src/HOL/Arith_Tools.thy
author wenzelm
Wed Sep 17 21:27:14 2008 +0200 (2008-09-17)
changeset 28263 69eaa97e7e96
parent 26462 dac4e2bce00d
child 28402 09e4aa3ddc25
permissions -rw-r--r--
moved global ML bindings to global place;
     1 (*  Title:      HOL/Arith_Tools.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Amine Chaieb, TU Muenchen
     5 *)
     6 
     7 header {* Setup of arithmetic tools *}
     8 
     9 theory Arith_Tools
    10 imports Groebner_Basis
    11 uses
    12   "~~/src/Provers/Arith/cancel_numeral_factor.ML"
    13   "~~/src/Provers/Arith/extract_common_term.ML"
    14   "int_factor_simprocs.ML"
    15   "nat_simprocs.ML"
    16 begin
    17 
    18 subsection {* Simprocs for the Naturals *}
    19 
    20 declaration {* K nat_simprocs_setup *}
    21 
    22 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
    23 
    24 text{*Where K above is a literal*}
    25 
    26 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
    27 by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
    28 
    29 text {*Now just instantiating @{text n} to @{text "number_of v"} does
    30   the right simplification, but with some redundant inequality
    31   tests.*}
    32 lemma neg_number_of_pred_iff_0:
    33   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
    34 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
    35 apply (simp only: less_Suc_eq_le le_0_eq)
    36 apply (subst less_number_of_Suc, simp)
    37 done
    38 
    39 text{*No longer required as a simprule because of the @{text inverse_fold}
    40    simproc*}
    41 lemma Suc_diff_number_of:
    42      "neg (number_of (uminus v)::int) ==>
    43       Suc m - (number_of v) = m - (number_of (Int.pred v))"
    44 apply (subst Suc_diff_eq_diff_pred)
    45 apply simp
    46 apply (simp del: nat_numeral_1_eq_1)
    47 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
    48                         neg_number_of_pred_iff_0)
    49 done
    50 
    51 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
    52 by (simp add: numerals split add: nat_diff_split)
    53 
    54 
    55 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
    56 
    57 lemma nat_case_number_of [simp]:
    58      "nat_case a f (number_of v) =
    59         (let pv = number_of (Int.pred v) in
    60          if neg pv then a else f (nat pv))"
    61 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
    62 
    63 lemma nat_case_add_eq_if [simp]:
    64      "nat_case a f ((number_of v) + n) =
    65        (let pv = number_of (Int.pred v) in
    66          if neg pv then nat_case a f n else f (nat pv + n))"
    67 apply (subst add_eq_if)
    68 apply (simp split add: nat.split
    69             del: nat_numeral_1_eq_1
    70             add: numeral_1_eq_Suc_0 [symmetric] Let_def
    71                  neg_imp_number_of_eq_0 neg_number_of_pred_iff_0)
    72 done
    73 
    74 lemma nat_rec_number_of [simp]:
    75      "nat_rec a f (number_of v) =
    76         (let pv = number_of (Int.pred v) in
    77          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
    78 apply (case_tac " (number_of v) ::nat")
    79 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
    80 apply (simp split add: split_if_asm)
    81 done
    82 
    83 lemma nat_rec_add_eq_if [simp]:
    84      "nat_rec a f (number_of v + n) =
    85         (let pv = number_of (Int.pred v) in
    86          if neg pv then nat_rec a f n
    87                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
    88 apply (subst add_eq_if)
    89 apply (simp split add: nat.split
    90             del: nat_numeral_1_eq_1
    91             add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0
    92                  neg_number_of_pred_iff_0)
    93 done
    94 
    95 
    96 subsubsection{*Various Other Lemmas*}
    97 
    98 text {*Evens and Odds, for Mutilated Chess Board*}
    99 
   100 text{*Lemmas for specialist use, NOT as default simprules*}
   101 lemma nat_mult_2: "2 * z = (z+z::nat)"
   102 proof -
   103   have "2*z = (1 + 1)*z" by simp
   104   also have "... = z+z" by (simp add: left_distrib)
   105   finally show ?thesis .
