matita/matita/contribs/lambdadelta/static_2/syntax/tdpos.ma

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  14
  15 include "static_2/notation/relations/positive_3.ma".
  16 include "static_2/syntax/item_sd.ma".
  17 include "static_2/syntax/term.ma".
  18
  19 (* DEGREE POSITIVITY ON TERMS ***********************************************)
  20
  21 inductive tdpos (h) (o): predicate term ≝
  22 | tdpos_sort: ∀s,d. deg h o s (↑d) → tdpos h o (⋆s)
  23 | tdpos_lref: ∀i. tdpos h o (#i)
  24 | tdpos_gref: ∀l. tdpos h o (§l)
  25 | tdpos_pair: ∀I,V,T. tdpos h o V → tdpos h o T → tdpos h o (②{I}V.T)
  26 .
  27
  28 interpretation
  29    "context-free degree positivity (term)"
  30    'Positive h o T = (tdpos h o T).
  31
  32 (* Basic inversion lemmas ***************************************************)
  33
  34 fact tdpos_inv_sort_aux (h) (o):
  35                         ∀X. 𝐏[h,o]⦃X⦄ → ∀s. X = ⋆s → ∃d. deg h o s (↑d).
  36 #h #o #H *
  37 [ #s #d #Hsd #x #H destruct /2 width=2 by ex_intro/
  38 | #i #x #H destruct
  39 | #l #x #H destruct
  40 | #I #V #T #_ #_ #x #H destruct
  41 ]
  42 qed-.
  43
  44 lemma tdpos_inv_sort (h) (o): ∀s. 𝐏[h,o]⦃⋆s⦄ → ∃d. deg h o s (↑d).
  45 /2 width=3 by tdpos_inv_sort_aux/ qed-.
  46
  47 fact tdpos_inv_pair_aux (h) (o): ∀X. 𝐏[h,o]⦃X⦄ → ∀I,V,T. X = ②{I}V.T →
  48                                  ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄.
  49 #h #o #H *
  50 [ #s #d #_ #Z #X1 #X2 #H destruct
  51 | #i #Z #X1 #X2 #H destruct
  52 | #l #Z #X1 #X2 #H destruct
  53 | #I #V #T #HV #HT #Z #X1 #X2 #H destruct /2 width=1 by conj/
  54 ]
  55 qed-.
  56
  57 lemma tdpos_inv_pair (h) (o): ∀I,V,T. 𝐏[h,o]⦃②{I}V.T⦄ →
  58                               ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄.
  59 /2 width=4 by tdpos_inv_pair_aux/ qed-.
  60
  61 (* Basic properties *********************************************************)
  62
  63 axiom tdpos_total (h): ∀T. ∃o. 𝐏[h,o]⦃T⦄.