matita/matita/contribs/lambdadelta/ground/arith/ynat_succ.ma

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  14
  15 include "ground/arith/nat_succ.ma".
  16 include "ground/arith/ynat_nat.ma".
  17
  18 (* SUCCESSOR FOR NON-NEGATIVE INTEGERS WITH INFINITY ************************)
  19
  20 definition ysucc_aux (n): ynat ≝
  21            yinj_nat (↑n).
  22
  23 (*** ysucc *)
  24 definition ysucc: ynat → ynat ≝
  25            ynat_bind_nat ysucc_aux (∞).
  26
  27 interpretation
  28   "successor (non-negative integers with infinity)"
  29   'UpArrow x = (ysucc x).
  30
  31 (* Constructions ************************************************************)
  32
  33 (*** ysucc_inj *)
  34 lemma ysucc_inj (n): yinj_nat (↑n) = ↑(yinj_nat n).
  35 @(ynat_bind_nat_inj ysucc_aux)
  36 qed.
  37
  38 (*** ysucc_Y *)
  39 lemma ysucc_inf: ∞ = ↑(∞).
  40 // qed.
  41
  42 (* Inversion lemmas *********************************************************)
  43
  44 (*** ysucc_inv_inj_sn *)
  45 lemma eq_inv_inj_ysucc (n1) (x2):
  46       yinj_nat n1 = ↑x2 →
  47       ∃∃n2. yinj_nat n2 = x2 & ↑n2 = n1.
  48 #n1 #x2 @(ynat_split_nat_inf … x2) -x2
  49 [ /3 width=3 by eq_inv_yinj_nat_bi, ex2_intro/
  50 | #H elim (eq_inv_yinj_nat_inf … H)
  51 ]
  52 qed-.
  53
  54 (*** ysucc_inv_inj_dx *)
  55 lemma eq_inv_ysucc_inj (x1) (n2):
  56       ↑x1 = yinj_nat n2  →
  57       ∃∃n1. yinj_nat n1 = x1 & ↑n1 = n2.
  58 /2 width=1 by eq_inv_inj_ysucc/ qed-.
  59
  60 (*** ysucc_inv_Y_sn *)
  61 lemma eq_inv_inf_ysucc (x2): ∞ = ↑x2 → ∞ = x2.
  62 #x2 @(ynat_split_nat_inf … x2) -x2 //
  63 #n1 <ysucc_inj #H elim (eq_inv_inf_yinj_nat … H)
  64 qed-.
  65
  66 (*** ysucc_inv_Y_dx *)
  67 lemma eq_inv_ysucc_inf (x1): ↑x1 = ∞ → ∞ = x1.
  68 /2 width=1 by eq_inv_inf_ysucc/ qed-.
  69
  70 (*** ysucc_inv_inj *)
  71 lemma eq_inv_ysucc_bi: injective … ysucc.
  72 #x1 @(ynat_split_nat_inf … x1) -x1
  73 [ #n1 #x2 <ysucc_inj #H
  74   elim (eq_inv_inj_ysucc … H) -H #n2 #H1 #H2 destruct
  75   <(eq_inv_nsucc_bi … H2) -H2 //
  76 | #x2 #H <(eq_inv_inf_ysucc … H) -H //
  77 ]
  78 qed-.
  79
  80 (*** ysucc_inv_refl *)
  81 lemma ysucc_inv_refl (x): x = ↑x → ∞ = x.
  82 #x @(ynat_split_nat_inf … x) -x //
  83 #n <ysucc_inj #H
  84 lapply (eq_inv_yinj_nat_bi … H) -H #H
  85 elim (nsucc_inv_refl … H)
  86 qed-.
  87
  88 (*** ysucc_inv_O_sn *)
  89 lemma eq_inv_zero_ysucc (x): 𝟎 = ↑x → ⊥.
  90 #x #H
  91 elim (eq_inv_inj_ysucc (𝟎) ? H) -H #n #_ #H
  92 /2 width=2 by eq_inv_zero_nsucc/
  93 qed-.
  94
  95 (*** ysucc_inv_O_dx *)
  96 lemma eq_inv_ysucc_zero (x): ↑x = 𝟎 → ⊥.
  97 /2 width=2 by eq_inv_zero_ysucc/ qed-.
  98
  99 (* Eliminators **************************************************************)
 100
 101 (*** ynat_ind *)
 102 lemma ynat_ind_succ (Q:predicate …):
 103       Q (𝟎) → (∀n:nat. Q (yinj_nat n) → Q (↑(yinj_nat n))) → Q (∞) → ∀x. Q x.
 104 #Q #IH1 #IH2 #IH3 #x @(ynat_split_nat_inf … x) -x //
 105 #n @(nat_ind_succ … n) -n /2 width=1 by/
 106 qed-.