matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma

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  14
  15 include "ground_2A/ynat/ynat_plus.ma".
  16
  17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
  18
  19 fact ymin_pre_dx_aux: ∀x,y. y ≤ x → x - (x - y) ≤ y.
  20 #x #y * -x -y
  21 [ #x #y #Hxy >yminus_inj
  22  /3 width=4 by yle_inj, monotonic_le_minus_l/
  23 | * //
  24 ]
  25 qed-.
  26
  27 lemma ymin_pre_sn: ∀x,y. x ≤ y → x - (x - y) = x.
  28 #x #y * -x -y //
  29 #x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
  30 qed-.
  31
  32 lemma ymin_pre_i_dx: ∀x,y. x - (x - y) ≤ y.
  33 #x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/
  34 #Hxy >(ymin_pre_sn … Hxy) //
  35 qed.
  36
  37 lemma ymin_pre_i_sn: ∀x,y. x - (x - y) ≤ x.
  38 // qed.
  39
  40 lemma ymin_pre_dx: ∀x,y. y ≤ yinj x → yinj x - (yinj x - y) = y.
  41 #x #y #H elim (yle_inv_inj2 … H) -H
  42 #z #Hzx #H destruct >yminus_inj
  43 /3 width=4 by minus_le_minus_minus_comm, eq_f/
  44 qed-.
  45
  46 lemma ymin_pre_e: ∀z,x. z ≤ yinj x → ∀y. z ≤ y →
  47                   z ≤ yinj x - (yinj x - y).
  48 #z #x #Hzx #y #Hzy elim (yle_split x y)
  49 [ #H >(ymin_pre_sn … H) -y //
  50 | #H >(ymin_pre_dx … H) -x //
  51 ]
  52 qed.