matita/matita/contribs/lambdadelta/ground/lib/arith_2b.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/lib/arith.ma".
  16
  17 (* ARITHMETICAL PROPERTIES FOR λδ-2B ****************************************)
  18
  19 lemma arith_l4 (m11) (m12) (m21) (m22):
  20                m21+m22-(m11+m12) = m21-m11-m12+(m22-(m11-m21)-(m12-(m21-m11))).
  21 #m11 #m12 #m21 #m22 >minus_plus
  22 elim (le_or_ge (m11+m12) m21) #Hm1121
  23 [ lapply (transitive_le m11 ??? Hm1121) // #Hm121
  24   lapply (le_plus_to_minus_l … Hm1121) #Hm12211
  25   <plus_minus // @eq_f2 // >(eq_minus_O m11 ?) // >(eq_minus_O m12 ?) //
  26 | >(eq_minus_O m21 ?) // <plus_O_n <minus_plus <commutative_plus
  27   elim (le_or_ge m11 m21) #Hm121
  28   [ lapply (le_plus_to_minus_comm … Hm1121) #Hm2112
  29     >(eq_minus_O m11 ?) // <plus_minus_associative // <minus_le_minus_minus_comm //
  30   | >(eq_minus_O m21 ?) // <minus_le_minus_minus_comm //
  31   ]
  32 ]
  33 qed.
  34
  35 lemma arith_l3 (m) (n1) (n2): n1+n2-m = n1-m+(n2-(m-n1)).
  36 // qed.
  37
  38 lemma arith_l2 (n1) (n2): ↑n2-n1 = 1-n1+(n2-(n1-1)).
  39 #n1 #n2 <arith_l3 //
  40 qed.
  41
  42 lemma arith_l1: ∀x. 1 = 1-x+(x-(x-1)).
  43 #x <arith_l2 //
  44 qed.