   106 qed
   107 
   108 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
   109 by (subst mult_commute, rule nat_mult_2)
   110 
   111 text{*Case analysis on @{term "n<2"}*}
   112 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
   113 by arith
   114 
   115 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
   116 by arith
   117 
   118 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
   119 by (simp add: nat_mult_2 [symmetric])
   120 
   121 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
   122 apply (subgoal_tac "m mod 2 < 2")
   123 apply (erule less_2_cases [THEN disjE])
   124 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
   125 done
   126 
   127 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
   128 apply (subgoal_tac "m mod 2 < 2")
   129 apply (force simp del: mod_less_divisor, simp)
   130 done
   131 
   132 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
   133 
   134 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
   135 by simp
   136 
   137 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
   138 by simp
   139 
   140 text{*Can be used to eliminate long strings of Sucs, but not by default*}
   141 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
   142 by simp
   143 
   144 
   145 text{*These lemmas collapse some needless occurrences of Suc:
   146     at least three Sucs, since two and fewer are rewritten back to Suc again!
   147     We already have some rules to simplify operands smaller than 3.*}
   148 
   149 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
   150 by (simp add: Suc3_eq_add_3)
   151 
   152 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
   153 by (simp add: Suc3_eq_add_3)
   154 
   155 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
   156 by (simp add: Suc3_eq_add_3)
   157 
   158 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
   159 by (simp add: Suc3_eq_add_3)
   160 
   161 lemmas Suc_div_eq_add3_div_number_of =
   162     Suc_div_eq_add3_div [of _ "number_of v", standard]
   163 declare Suc_div_eq_add3_div_number_of [simp]
   164 
   165 lemmas Suc_mod_eq_add3_mod_number_of =
   166     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
   167 declare Suc_mod_eq_add3_mod_number_of [simp]
   168 
   169 
   170 subsubsection{*Special Simplification for Constants*}
   171 
   172 text{*These belong here, late in the development of HOL, to prevent their
   173 interfering with proofs of abstract properties of instances of the function
   174 @{term number_of}*}
   175 
   176 text{*These distributive laws move literals inside sums and differences.*}
   177 lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard]
   178 declare left_distrib_number_of [simp]
   179 
   180 lemmas right_distrib_number_of = right_distrib [of "number_of v", standard]
   181 declare right_distrib_number_of [simp]
   182 
   183 
   184 lemmas left_diff_distrib_number_of =
   185     left_diff_distrib [of _ _ "number_of v", standard]
   186 declare left_diff_distrib_number_of [simp]
   187 
   188 lemmas right_diff_distrib_number_of =
   189     right_diff_distrib [of "number_of v", standard]
   190 declare right_diff_distrib_number_of [simp]
   191 
   192 
   193 text{*These are actually for fields, like real: but where else to put them?*}
   194 lemmas zero_less_divide_iff_number_of =
   195     zero_less_divide_iff [of "number_of w", standard]
   196 declare zero_less_divide_iff_number_of [simp,noatp]
   197 
   198 lemmas divide_less_0_iff_number_of =
   199     divide_less_0_iff [of "number_of w", standard]
   200 declare divide_less_0_iff_number_of [simp,noatp]
   201 
   202 lemmas zero_le_divide_iff_number_of =
   203     zero_le_divide_iff [of "number_of w", standard]
   204 declare zero_le_divide_iff_number_of [simp,noatp]
   205 
   206 lemmas divide_le_0_iff_number_of =
   207     divide_le_0_iff [of "number_of w", standard]
   208 declare divide_le_0_iff_number_of [simp,noatp]
   209 
   210 
   211 (****
   212 IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
   213 then these special-case declarations may be useful.
