src/HOL/Numeral_Simprocs.thy
changeset 33366 b0096ac3b731
child 37886 2f9d3fc1a8ac
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Numeral_Simprocs.thy	Fri Oct 30 18:33:07 2009 +0100
     1.3 @@ -0,0 +1,120 @@
     1.4 +(* Author: Various *)
     1.5 +
     1.6 +header {* Combination and Cancellation Simprocs for Numeral Expressions *}
     1.7 +
     1.8 +theory Numeral_Simprocs
     1.9 +imports Divides
    1.10 +uses
    1.11 +  "~~/src/Provers/Arith/assoc_fold.ML"
    1.12 +  "~~/src/Provers/Arith/cancel_numerals.ML"
    1.13 +  "~~/src/Provers/Arith/combine_numerals.ML"
    1.14 +  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
    1.15 +  "~~/src/Provers/Arith/extract_common_term.ML"
    1.16 +  ("Tools/numeral_simprocs.ML")
    1.17 +  ("Tools/nat_numeral_simprocs.ML")
    1.18 +begin
    1.19 +
    1.20 +declare split_div [of _ _ "number_of k", standard, arith_split]
    1.21 +declare split_mod [of _ _ "number_of k", standard, arith_split]
    1.22 +
    1.23 +text {* For @{text combine_numerals} *}
    1.24 +
    1.25 +lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
    1.26 +by (simp add: add_mult_distrib)
    1.27 +
    1.28 +text {* For @{text cancel_numerals} *}
    1.29 +
    1.30 +lemma nat_diff_add_eq1:
    1.31 +     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
    1.32 +by (simp split add: nat_diff_split add: add_mult_distrib)
    1.33 +
    1.34 +lemma nat_diff_add_eq2:
    1.35 +     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
    1.36 +by (simp split add: nat_diff_split add: add_mult_distrib)
    1.37 +
    1.38 +lemma nat_eq_add_iff1:
    1.39 +     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
    1.40 +by (auto split add: nat_diff_split simp add: add_mult_distrib)
    1.41 +
    1.42 +lemma nat_eq_add_iff2:
    1.43 +     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
    1.44 +by (auto split add: nat_diff_split simp add: add_mult_distrib)
    1.45 +
    1.46 +lemma nat_less_add_iff1:
    1.47 +     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
    1.48 +by (auto split add: nat_diff_split simp add: add_mult_distrib)
    1.49 +
    1.50 +lemma nat_less_add_iff2:
    1.51 +     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
    1.52 +by (auto split add: nat_diff_split simp add: add_mult_distrib)
    1.53 +
    1.54 +lemma nat_le_add_iff1:
    1.55 +     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
    1.56 +by (auto split add: nat_diff_split simp add: add_mult_distrib)
    1.57 +
    1.58 +lemma nat_le_add_iff2:
    1.59 +     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
    1.60 +by (auto split add: nat_diff_split simp add: add_mult_distrib)
    1.61 +
    1.62 +text {* For @{text cancel_numeral_factors} *}
    1.63 +
    1.64 +lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
    1.65 +by auto
    1.66 +
    1.67 +lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
    1.68 +by auto
    1.69 +
    1.70 +lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
    1.71 +by auto
    1.72 +
    1.73 +lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
    1.74 +by auto
    1.75 +
    1.76 +lemma nat_mult_dvd_cancel_disj[simp]:
    1.77 +  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
    1.78 +by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
    1.79 +
    1.80 +lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
    1.81 +by(auto)
    1.82 +
    1.83 +text {* For @{text cancel_factor} *}
    1.84 +
    1.85 +lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
    1.86 +by auto
    1.87 +
    1.88 +lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
    1.89 +by auto
    1.90 +
    1.91 +lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
    1.92 +by auto
    1.93 +
    1.94 +lemma nat_mult_div_cancel_disj[simp]:
    1.95 +     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
    1.96 +by (simp add: nat_mult_div_cancel1)
    1.97 +
    1.98 +
    1.99 +use "Tools/numeral_simprocs.ML"
   1.100 +
   1.101 +use "Tools/nat_numeral_simprocs.ML"
   1.102 +
   1.103 +declaration {* 
   1.104 +  K (Lin_Arith.add_simps (@{thms neg_simps} @ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}])
   1.105 +  #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1},
   1.106 +     @{thm nat_0}, @{thm nat_1},
   1.107 +     @{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of},
   1.108 +     @{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less},
   1.109 +     @{thm le_Suc_number_of}, @{thm le_number_of_Suc},
   1.110 +     @{thm less_Suc_number_of}, @{thm less_number_of_Suc},
   1.111 +     @{thm Suc_eq_number_of}, @{thm eq_number_of_Suc},
   1.112 +     @{thm mult_Suc}, @{thm mult_Suc_right},
   1.113 +     @{thm add_Suc}, @{thm add_Suc_right},
   1.114 +     @{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of},
   1.115 +     @{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of},
   1.116 +     @{thm if_True}, @{thm if_False}])
   1.117 +  #> Lin_Arith.add_simprocs (Numeral_Simprocs.assoc_fold_simproc
   1.118 +      :: Numeral_Simprocs.combine_numerals
   1.119 +      :: Numeral_Simprocs.cancel_numerals)
   1.120 +  #> Lin_Arith.add_simprocs (Nat_Numeral_Simprocs.combine_numerals :: Nat_Numeral_Simprocs.cancel_numerals))
   1.121 +*}
   1.122 +
   1.123 +end
   1.124 \ No newline at end of file