src/HOL/Arith_Tools.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 24286 7619080e49f0
child 25128 962e4f4142fa
permissions -rw-r--r--
moved Finite_Set before Datatype
     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 Dense_Linear_Order
    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 (Numeral.pred v)::int) = (number_of v = (0::nat))"
    34 apply (subgoal_tac "neg (number_of (Numeral.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 (Numeral.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 (Numeral.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 (Numeral.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 (Numeral.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 (Numeral.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_of = le_divide_eq [of _ _ "number_of w", standard]
   331 declare le_divide_eq_number_of [simp]
   332 
   333 lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard]
   334 declare divide_le_eq_number_of [simp]
   335 
   336 lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard]
   337 declare less_divide_eq_number_of [simp]
   338 
   339 lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard]
   340 declare divide_less_eq_number_of [simp]
   341 
   342 lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard]
   343 declare eq_divide_eq_number_of [simp]
   344 
   345 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard]
   346 declare divide_eq_eq_number_of [simp]
   347 
   348 
   349 
   350 subsubsection{*Optional Simplification Rules Involving Constants*}
   351 
   352 text{*Simplify quotients that are compared with a literal constant.*}
   353 
   354 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
   355 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
   356 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
   357 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
   358 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
   359 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
   360 
   361 
   362 text{*Not good as automatic simprules because they cause case splits.*}
   363 lemmas divide_const_simps =
   364   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
   365   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
   366   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
   367 
   368 text{*Division By @{text "-1"}*}
   369 
   370 lemma divide_minus1 [simp]:
   371      "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
   372 by simp
   373 
   374 lemma minus1_divide [simp]:
   375      "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
   376 by (simp add: divide_inverse inverse_minus_eq)
   377 
   378 lemma half_gt_zero_iff:
   379      "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
   380 by auto
   381 
   382 lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
   383 declare half_gt_zero [simp]
   384 
   385 (* The following lemma should appear in Divides.thy, but there the proof
   386    doesn't work. *)
   387 
   388 lemma nat_dvd_not_less:
   389   "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
   390   by (unfold dvd_def) auto
   391 
   392 ML {*
   393 val divide_minus1 = @{thm divide_minus1};
   394 val minus1_divide = @{thm minus1_divide};
   395 *}
   396 
   397 
   398 subsection{* Groebner Bases for fields *}
   399 
   400 interpretation class_fieldgb:
   401   fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
   402 
   403 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
   404 lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
   405   by simp
   406 lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
   407   by simp
   408 lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   409   by simp
   410 lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   411   by simp
   412 
   413 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
   414 
   415 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
   416   by (simp add: add_divide_distrib)
   417 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
   418   by (simp add: add_divide_distrib)
   419 
   420 declaration{*
   421 let
   422  val zr = @{cpat "0"}
   423  val zT = ctyp_of_term zr
   424  val geq = @{cpat "op ="}
   425  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
   426  val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
   427  val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
   428  val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
   429 
   430  fun prove_nz ctxt =
   431   let val ss = local_simpset_of ctxt
   432   in fn T => fn t =>
   433     let
   434       val z = instantiate_cterm ([(zT,T)],[]) zr
   435       val eq = instantiate_cterm ([(eqT,T)],[]) geq
   436       val th = Simplifier.rewrite (ss addsimps simp_thms)
   437            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
   438                   (Thm.capply (Thm.capply eq t) z)))
   439     in equal_elim (symmetric th) TrueI
   440     end
   441   end
   442 
   443  fun proc ctxt phi ss ct =
   444   let
   445     val ((x,y),(w,z)) =
   446          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
   447     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
   448     val T = ctyp_of_term x
   449     val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
   450     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
   451   in SOME (implies_elim (implies_elim th y_nz) z_nz)
   452   end
   453   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
   454 
   455  fun proc2 ctxt phi ss ct =
   456   let
   457     val (l,r) = Thm.dest_binop ct
   458     val T = ctyp_of_term l
   459   in (case (term_of l, term_of r) of
   460       (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
   461         let val (x,y) = Thm.dest_binop l val z = r
   462             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
   463             val ynz = prove_nz ctxt T y
   464         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
   465         end
   466      | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
   467         let val (x,y) = Thm.dest_binop r val z = l
   468             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
   469             val ynz = prove_nz ctxt T y
   470         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
   471         end
   472      | _ => NONE)
   473   end
   474   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
   475 
   476  fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
   477    | is_number t = can HOLogic.dest_number t
   478 
   479  val is_number = is_number o term_of
   480 