   214 
   215 text{*These simprules move numerals into numerators and denominators.*}
   216 lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
   217 by (simp add: times_divide_eq)
   218 
   219 lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
   220 by (simp add: times_divide_eq)
   221 
   222 lemmas times_divide_eq_right_number_of =
   223     times_divide_eq_right [of "number_of w", standard]
   224 declare times_divide_eq_right_number_of [simp]
   225 
   226 lemmas times_divide_eq_right_number_of =
   227     times_divide_eq_right [of _ _ "number_of w", standard]
   228 declare times_divide_eq_right_number_of [simp]
   229 
   230 lemmas times_divide_eq_left_number_of =
   231     times_divide_eq_left [of _ "number_of w", standard]
   232 declare times_divide_eq_left_number_of [simp]
   233 
   234 lemmas times_divide_eq_left_number_of =
   235     times_divide_eq_left [of _ _ "number_of w", standard]
   236 declare times_divide_eq_left_number_of [simp]
   237 
   238 ****)
   239 
   240 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
   241   strange, but then other simprocs simplify the quotient.*}
   242 
   243 lemmas inverse_eq_divide_number_of =
   244     inverse_eq_divide [of "number_of w", standard]
   245 declare inverse_eq_divide_number_of [simp]
   246 
   247 
   248 text {*These laws simplify inequalities, moving unary minus from a term
   249 into the literal.*}
   250 lemmas less_minus_iff_number_of =
   251     less_minus_iff [of "number_of v", standard]
   252 declare less_minus_iff_number_of [simp,noatp]
   253 
   254 lemmas le_minus_iff_number_of =
   255     le_minus_iff [of "number_of v", standard]
   256 declare le_minus_iff_number_of [simp,noatp]
   257 
   258 lemmas equation_minus_iff_number_of =
   259     equation_minus_iff [of "number_of v", standard]
   260 declare equation_minus_iff_number_of [simp,noatp]
   261 
   262 
   263 lemmas minus_less_iff_number_of =
   264     minus_less_iff [of _ "number_of v", standard]
   265 declare minus_less_iff_number_of [simp,noatp]
   266 
   267 lemmas minus_le_iff_number_of =
   268     minus_le_iff [of _ "number_of v", standard]
   269 declare minus_le_iff_number_of [simp,noatp]
   270 
   271 lemmas minus_equation_iff_number_of =
   272     minus_equation_iff [of _ "number_of v", standard]
   273 declare minus_equation_iff_number_of [simp,noatp]
   274 
   275 
   276 text{*To Simplify Inequalities Where One Side is the Constant 1*}
   277 
   278 lemma less_minus_iff_1 [simp,noatp]:
   279   fixes b::"'b::{ordered_idom,number_ring}"
   280   shows "(1 < - b) = (b < -1)"
   281 by auto
   282 
   283 lemma le_minus_iff_1 [simp,noatp]:
   284   fixes b::"'b::{ordered_idom,number_ring}"
   285   shows "(1 \<le> - b) = (b \<le> -1)"
   286 by auto
   287 
   288 lemma equation_minus_iff_1 [simp,noatp]:
   289   fixes b::"'b::number_ring"
   290   shows "(1 = - b) = (b = -1)"
   291 by (subst equation_minus_iff, auto)
   292 
   293 lemma minus_less_iff_1 [simp,noatp]:
   294   fixes a::"'b::{ordered_idom,number_ring}"
   295   shows "(- a < 1) = (-1 < a)"
   296 by auto
   297 
   298 lemma minus_le_iff_1 [simp,noatp]:
   299   fixes a::"'b::{ordered_idom,number_ring}"
   300   shows "(- a \<le> 1) = (-1 \<le> a)"
   301 by auto
   302 
   303 lemma minus_equation_iff_1 [simp,noatp]:
   304   fixes a::"'b::number_ring"
   305   shows "(- a = 1) = (a = -1)"
   306 by (subst minus_equation_iff, auto)
   307 
   308 
   309 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
   310 
   311 lemmas mult_less_cancel_left_number_of =
   312     mult_less_cancel_left [of "number_of v", standard]
   313 declare mult_less_cancel_left_number_of [simp,noatp]
   314 
   315 lemmas mult_less_cancel_right_number_of =
   316     mult_less_cancel_right [of _ "number_of v", standard]
   317 declare mult_less_cancel_right_number_of [simp,noatp]
   318 
   319 lemmas mult_le_cancel_left_number_of =
   320     mult_le_cancel_left [of "number_of v", standard]
   321 declare mult_le_cancel_left_number_of [simp,noatp]
   322 
   323 lemmas mult_le_cancel_right_number_of =
   324     mult_le_cancel_right [of _ "number_of v", standard]
   325 declare mult_le_cancel_right_number_of [simp,noatp]
   326 
   327 
   328 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
   329 
   330 lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
   331 lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
   332 lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
   333 lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
   334 lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
   335 lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
   336 
   337 
   338 subsubsection{*Optional Simplification Rules Involving Constants*}
   339 
   340 text{*Simplify quotients that are compared with a literal constant.*}
   341 
   342 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
   343 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
   344 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
   345 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
   346 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
   347 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
   348 
   349 
   350 text{*Not good as automatic simprules because they cause case splits.*}
