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

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  14
  15 include "ground/arith/pnat_iter.ma".
  16
  17 (* POSITIVE INTEGERS ********************************************************)
  18
  19 definition pplus: pnat → pnat → pnat ≝
  20            λp,q. psucc^q p.
  21
  22 interpretation
  23   "plus (positive integers"
  24   'plus p q = (pplus p q).
  25
  26 (* Basic rewrites ***********************************************************)
  27
  28 lemma pplus_one_dx (p): ↑p = p + 𝟏.
  29 // qed.
  30
  31 lemma pplus_succ_dx (p) (q): ↑(p+q) = p + ↑q.
  32 // qed.
  33
  34 (* Semigroup properties *****************************************************)
  35
  36 lemma pplus_succ_sn (p) (q): ↑(p+q) = ↑p + q.
  37 #p #q @(piter_appl … psucc)
  38 qed.
  39
  40 lemma pplus_one_sn (p): ↑p = 𝟏 + p.
  41 #p elim p -p //
  42 qed.
  43
  44 lemma pplus_comm: commutative … pplus.
  45 #p elim p -p //
  46 qed.
  47
  48 lemma pplus_assoc: associative … pplus.
  49 #p #q #r elim r -r //
  50 #r #IH <pplus_succ_dx <pplus_succ_dx <IH -IH //
  51 qed.