src/HOL/Groebner_Basis.thy
author huffman
Thu Dec 04 16:05:45 2008 -0800 (2008-12-04)
changeset 28987 dc0ab579a5ca
parent 28986 1ff53ff7041d
child 29039 8b9207f82a78
child 29223 e09c53289830
permissions -rw-r--r--
remove duplicated lemmas
     1 (*  Title:      HOL/Groebner_Basis.thy
     2     ID:         $Id$
     3     Author:     Amine Chaieb, TU Muenchen
     4 *)
     5 
     6 header {* Semiring normalization and Groebner Bases *}
     7 
     8 theory Groebner_Basis
     9 imports Arith_Tools
    10 uses
    11   "Tools/Groebner_Basis/misc.ML"
    12   "Tools/Groebner_Basis/normalizer_data.ML"
    13   ("Tools/Groebner_Basis/normalizer.ML")
    14   ("Tools/Groebner_Basis/groebner.ML")
    15 begin
    16 
    17 subsection {* Semiring normalization *}
    18 
    19 setup NormalizerData.setup
    20 
    21 
    22 locale gb_semiring =
    23   fixes add mul pwr r0 r1
    24   assumes add_a:"(add x (add y z) = add (add x y) z)"
    25     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
    26     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
    27     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
    28     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
    29     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
    30 begin
    31 
    32 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
    33 proof (induct p)
    34   case 0
    35   then show ?case by (auto simp add: pwr_0 mul_1)
    36 next
    37   case Suc
    38   from this [symmetric] show ?case
    39     by (auto simp add: pwr_Suc mul_1 mul_a)
    40 qed
    41 
    42 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    43 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
    44   fix q x y
    45   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    46   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
    47     by (simp add: mul_a)
    48   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
    49   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
    50   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
    51     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
    52 qed
    53 
    54 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
    55 proof (induct p arbitrary: q)
    56   case 0
    57   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
    58 next
    59   case Suc
    60   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
    61 qed
    62 
    63 
    64 subsubsection {* Declaring the abstract theory *}
    65 
    66 lemma semiring_ops:
    67   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
    68     and "TERM r0" and "TERM r1" .
    69 
    70 lemma semiring_rules:
    71   "add (mul a m) (mul b m) = mul (add a b) m"
    72   "add (mul a m) m = mul (add a r1) m"
    73   "add m (mul a m) = mul (add a r1) m"
    74   "add m m = mul (add r1 r1) m"
    75   "add r0 a = a"
    76   "add a r0 = a"
    77   "mul a b = mul b a"
    78   "mul (add a b) c = add (mul a c) (mul b c)"
    79   "mul r0 a = r0"
    80   "mul a r0 = r0"
    81   "mul r1 a = a"
    82   "mul a r1 = a"
    83   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
    84   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
    85   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
    86   "mul (mul lx ly) rx = mul (mul lx rx) ly"
    87   "mul (mul lx ly) rx = mul lx (mul ly rx)"
    88   "mul lx (mul rx ry) = mul (mul lx rx) ry"
    89   "mul lx (mul rx ry) = mul rx (mul lx ry)"
    90   "add (add a b) (add c d) = add (add a c) (add b d)"
    91   "add (add a b) c = add a (add b c)"
    92   "add a (add c d) = add c (add a d)"
    93   "add (add a b) c = add (add a c) b"
    94   "add a c = add c a"
    95   "add a (add c d) = add (add a c) d"
    96   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
    97   "mul x (pwr x q) = pwr x (Suc q)"
    98   "mul (pwr x q) x = pwr x (Suc q)"
    99   "mul x x = pwr x 2"
   100   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
   101   "pwr (pwr x p) q = pwr x (p * q)"
   102   "pwr x 0 = r1"
   103   "pwr x 1 = x"
   104   "mul x (add y z) = add (mul x y) (mul x z)"
   105   "pwr x (Suc q) = mul x (pwr x q)"
   106   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
   107   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
   108 proof -
   109   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
   110 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
   111 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
   112 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
   113 next show "add r0 a = a" using add_0 by simp
   114 next show "add a r0 = a" using add_0 add_c by simp
   115 next show "mul a b = mul b a" using mul_c by simp
   116 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
   117 next show "mul r0 a = r0" using mul_0 by simp
   118 next show "mul a r0 = r0" using mul_0 mul_c by simp
   119 next show "mul r1 a = a" using mul_1 by simp
   120 next show "mul a r1 = a" using mul_1 mul_c by simp
   121 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
   122     using mul_c mul_a by simp
   123 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
   124     using mul_a by simp
   125 next
   126   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
   127   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
   128   finally
   129   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
   130     using mul_c by simp
   131 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
   132 next
   133   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
   134 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
   135 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
   136 next show "add (add a b) (add c d) = add (add a c) (add b d)"
   137     using add_c add_a by simp
   138 next show "add (add a b) c = add a (add b c)" using add_a by simp
   139 next show "add a (add c d) = add c (add a d)"
   140     apply (simp add: add_a) by (simp only: add_c)
   141 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
   142 next show "add a c = add c a" by (rule add_c)
   143 next show "add a (add c d) = add (add a c) d" using add_a by simp
   144 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
   145 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
   146 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
   147 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
   148 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
   149 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
   150 next show "pwr x 0 = r1" using pwr_0 .
