matita/matita/contribs/lambdadelta/ground_2/etc/ynat/ynat_rpm.etc

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  14
  15 include "ground_2/notation/relations/rplusminus_4.ma".
  16 include "ground_2/ynat/ynat_plus.ma".
  17
  18 (* NATURAL NUMBERS WITH INFINITY ********************************************)
  19
  20 (* algebraic x + y1 - y2 = z *)
  21 inductive yrpm (x:ynat) (y1:ynat) (y2:ynat): predicate ynat ≝
  22 | yrpm_ge: y2 ≤ y1 → yrpm x y1 y2 (x + (y1 - y2))
  23 | yrpm_lt: y1 < y2 → yrpm x y1 y2 (x - (y2 - y1))
  24 .
  25
  26 interpretation "ynat 'algebraic plus-minus' (relational)"
  27    'RPlusMinus x y1 y2 z = (yrpm x y1 y2 z).
  28
  29 (* Basic inversion lemmas ***************************************************)
  30
  31 lemma ypm_inv_ge: ∀x,y1,y2,z. x ⊞ y1 ⊟ y2 ≡ z →
  32                   y2 ≤ y1 → z = x + (y1 - y2).
  33 #x #y1 #y2 #z * -z //
  34 #Hy12 #H elim (ylt_yle_false … H) -H //
  35 qed-.
  36
  37 lemma ypm_inv_lt: ∀x,y1,y2,z. x ⊞ y1 ⊟ y2 ≡ z →
  38                   y1 < y2 → z = x - (y2 - y1).
  39 #x #y1 #y2 #z * -z //
  40 #Hy21 #H elim (ylt_yle_false … H) -H //
  41 qed-.
  42
  43 (* Advanced inversion lemmas ************************************************)
  44
  45 lemma ypm_inv_le: ∀x,y1,y2,z. x ⊞ y1 ⊟ y2 ≡ z →
  46                   y1 ≤ y2 → z = x - (y2 - y1).
  47 #x #y1 #y2 #z #H #Hy12 elim (yle_split_eq … Hy12) -Hy12 #Hy12
  48 [ /2 width=1 by ypm_inv_lt/
  49 | >(ypm_inv_ge … H) -H // destruct >yminus_refl //
  50 ]
  51 qed-.