   351 lemmas divide_const_simps =
   352   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
   353   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
   354   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
   355 
   356 text{*Division By @{text "-1"}*}
   357 
   358 lemma divide_minus1 [simp]:
   359      "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
   360 by simp
   361 
   362 lemma minus1_divide [simp]:
   363      "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
   364 by (simp add: divide_inverse inverse_minus_eq)
   365 
   366 lemma half_gt_zero_iff:
   367      "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
   368 by auto
   369 
   370 lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
   371 declare half_gt_zero [simp]
   372 
   373 (* The following lemma should appear in Divides.thy, but there the proof
   374    doesn't work. *)
   375 
   376 lemma nat_dvd_not_less:
   377   "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
   378   by (unfold dvd_def) auto
   379 
   380 ML {*
   381 val divide_minus1 = @{thm divide_minus1};
   382 val minus1_divide = @{thm minus1_divide};
   383 *}
   384 
   385 
   386 subsection{* Groebner Bases for fields *}
   387 
   388 interpretation class_fieldgb:
   389   fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
   390 
   391 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
   392 lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
   393   by simp
   394 lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
   395   by simp
   396 lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   397   by simp
   398 lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   399   by simp
   400 
   401 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
   402 
   403 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
   404   by (simp add: add_divide_distrib)
   405 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
   406   by (simp add: add_divide_distrib)
   407 
   408 
   409 ML{* 
   410 local
   411  val zr = @{cpat "0"}
   412  val zT = ctyp_of_term zr
   413  val geq = @{cpat "op ="}
   414  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
   415  val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
   416  val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
   417  val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
   418 
   419  fun prove_nz ss T t =
   420     let
   421       val z = instantiate_cterm ([(zT,T)],[]) zr
   422       val eq = instantiate_cterm ([(eqT,T)],[]) geq
   423       val th = Simplifier.rewrite (ss addsimps simp_thms)
   424            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
   425                   (Thm.capply (Thm.capply eq t) z)))
   426     in equal_elim (symmetric th) TrueI
   427     end
   428 
   429  fun proc phi ss ct =
   430   let
   431     val ((x,y),(w,z)) =
   432          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
   433     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
   434     val T = ctyp_of_term x
   435     val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
   436     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
   437   in SOME (implies_elim (implies_elim th y_nz) z_nz)
   438   end
   439   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
   440 
   441  fun proc2 phi ss ct =
   442   let
   443     val (l,r) = Thm.dest_binop ct
   444     val T = ctyp_of_term l
   445   in (case (term_of l, term_of r) of
   446       (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
   447         let val (x,y) = Thm.dest_binop l val z = r
   448             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
   449             val ynz = prove_nz ss T y
   450         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
   451         end
   452      | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
   453         let val (x,y) = Thm.dest_binop r val z = l
   454             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
   455             val ynz = prove_nz ss T y
   456         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
   457         end
   458      | _ => NONE)
   459   end
   460   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
   461 
   462  fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
   463    | is_number t = can HOLogic.dest_number t
   464 
   465  val is_number = is_number o term_of
   466 
   467  fun proc3 phi ss ct =
   468   (case term_of ct of
   469     Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   470       let
   471         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   472         val _ = map is_number [a,b,c]
   473         val T = ctyp_of_term c
   474         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
   475       in SOME (mk_meta_eq th) end
   476   | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   477       let
   478         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   479         val _ = map is_number [a,b,c]
   480         val T = ctyp_of_term c
   481         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
   482       in SOME (mk_meta_eq th) end
   483   | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   484       let
   485         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   486         val _ = map is_number [a,b,c]