   481  fun proc3 phi ss ct =
   482   (case term_of ct of
   483     Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   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_less_eq"}
   489       in SOME (mk_meta_eq th) end
   490   | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   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 "divide_le_eq"}
   496       in SOME (mk_meta_eq th) end
   497   | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   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 "divide_eq_eq"}
   503       in SOME (mk_meta_eq th) end
   504   | Const(@{const_name HOL.less},_)$_$(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 "less_divide_eq"}
   510       in SOME (mk_meta_eq th) end
   511   | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   512     let
   513       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   514         val _ = map is_number [a,b,c]
   515         val T = ctyp_of_term c
   516         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
   517       in SOME (mk_meta_eq th) end
   518   | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   519     let
   520       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   521         val _ = map is_number [a,b,c]
   522         val T = ctyp_of_term c
   523         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
   524       in SOME (mk_meta_eq th) end
   525   | _ => NONE)
   526   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
   527 
   528 fun add_frac_frac_simproc ctxt =
   529        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
   530                      name = "add_frac_frac_simproc",
   531                      proc = proc ctxt, identifier = []}
   532 
   533 fun add_frac_num_simproc ctxt =
   534        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
   535                      name = "add_frac_num_simproc",
   536                      proc = proc2 ctxt, identifier = []}
   537 
   538 val ord_frac_simproc =
   539   make_simproc
   540     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
   541              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
   542              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
   543              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
   544              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
   545              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
   546              name = "ord_frac_simproc", proc = proc3, identifier = []}
   547 
   548 val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
   549                "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]
   550 
   551 val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
   552                  "add_Suc", "add_number_of_left", "mult_number_of_left",
   553                  "Suc_eq_add_numeral_1"])@
   554                  (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
   555                  @ arith_simps@ nat_arith @ rel_simps
   556 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
   557            @{thm "divide_Numeral1"},
   558            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
   559            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
   560            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
   561            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
   562            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
   563            @{thm "diff_def"}, @{thm "minus_divide_left"},
   564            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
   565 
   566 local
   567 open Conv
   568 in
   569 fun comp_conv ctxt = (Simplifier.rewrite
   570 (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
   571               addsimps ths addsimps comp_arith addsimps simp_thms
   572               addsimprocs field_cancel_numeral_factors
   573                addsimprocs [add_frac_frac_simproc ctxt, add_frac_num_simproc ctxt,
   574                             ord_frac_simproc]
   575                 addcongs [@{thm "if_weak_cong"}]))
   576 then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
   577   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
   578 end
   579 
   580 fun numeral_is_const ct =
   581   case term_of ct of
   582    Const (@{const_name "HOL.divide"},_) $ a $ b =>
   583      numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
   584  | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
   585  | t => can HOLogic.dest_number t
   586 
   587 fun dest_const ct = case term_of ct of
   588    Const (@{const_name "HOL.divide"},_) $ a $ b=>
   589     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
   590  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
   591 
   592 fun mk_const phi cT x =
   593  let val (a, b) = Rat.quotient_of_rat x
   594  in if b = 1 then Numeral.mk_cnumber cT a
   595     else Thm.capply
   596          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
   597                      (Numeral.mk_cnumber cT a))
   598          (Numeral.mk_cnumber cT b)
   599   end
   600 
   601 in
   602  NormalizerData.funs @{thm class_fieldgb.axioms}
   603    {is_const = K numeral_is_const,
   604     dest_const = K dest_const,
   605     mk_const = mk_const,
   606     conv = K comp_conv}
   607 end
   608 
   609 *}
   610 
   611 
   612 subsection {* Ferrante and Rackoff algorithm over ordered fields *}
   613 
   614 lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
   615 proof-
   616   assume H: "c < 0"
   617   have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
   618   also have "\<dots> = (0 < x)" by simp
   619   finally show  "(c*x < 0) == (x > 0)" by simp
   620 qed
   621 
   622 lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
   623 proof-
   624   assume H: "c > 0"
   625   hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
   626   also have "\<dots> = (0 > x)" by simp
   627   finally show  "(c*x < 0) == (x < 0)" by simp
   628 qed
   629 
   630 lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
   631 proof-
   632   assume H: "c < 0"
   633   have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
   634   also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
   635   also have "\<dots> = ((- 1/c)*t < x)" by simp
   636   finally show  "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
   637 qed
   638 
   639 lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
   640 proof-
   641   assume H: "c > 0"
   642   have "c*x + t< 0 = (c*x < -t)"  by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
