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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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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] → False.
  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] → False.
  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] → False.
  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.