   487         val T = ctyp_of_term c
   488         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
   489       in SOME (mk_meta_eq th) end
   490   | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   491     let
   492       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   493         val _ = map is_number [a,b,c]
   494         val T = ctyp_of_term c
   495         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
   496       in SOME (mk_meta_eq th) end
   497   | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   498     let
   499       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   500         val _ = map is_number [a,b,c]
   501         val T = ctyp_of_term c
   502         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
   503       in SOME (mk_meta_eq th) end
   504   | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   505     let
   506       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   507         val _ = map is_number [a,b,c]
   508         val T = ctyp_of_term c
   509         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
   510       in SOME (mk_meta_eq th) end
   511   | _ => NONE)
   512   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
   513 
   514 val add_frac_frac_simproc =
   515        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
   516                      name = "add_frac_frac_simproc",
   517                      proc = proc, identifier = []}
   518 
   519 val add_frac_num_simproc =
   520        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
   521                      name = "add_frac_num_simproc",
   522                      proc = proc2, identifier = []}
   523 
   524 val ord_frac_simproc =
   525   make_simproc
   526     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
   527              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
   528              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
   529              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
   530              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
   531              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
   532              name = "ord_frac_simproc", proc = proc3, identifier = []}
   533 
   534 val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
   535                "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]
   536 
   537 val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
   538                  "add_Suc", "add_number_of_left", "mult_number_of_left",
   539                  "Suc_eq_add_numeral_1"])@
   540                  (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
   541                  @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps}
   542 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
   543            @{thm "divide_Numeral1"},
   544            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
   545            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
   546            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
   547            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
   548            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
   549            @{thm "diff_def"}, @{thm "minus_divide_left"},
   550            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
   551 
   552 local
   553 open Conv
   554 in
   555 val comp_conv = (Simplifier.rewrite
   556 (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
   557               addsimps ths addsimps comp_arith addsimps simp_thms
   558               addsimprocs field_cancel_numeral_factors
   559                addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
   560                             ord_frac_simproc]
   561                 addcongs [@{thm "if_weak_cong"}]))
   562 then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
   563   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
   564 end
   565 
   566 fun numeral_is_const ct =
   567   case term_of ct of
   568    Const (@{const_name "HOL.divide"},_) $ a $ b =>
   569      numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
   570  | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
   571  | t => can HOLogic.dest_number t
   572 
   573 fun dest_const ct = ((case term_of ct of
   574    Const (@{const_name "HOL.divide"},_) $ a $ b=>
   575     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
   576  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
   577    handle TERM _ => error "ring_dest_const")
   578 
   579 fun mk_const phi cT x =
   580  let val (a, b) = Rat.quotient_of_rat x
   581  in if b = 1 then Numeral.mk_cnumber cT a
   582     else Thm.capply
   583          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
   584                      (Numeral.mk_cnumber cT a))
   585          (Numeral.mk_cnumber cT b)
   586   end
   587 
   588 in
   589  val field_comp_conv = comp_conv;
   590  val fieldgb_declaration = 
   591   NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
   592    {is_const = K numeral_is_const,
   593     dest_const = K dest_const,
   594     mk_const = mk_const,
   595     conv = K (K comp_conv)}
   596 end;
   597 *}
   598 
   599 declaration{* fieldgb_declaration *}
   600 end