   643   also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
   644   also have "\<dots> = ((- 1/c)*t > x)" by simp
   645   finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
   646 qed
   647 
   648 lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
   649   using less_diff_eq[where a= x and b=t and c=0] by simp
   650 
   651 lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
   652 proof-
   653   assume H: "c < 0"
   654   have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
   655   also have "\<dots> = (0 <= x)" by simp
   656   finally show  "(c*x <= 0) == (x >= 0)" by simp
   657 qed
   658 
   659 lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
   660 proof-
   661   assume H: "c > 0"
   662   hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
   663   also have "\<dots> = (0 >= x)" by simp
   664   finally show  "(c*x <= 0) == (x <= 0)" by simp
   665 qed
   666 
   667 lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
   668 proof-
   669   assume H: "c < 0"
   670   have "c*x + t <= 0 = (c*x <= -t)"  by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
   671   also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
   672   also have "\<dots> = ((- 1/c)*t <= x)" by simp
   673   finally show  "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
   674 qed
   675 
   676 lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
   677 proof-
   678   assume H: "c > 0"
   679   have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
   680   also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
   681   also have "\<dots> = ((- 1/c)*t >= x)" by simp
   682   finally show  "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
   683 qed
   684 
   685 lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
   686   using le_diff_eq[where a= x and b=t and c=0] by simp
   687 
   688 lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
   689 lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
   690 proof-
   691   assume H: "c \<noteq> 0"
   692   have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
   693   also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_simps)
   694   finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
   695 qed
   696 lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
   697   using eq_diff_eq[where a= x and b=t and c=0] by simp
   698 
   699 
   700 interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order
   701  ["op <=" "op <"
   702    "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
   703 proof (unfold_locales, dlo, dlo, auto)
   704   fix x y::'a assume lt: "x < y"
   705   from  less_half_sum[OF lt] show "x < (x + y) /2" by simp
   706 next
   707   fix x y::'a assume lt: "x < y"
   708   from  gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
   709 qed
   710 
   711 declaration{*
   712 let
   713 fun earlier [] x y = false
   714         | earlier (h::t) x y =
   715     if h aconvc y then false else if h aconvc x then true else earlier t x y;
   716 
   717 fun dest_frac ct = case term_of ct of
   718    Const (@{const_name "HOL.divide"},_) $ a $ b=>
   719     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
   720  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
   721 
   722 fun mk_frac phi cT x =
   723  let val (a, b) = Rat.quotient_of_rat x
   724  in if b = 1 then Numeral.mk_cnumber cT a
   725     else Thm.capply
   726          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
   727                      (Numeral.mk_cnumber cT a))
   728          (Numeral.mk_cnumber cT b)
   729  end
   730 
   731 fun whatis x ct = case term_of ct of
   732   Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
   733      if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
   734      else ("Nox",[])
   735 | Const(@{const_name "HOL.plus"}, _)$y$_ =>
   736      if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
   737      else ("Nox",[])
   738 | Const(@{const_name "HOL.times"}, _)$_$y =>
   739      if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
   740      else ("Nox",[])
   741 | t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
   742 
   743 fun xnormalize_conv ctxt [] ct = reflexive ct
   744 | xnormalize_conv ctxt (vs as (x::_)) ct =
   745    case term_of ct of
   746    Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) =>
   747     (case whatis x (Thm.dest_arg1 ct) of
   748     ("c*x+t",[c,t]) =>
   749        let
   750         val cr = dest_frac c
   751         val clt = Thm.dest_fun2 ct
   752         val cz = Thm.dest_arg ct
   753         val neg = cr </ Rat.zero
   754         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   755                (Thm.capply @{cterm "Trueprop"}
   756                   (if neg then Thm.capply (Thm.capply clt c) cz
   757                     else Thm.capply (Thm.capply clt cz) c))
   758         val cth = equal_elim (symmetric cthp) TrueI
   759         val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
   760              (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
   761         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   762                    (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   763       in rth end
   764     | ("x+t",[t]) =>
   765        let
   766         val T = ctyp_of_term x
   767         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
   768         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   769               (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   770        in  rth end
   771     | ("c*x",[c]) =>
   772        let
   773         val cr = dest_frac c
   774         val clt = Thm.dest_fun2 ct
   775         val cz = Thm.dest_arg ct
   776         val neg = cr </ Rat.zero
   777         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   778                (Thm.capply @{cterm "Trueprop"}
   779                   (if neg then Thm.capply (Thm.capply clt c) cz
   780                     else Thm.capply (Thm.capply clt cz) c))
   781         val cth = equal_elim (symmetric cthp) TrueI
   782         val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
   783              (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
   784         val rth = th
   785       in rth end
   786     | _ => reflexive ct)
   787 
   788 
   789 |  Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) =>
   790    (case whatis x (Thm.dest_arg1 ct) of
   791     ("c*x+t",[c,t]) =>
   792        let
   793         val T = ctyp_of_term x
   794         val cr = dest_frac c
   795         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
