tif_inv_cast: ∀V,T. 𝐈⦃ⓝV.T⦄ → ⊥.
  39 /2 width=1/ qed-.
  40
  41 (* Basic properties *********************************************************)
  42
  43 lemma tif_atom: ∀I. 𝐈⦃⓪{I}⦄.
  44 /2 width=4/ qed.
  45
  46 lemma tif_abst: ∀V,T. 𝐈⦃V⦄ →  𝐈⦃T⦄ →  𝐈⦃ⓛV.T⦄.
  47 #V #T #HV #HT #H
  48 elim (trf_inv_abst … H) -H /2 width=1/
  49 qed.
  50
  51 lemma tif_appl: ∀V,T. 𝐈⦃V⦄ →  𝐈⦃T⦄ →  𝐒⦃T⦄ → 𝐈⦃ⓐV.T⦄.
  52 #V #T #HV #HT #S #H
  53 elim (trf_inv_appl … H) -H /2 width=1/
  54 qed.