src/HOL/Numeral_Simprocs.thy
author huffman
Sun Apr 01 16:09:58 2012 +0200 (2012-04-01)
changeset 47255 30a1692557b0
parent 47159 978c00c20a59
child 48891 c0eafbd55de3
permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
     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 lemmas semiring_norm =
    18   Let_def arith_simps nat_arith rel_simps
    19   if_False if_True
    20   add_0 add_Suc add_numeral_left
    21   add_neg_numeral_left mult_numeral_left
    22   numeral_1_eq_1 [symmetric] Suc_eq_plus1
    23   eq_numeral_iff_iszero not_iszero_Numeral1
    24 
    25 declare split_div [of _ _ "numeral k", arith_split] for k
    26 declare split_mod [of _ _ "numeral k", arith_split] for k
    27 
    28 text {* For @{text combine_numerals} *}
    29 
    30 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
    31 by (simp add: add_mult_distrib)
    32 
    33 text {* For @{text cancel_numerals} *}
    34 
    35 lemma nat_diff_add_eq1:
    36      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
    37 by (simp split add: nat_diff_split add: add_mult_distrib)
    38 
    39 lemma nat_diff_add_eq2:
    40      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
    41 by (simp split add: nat_diff_split add: add_mult_distrib)
    42 
    43 lemma nat_eq_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_eq_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_less_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_less_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 lemma nat_le_add_iff1:
    60      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
    61 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    62 
    63 lemma nat_le_add_iff2:
    64      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
    65 by (auto split add: nat_diff_split simp add: add_mult_distrib)
    66 
    67 text {* For @{text cancel_numeral_factors} *}
    68 
    69 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
    70 by auto
    71 
    72 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
    73 by auto
    74 
    75 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
    76 by auto
    77 
    78 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
    79 by auto
    80 
    81 lemma nat_mult_dvd_cancel_disj[simp]:
    82   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
    83 by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1)
    84 
    85 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
    86 by(auto)
    87 
    88 text {* For @{text cancel_factor} *}
    89 
    90 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
    91 by auto
    92 
    93 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
    94 by auto
    95 
    96 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
    97 by auto
    98 
    99 lemma nat_mult_div_cancel_disj[simp]:
   100      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
   101 by (simp add: nat_mult_div_cancel1)
   102 
   103 use "Tools/numeral_simprocs.ML"
   104 
   105 simproc_setup semiring_assoc_fold
   106   ("(a::'a::comm_semiring_1_cancel) * b") =
   107   {* fn phi => Numeral_Simprocs.assoc_fold *}
   108 
   109 (* TODO: see whether the type class can be generalized further *)
   110 simproc_setup int_combine_numerals
   111   ("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") =
   112   {* fn phi => Numeral_Simprocs.combine_numerals *}
   113 
   114 simproc_setup field_combine_numerals
   115   ("(i::'a::{field_inverse_zero,ring_char_0}) + j"
   116   |"(i::'a::{field_inverse_zero,ring_char_0}) - j") =
   117   {* fn phi => Numeral_Simprocs.field_combine_numerals *}
   118 
   119 simproc_setup inteq_cancel_numerals
   120   ("(l::'a::comm_ring_1) + m = n"
   121   |"(l::'a::comm_ring_1) = m + n"
   122   |"(l::'a::comm_ring_1) - m = n"
   123   |"(l::'a::comm_ring_1) = m - n"
   124   |"(l::'a::comm_ring_1) * m = n"
   125   |"(l::'a::comm_ring_1) = m * n"
   126   |"- (l::'a::comm_ring_1) = m"
   127   |"(l::'a::comm_ring_1) = - m") =
   128   {* fn phi => Numeral_Simprocs.eq_cancel_numerals *}
   129 
   130 simproc_setup intless_cancel_numerals
   131   ("(l::'a::linordered_idom) + m < n"
   132   |"(l::'a::linordered_idom) < m + n"
