matita/matita/contribs/lambdadelta/basic_2/unfold/fsupp.ma

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  14
  15 include "basic_2/substitution/fsup.ma".
  16
  17 (* PLUS-ITERATED SUPCLOSURE *************************************************)
  18
  19 definition fsupp: bi_relation lenv term ≝ bi_TC … fsup.
  20
  21 interpretation "plus-iterated structural successor (closure)"
  22    'SupTermPlus L1 T1 L2 T2 = (fsupp L1 T1 L2 T2).
  23
  24 (* Basic eliminators ********************************************************)
  25
  26 lemma fsupp_ind: ∀L1,T1. ∀R:relation2 lenv term.
  27                  (∀L2,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → R L2 T2) →
  28                  (∀L,T,L2,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ → R L T → R L2 T2) →
  29                  ∀L2,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → R L2 T2.
  30 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
  31 @(bi_TC_ind … IH1 IH2 ? ? H)
  32 qed-.
  33
  34 lemma fsupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term.
  35                     (∀L1,T1. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → R L1 T1) →
  36                     (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ → R L T → R L1 T1) →
  37                     ∀L1,T1. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → R L1 T1.
  38 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
  39 @(bi_TC_ind_dx … IH1 IH2 ? ? H)
  40 qed-.
  41
  42 (* Basic properties *********************************************************)
  43
  44 lemma fsup_fsupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
  45 /2 width=1/ qed.
  46
  47 lemma fsupp_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ →
  48                     ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
  49 /2 width=4/ qed.
  50
  51 lemma fsupp_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ →
  52                     ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
  53 /2 width=4/ qed.
  54
  55 (* Basic forward lemmas *****************************************************)
  56
  57 lemma fsupp_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
  58 #L1 #L2 #T1 #T2 #H @(fsupp_ind … H) -L2 -T2
  59 /3 width=3 by fsup_fwd_cw, transitive_lt/
  60 qed-.