matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_lift.ma

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  14
  15 include "ground_2/notation/functions/lift_1.ma".
  16 include "ground_2/relocation/nstream.ma".
  17
  18 (* RELOCATION N-STREAM ******************************************************)
  19
  20 definition push: rtmap → rtmap ≝ λf. 0@f.
  21
  22 interpretation "push (nstream)" 'Lift f = (push f).
  23
  24 definition next: rtmap → rtmap.
  25 * #n #f @(⫯n@f)
  26 qed.
  27
  28 interpretation "next (nstream)" 'Successor f = (next f).
  29
  30 (* Basic properties *********************************************************)
  31
  32 lemma push_rew: ∀f. 0@f = ↑f.
  33 // qed.
  34
  35 lemma next_rew: ∀f,n. (⫯n)@f = ⫯(n@f).
  36 // qed.
  37 (*
  38 lemma next_rew_sn: ∀f,n1,n2. n1 = ⫯n2 → n1@f = ⫯(n2@f).
  39 // qed.
  40 *)
  41 (* Basic inversion lemmas ***************************************************)
  42
  43 lemma injective_push: injective ? ? push.
  44 #f1 #f2 normalize #H destruct //
  45 qed-.
  46
  47 lemma discr_push_next: ∀f1,f2. ↑f1 = ⫯f2 → ⊥.
  48 #f1 * #n2 #f2 normalize #H destruct
  49 qed-.
  50
  51 lemma discr_next_push: ∀f1,f2. ⫯f1 = ↑f2 → ⊥.
  52 * #n1 #f1 #f2 normalize #H destruct
  53 qed-.
  54
  55 lemma injective_next: injective ? ? next.
  56 * #n1 #f1 * #n2 #f2 normalize #H destruct //
  57 qed-.
  58
  59 lemma push_inv_seq_sn: ∀f,g,n. n@g = ↑f → 0 = n ∧ g = f.
  60 #f #g #n <push_rew #H destruct /2 width=1 by conj/
  61 qed-.
  62
  63 lemma push_inv_seq_dx: ∀f,g,n. ↑f = n@g → 0 = n ∧ g = f.
  64 #f #g #n <push_rew #H destruct /2 width=1 by conj/
  65 qed-.
  66
  67 lemma next_inv_seq_sn: ∀f,g,n. n@g = ⫯f → ∃∃m. m@g = f & ⫯m = n.
  68 * #m #f #g #n <next_rew #H destruct /2 width=3 by ex2_intro/
  69 qed-.
  70
  71 lemma next_inv_seq_dx: ∀f,g,n. ⫯f = n@g → ∃∃m. m@g = f & ⫯m = n.
  72 * #m #f #g #n <next_rew #H destruct /2 width=3 by ex2_intro/
  73 qed-.
  74
  75 lemma pn_split: ∀f. (∃g. ↑g = f) ∨ (∃g. ⫯g = f).
  76 * * /3 width=2 by or_introl, or_intror, ex_intro/
  77 qed-.