   133   |"(l::'a::linordered_idom) - m < n"
   134   |"(l::'a::linordered_idom) < m - n"
   135   |"(l::'a::linordered_idom) * m < n"
   136   |"(l::'a::linordered_idom) < m * n"
   137   |"- (l::'a::linordered_idom) < m"
   138   |"(l::'a::linordered_idom) < - m") =
   139   {* fn phi => Numeral_Simprocs.less_cancel_numerals *}
   140 
   141 simproc_setup intle_cancel_numerals
   142   ("(l::'a::linordered_idom) + m \<le> n"
   143   |"(l::'a::linordered_idom) \<le> m + n"
   144   |"(l::'a::linordered_idom) - m \<le> n"
   145   |"(l::'a::linordered_idom) \<le> m - n"
   146   |"(l::'a::linordered_idom) * m \<le> n"
   147   |"(l::'a::linordered_idom) \<le> m * n"
   148   |"- (l::'a::linordered_idom) \<le> m"
   149   |"(l::'a::linordered_idom) \<le> - m") =
   150   {* fn phi => Numeral_Simprocs.le_cancel_numerals *}
   151 
   152 simproc_setup ring_eq_cancel_numeral_factor
   153   ("(l::'a::{idom,ring_char_0}) * m = n"
   154   |"(l::'a::{idom,ring_char_0}) = m * n") =
   155   {* fn phi => Numeral_Simprocs.eq_cancel_numeral_factor *}
   156 
   157 simproc_setup ring_less_cancel_numeral_factor
   158   ("(l::'a::linordered_idom) * m < n"
   159   |"(l::'a::linordered_idom) < m * n") =
   160   {* fn phi => Numeral_Simprocs.less_cancel_numeral_factor *}
   161 
   162 simproc_setup ring_le_cancel_numeral_factor
   163   ("(l::'a::linordered_idom) * m <= n"
   164   |"(l::'a::linordered_idom) <= m * n") =
   165   {* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *}
   166 
   167 (* TODO: remove comm_ring_1 constraint if possible *)
   168 simproc_setup int_div_cancel_numeral_factors
   169   ("((l::'a::{semiring_div,comm_ring_1,ring_char_0}) * m) div n"
   170   |"(l::'a::{semiring_div,comm_ring_1,ring_char_0}) div (m * n)") =
   171   {* fn phi => Numeral_Simprocs.div_cancel_numeral_factor *}
   172 
   173 simproc_setup divide_cancel_numeral_factor
   174   ("((l::'a::{field_inverse_zero,ring_char_0}) * m) / n"
   175   |"(l::'a::{field_inverse_zero,ring_char_0}) / (m * n)"
   176   |"((numeral v)::'a::{field_inverse_zero,ring_char_0}) / (numeral w)") =
   177   {* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *}
   178 
   179 simproc_setup ring_eq_cancel_factor
   180   ("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =
   181   {* fn phi => Numeral_Simprocs.eq_cancel_factor *}
   182 
   183 simproc_setup linordered_ring_le_cancel_factor
   184   ("(l::'a::linordered_idom) * m <= n"
   185   |"(l::'a::linordered_idom) <= m * n") =
   186   {* fn phi => Numeral_Simprocs.le_cancel_factor *}
   187 
   188 simproc_setup linordered_ring_less_cancel_factor
   189   ("(l::'a::linordered_idom) * m < n"
   190   |"(l::'a::linordered_idom) < m * n") =
   191   {* fn phi => Numeral_Simprocs.less_cancel_factor *}
   192 
   193 simproc_setup int_div_cancel_factor
   194   ("((l::'a::semiring_div) * m) div n"
   195   |"(l::'a::semiring_div) div (m * n)") =
   196   {* fn phi => Numeral_Simprocs.div_cancel_factor *}
   197 
   198 simproc_setup int_mod_cancel_factor
   199   ("((l::'a::semiring_div) * m) mod n"
   200   |"(l::'a::semiring_div) mod (m * n)") =
   201   {* fn phi => Numeral_Simprocs.mod_cancel_factor *}
   202 
   203 simproc_setup dvd_cancel_factor
   204   ("((l::'a::idom) * m) dvd n"
   205   |"(l::'a::idom) dvd (m * n)") =
   206   {* fn phi => Numeral_Simprocs.dvd_cancel_factor *}
   207 
   208 simproc_setup divide_cancel_factor
   209   ("((l::'a::field_inverse_zero) * m) / n"
   210   |"(l::'a::field_inverse_zero) / (m * n)") =
   211   {* fn phi => Numeral_Simprocs.divide_cancel_factor *}
   212 
   213 use "Tools/nat_numeral_simprocs.ML"
   214 
   215 simproc_setup nat_combine_numerals
   216   ("(i::nat) + j" | "Suc (i + j)") =
   217   {* fn phi => Nat_Numeral_Simprocs.combine_numerals *}
   218 
   219 simproc_setup nateq_cancel_numerals
   220   ("(l::nat) + m = n" | "(l::nat) = m + n" |
   221    "(l::nat) * m = n" | "(l::nat) = m * n" |
   222    "Suc m = n" | "m = Suc n") =
   223   {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *}
   224 
   225 simproc_setup natless_cancel_numerals
   226   ("(l::nat) + m < n" | "(l::nat) < m + n" |
   227    "(l::nat) * m < n" | "(l::nat) < m * n" |
