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

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   3 (*      ||M||                                                             *)
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  14
  15 include "ground_2A/ynat/ynat_plus.ma".
  16
  17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
  18
  19 lemma ymax_pre_dx: ∀x,y. x ≤ y → x - y + y = y.
  20 #x #y * -x -y //
  21 #x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
  22 qed-.
  23
  24 lemma ymax_pre_sn: ∀x,y. y ≤ x → x - y + y = x.
  25 #x #y * -x -y
  26 [ #x #y #Hxy >yminus_inj /3 width=3 by plus_minus, eq_f/
  27 | * //
  28 ]
  29 qed-.
  30
  31 lemma ymax_pre_i_dx: ∀y,x. y ≤ x - y + y.
  32 // qed.
  33
  34 lemma ymax_pre_i_sn: ∀y,x. x ≤ x - y + y.
  35 * // #y * /2 width=1 by yle_inj/
  36 qed.
  37
  38 lemma ymax_pre_e: ∀x,z. x ≤ z → ∀y. y ≤ z → x - y + y ≤ z.
  39 #x #z #Hxz #y #Hyz elim (yle_split x y)
  40 [ #Hxy >(ymax_pre_dx … Hxy) -x //
  41 | #Hyx >(ymax_pre_sn … Hyx) -y //
  42 ]
  43 qed.
  44
  45 lemma ymax_pre_dx_comm: ∀x,y. x ≤ y → y + (x - y) = y.
  46 /2 width=1 by ymax_pre_dx/ qed-.
  47
  48 lemma ymax_pre_sn_comm: ∀x,y. y ≤ x → y + (x - y) = x.
  49 /2 width=1 by ymax_pre_sn/ qed-.
  50
  51 lemma ymax_pre_i_dx_comm: ∀y,x. y ≤ y + (x - y).
  52 // qed.
  53
  54 lemma ymax_pre_i_sn_comm: ∀y,x. x ≤ y + (x - y).
  55 /2 width=1 by ymax_pre_i_sn/ qed.
  56
  57 lemma ymax_pre_e_comm: ∀x,z. x ≤ z → ∀y. y ≤ z → y + (x - y) ≤ z.
  58 /2 width=1 by ymax_pre_e/ qed.