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

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  14
  15 include "ground/arith/nat_le_minus_plus.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 >nminus_plus_assoc
  22 elim (nat_split_le_ge (m11+m12) m21) #Hm1121
  23 [ lapply (nle_trans m11 ??? Hm1121) // #Hm121
  24   lapply (nle_minus_dx_dx … Hm1121) #Hm12211
  25   <nminus_plus_comm_23 // @eq_f2 //
  26   <(nle_inv_eq_zero_minus m11 ?) // <(nle_inv_eq_zero_minus m12 ?) //
  27 | <(nle_inv_eq_zero_minus m21 ?) // <nplus_zero_sn <nminus_plus_assoc <nplus_comm
  28   elim (nat_split_le_ge m11 m21) #Hm121
  29   [ lapply (nle_minus_sn_dx … Hm1121) #Hm2112
  30     <(nle_inv_eq_zero_minus m11 ?) // >nplus_minus_assoc // >nminus_assoc_comm_23 //
  31   | <(nle_inv_eq_zero_minus m21 ?) // >nminus_assoc_comm_23 //
  32   ]
  33 ]
  34 qed.
  35
  36 lemma arith_l3 (m) (n1) (n2): n1+n2-m = n1-m+(n2-(m-n1)).
  37 // qed.
  38
  39 lemma arith_l2 (n1) (n2): ↑n2-n1 = 𝟏-n1+(n2-(n1-𝟏)).
  40 #n1 #n2 <arith_l3 //
  41 qed.
  42
  43 lemma arith_l1 (n): ninj (𝟏) = 𝟏-n+(n-(n-𝟏)).
  44 #n <arith_l2 //
  45 qed.