   228    "Suc m < n" | "m < Suc n") =
   229   {* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *}
   230 
   231 simproc_setup natle_cancel_numerals
   232   ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" |
   233    "(l::nat) * m \<le> n" | "(l::nat) \<le> m * n" |
   234    "Suc m \<le> n" | "m \<le> Suc n") =
   235   {* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *}
   236 
   237 simproc_setup natdiff_cancel_numerals
   238   ("((l::nat) + m) - n" | "(l::nat) - (m + n)" |
   239    "(l::nat) * m - n" | "(l::nat) - m * n" |
   240    "Suc m - n" | "m - Suc n") =
   241   {* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *}
   242 
   243 simproc_setup nat_eq_cancel_numeral_factor
   244   ("(l::nat) * m = n" | "(l::nat) = m * n") =
   245   {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *}
   246 
   247 simproc_setup nat_less_cancel_numeral_factor
   248   ("(l::nat) * m < n" | "(l::nat) < m * n") =
   249   {* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *}
   250 
   251 simproc_setup nat_le_cancel_numeral_factor
   252   ("(l::nat) * m <= n" | "(l::nat) <= m * n") =
   253   {* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *}
   254 
   255 simproc_setup nat_div_cancel_numeral_factor
   256   ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
   257   {* fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor *}
   258 
   259 simproc_setup nat_dvd_cancel_numeral_factor
   260   ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
   261   {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *}
   262 
   263 simproc_setup nat_eq_cancel_factor
   264   ("(l::nat) * m = n" | "(l::nat) = m * n") =
   265   {* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *}
   266 
   267 simproc_setup nat_less_cancel_factor
   268   ("(l::nat) * m < n" | "(l::nat) < m * n") =
   269   {* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *}
   270 
   271 simproc_setup nat_le_cancel_factor
   272   ("(l::nat) * m <= n" | "(l::nat) <= m * n") =
   273   {* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *}
   274 
   275 simproc_setup nat_div_cancel_factor
   276   ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
   277   {* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *}
   278 
   279 simproc_setup nat_dvd_cancel_factor
   280   ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
   281   {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *}
   282 
   283 (* FIXME: duplicate rule warnings for:
   284   ring_distribs
   285   numeral_plus_numeral numeral_times_numeral
   286   numeral_eq_iff numeral_less_iff numeral_le_iff
   287   numeral_neq_zero zero_neq_numeral zero_less_numeral
   288   if_True if_False *)
   289 declaration {* 
   290   K (Lin_Arith.add_simps ([@{thm Suc_numeral}, @{thm int_numeral}])
   291   #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},
   292      @{thm nat_0}, @{thm nat_1},
   293      @{thm numeral_plus_numeral}, @{thm diff_nat_numeral}, @{thm numeral_times_numeral},
   294      @{thm numeral_eq_iff}, @{thm numeral_less_iff}, @{thm numeral_le_iff},
   295      @{thm le_Suc_numeral}, @{thm le_numeral_Suc},
   296      @{thm less_Suc_numeral}, @{thm less_numeral_Suc},
   297      @{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},
   298      @{thm mult_Suc}, @{thm mult_Suc_right},
   299      @{thm add_Suc}, @{thm add_Suc_right},
   300      @{thm numeral_neq_zero}, @{thm zero_neq_numeral}, @{thm zero_less_numeral},
   301      @{thm of_int_numeral}, @{thm of_nat_numeral}, @{thm nat_numeral},
   302      @{thm if_True}, @{thm if_False}])
   303   #> Lin_Arith.add_simprocs
   304       [@{simproc semiring_assoc_fold},
   305        @{simproc int_combine_numerals},
   306        @{simproc inteq_cancel_numerals},
   307        @{simproc intless_cancel_numerals},
   308        @{simproc intle_cancel_numerals}]
   309   #> Lin_Arith.add_simprocs
   310       [@{simproc nat_combine_numerals},
   311        @{simproc nateq_cancel_numerals},
   312        @{simproc natless_cancel_numerals},
   313        @{simproc natle_cancel_numerals},
   314        @{simproc natdiff_cancel_numerals}])
   315 *}
   316 
   317 end