src/HOL/Numeral_Simprocs.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 37886 2f9d3fc1a8ac
child 45284 ae78a4ffa81d
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
     1 (* Author: Various *)
     2 
     3 header {* Combination and Cancellation Simprocs for Numeral Expressions *}
     4 
     5 theory Numeral_Simprocs
     6 imports Divides
     7 uses
     8   "~~/src/Provers/Arith/assoc_fold.ML"
     9   "~~/src/Provers/Arith/cancel_numerals.ML"
    10   "~~/src/Provers/Arith/combine_numerals.ML"
    11   "~~/src/Provers/Arith/cancel_numeral_factor.ML"
    12   "~~/src/Provers/Arith/extract_common_term.ML"
    13   ("Tools/numeral_simprocs.ML")
    14   ("Tools/nat_numeral_simprocs.ML")
    15 begin
    16 
    17 declare split_div [of _ _ "number_of k", standard, arith_split]
    18 declare split_mod [of _ _ "number_of k", standard, arith_split]
    19 
    20 text {* For @{text combine_numerals} *}
    21 
    22 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
    23 by (simp add: add_mult_distrib)
    24 
    25 text {* For @{text cancel_numerals} *}
    26 
    27 lemma nat_diff_add_eq1:
    28      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
    29 by (simp split add: nat_diff_split add: add_mult_distrib)
    30 
    31 lemma nat_diff_add_eq2:
    32      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
    33 by (simp split add: nat_diff_split add: add_mult_distrib)
    34 
    35 lemma nat_eq_add_iff1:
    36      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
    37 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    38 
    39 lemma nat_eq_add_iff2:
    40      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
    41 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    42 
    43 lemma nat_less_add_iff1:
    44      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
    45 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    46 
    47 lemma nat_less_add_iff2:
    48      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
    49 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    50 
    51 lemma nat_le_add_iff1:
    52      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
    53 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    54 
    55 lemma nat_le_add_iff2:
    56      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
    57 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    58 
    59 text {* For @{text cancel_numeral_factors} *}
    60 
    61 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
    62 by auto
    63 
    64 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
    65 by auto
    66 
    67 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
    68 by auto
    69 
    70 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
    71 by auto
    72 
    73 lemma nat_mult_dvd_cancel_disj[simp]:
    74   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
    75 by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
    76 
    77 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
    78 by(auto)
    79 
    80 text {* For @{text cancel_factor} *}
    81 
    82 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
    83 by auto
    84 
    85 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
    86 by auto
    87 
    88 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
    89 by auto
    90 
    91 lemma nat_mult_div_cancel_disj[simp]:
    92      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
    93 by (simp add: nat_mult_div_cancel1)
    94 
    95 use "Tools/numeral_simprocs.ML"
    96 
    97 use "Tools/nat_numeral_simprocs.ML"
    98 
    99 declaration {* 
   100   K (Lin_Arith.add_simps (@{thms neg_simps} @ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}])
   101   #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1},
   102      @{thm nat_0}, @{thm nat_1},
   103      @{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of},
   104      @{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less},
   105      @{thm le_Suc_number_of}, @{thm le_number_of_Suc},
   106      @{thm less_Suc_number_of}, @{thm less_number_of_Suc},
   107      @{thm Suc_eq_number_of}, @{thm eq_number_of_Suc},
   108      @{thm mult_Suc}, @{thm mult_Suc_right},
   109      @{thm add_Suc}, @{thm add_Suc_right},
   110      @{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of},
   111      @{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of},
   112      @{thm if_True}, @{thm if_False}])
   113   #> Lin_Arith.add_simprocs (Numeral_Simprocs.assoc_fold_simproc
   114       :: Numeral_Simprocs.combine_numerals
   115       :: Numeral_Simprocs.cancel_numerals)
   116   #> Lin_Arith.add_simprocs (Nat_Numeral_Simprocs.combine_numerals :: Nat_Numeral_Simprocs.cancel_numerals))
   117 *}
   118 
   119 end