src/HOL/Arith_Tools.thy
 author wenzelm Thu Oct 04 20:29:42 2007 +0200 (2007-10-04) changeset 24850 0cfd722ab579 parent 24286 7619080e49f0 child 25128 962e4f4142fa permissions -rw-r--r--
Name.uu, Name.aT;
```     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
```