   796         val cz = Thm.dest_arg ct
   797         val neg = cr </ Rat.zero
   798         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   799                (Thm.capply @{cterm "Trueprop"}
   800                   (if neg then Thm.capply (Thm.capply clt c) cz
   801                     else Thm.capply (Thm.capply clt cz) c))
   802         val cth = equal_elim (symmetric cthp) TrueI
   803         val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
   804              (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
   805         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   806                    (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   807       in rth end
   808     | ("x+t",[t]) =>
   809        let
   810         val T = ctyp_of_term x
   811         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
   812         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   813               (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   814        in  rth end
   815     | ("c*x",[c]) =>
   816        let
   817         val T = ctyp_of_term x
   818         val cr = dest_frac c
   819         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
   820         val cz = Thm.dest_arg ct
   821         val neg = cr </ Rat.zero
   822         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   823                (Thm.capply @{cterm "Trueprop"}
   824                   (if neg then Thm.capply (Thm.capply clt c) cz
   825                     else Thm.capply (Thm.capply clt cz) c))
   826         val cth = equal_elim (symmetric cthp) TrueI
   827         val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
   828              (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
   829         val rth = th
   830       in rth end
   831     | _ => reflexive ct)
   832 
   833 |  Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
   834    (case whatis x (Thm.dest_arg1 ct) of
   835     ("c*x+t",[c,t]) =>
   836        let
   837         val T = ctyp_of_term x
   838         val cr = dest_frac c
   839         val ceq = Thm.dest_fun2 ct
   840         val cz = Thm.dest_arg ct
   841         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   842             (Thm.capply @{cterm "Trueprop"}
   843              (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
   844         val cth = equal_elim (symmetric cthp) TrueI
   845         val th = implies_elim
   846                  (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
   847         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   848                    (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   849       in rth end
   850     | ("x+t",[t]) =>
   851        let
   852         val T = ctyp_of_term x
   853         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
   854         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
   855               (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
   856        in  rth end
   857     | ("c*x",[c]) =>
   858        let
   859         val T = ctyp_of_term x
   860         val cr = dest_frac c
   861         val ceq = Thm.dest_fun2 ct
   862         val cz = Thm.dest_arg ct
   863         val cthp = Simplifier.rewrite (local_simpset_of ctxt)
   864             (Thm.capply @{cterm "Trueprop"}
   865              (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
   866         val cth = equal_elim (symmetric cthp) TrueI
   867         val rth = implies_elim
   868                  (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
   869       in rth end
   870     | _ => reflexive ct);
   871 
   872 local
   873   val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
   874   val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
   875   val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
   876 in
   877 fun field_isolate_conv phi ctxt vs ct = case term_of ct of
   878   Const(@{const_name HOL.less},_)$a$b =>
   879    let val (ca,cb) = Thm.dest_binop ct
   880        val T = ctyp_of_term ca
   881        val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
   882        val nth = Conv.fconv_rule
   883          (Conv.arg_conv (Conv.arg1_conv
   884               (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
   885        val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   886    in rth end
   887 | Const(@{const_name HOL.less_eq},_)$a$b =>
   888    let val (ca,cb) = Thm.dest_binop ct
   889        val T = ctyp_of_term ca
   890        val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
   891        val nth = Conv.fconv_rule
   892          (Conv.arg_conv (Conv.arg1_conv
   893               (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
   894        val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   895    in rth end
   896 
   897 | Const("op =",_)$a$b =>
   898    let val (ca,cb) = Thm.dest_binop ct
   899        val T = ctyp_of_term ca
   900        val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
   901        val nth = Conv.fconv_rule
   902          (Conv.arg_conv (Conv.arg1_conv
   903               (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
   904        val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   905    in rth end
   906 | @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
   907 | _ => reflexive ct
   908 end;
   909 
   910 fun classfield_whatis phi =
   911  let
   912   fun h x t =
   913    case term_of t of
   914      Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
   915                             else Ferrante_Rackoff_Data.Nox
   916    | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
   917                             else Ferrante_Rackoff_Data.Nox
   918    | Const(@{const_name HOL.less},_)$y$z =>
   919        if term_of x aconv y then Ferrante_Rackoff_Data.Lt
   920         else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
   921         else Ferrante_Rackoff_Data.Nox
   922    | Const (@{const_name HOL.less_eq},_)$y$z =>
   923          if term_of x aconv y then Ferrante_Rackoff_Data.Le
   924          else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
   925          else Ferrante_Rackoff_Data.Nox
   926    | _ => Ferrante_Rackoff_Data.Nox
   927  in h end;
   928 fun class_field_ss phi =
   929    HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
   930    addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
   931 
   932 in
   933 Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
   934   {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
   935 end
   936 *}
   937 
   938 end