   151 next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
   152 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
   153 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
   154 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
   155 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
   156     by (simp add: nat_number pwr_Suc mul_pwr)
   157 qed
   158 
   159 
   160 lemmas gb_semiring_axioms' =
   161   gb_semiring_axioms [normalizer
   162     semiring ops: semiring_ops
   163     semiring rules: semiring_rules]
   164 
   165 end
   166 
   167 interpretation class_semiring: gb_semiring
   168     ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
   169   proof qed (auto simp add: ring_simps power_Suc)
   170 
   171 lemmas nat_arith =
   172   add_nat_number_of
   173   diff_nat_number_of
   174   mult_nat_number_of
   175   eq_nat_number_of
   176   less_nat_number_of
   177 
   178 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
   179   by (simp add: numeral_1_eq_1)
   180 
   181 lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
   182   if_True add_0 add_Suc add_number_of_left mult_number_of_left
   183   numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
   184   numeral_0_eq_0[symmetric] numerals[symmetric]
   185   iszero_simps not_iszero_Numeral1
   186 
   187 lemmas semiring_norm = comp_arith
   188 
   189 ML {*
   190 local
   191 
   192 open Conv;
   193 
   194 fun numeral_is_const ct =
   195   can HOLogic.dest_number (Thm.term_of ct);
   196 
   197 fun int_of_rat x =
   198   (case Rat.quotient_of_rat x of (i, 1) => i
   199   | _ => error "int_of_rat: bad int");
   200 
   201 val numeral_conv =
   202   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
   203   Simplifier.rewrite (HOL_basic_ss addsimps
   204     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
   205 
   206 in
   207 
   208 fun normalizer_funs key =
   209   NormalizerData.funs key
   210    {is_const = fn phi => numeral_is_const,
   211     dest_const = fn phi => fn ct =>
   212       Rat.rat_of_int (snd
   213         (HOLogic.dest_number (Thm.term_of ct)
   214           handle TERM _ => error "ring_dest_const")),
   215     mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
   216     conv = fn phi => K numeral_conv}
   217 
   218 end
   219 *}
   220 
   221 declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
   222 
   223 
   224 locale gb_ring = gb_semiring +
   225   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   226     and neg :: "'a \<Rightarrow> 'a"
   227   assumes neg_mul: "neg x = mul (neg r1) x"
   228     and sub_add: "sub x y = add x (neg y)"
   229 begin
   230 
   231 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
   232 
   233 lemmas ring_rules = neg_mul sub_add
   234 
   235 lemmas gb_ring_axioms' =
   236   gb_ring_axioms [normalizer
   237     semiring ops: semiring_ops
   238     semiring rules: semiring_rules
   239     ring ops: ring_ops
   240     ring rules: ring_rules]
   241 
   242 end
   243 
   244 
   245 interpretation class_ring: gb_ring ["op +" "op *" "op ^"
   246     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
   247   proof qed simp_all
   248 
   249 
   250 declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
   251 
   252 use "Tools/Groebner_Basis/normalizer.ML"
   253 
   254 
   255 method_setup sring_norm = {*
   256   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
   257 *} "semiring normalizer"
   258 
   259 
   260 locale gb_field = gb_ring +
   261   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   262     and inverse:: "'a \<Rightarrow> 'a"
   263   assumes divide: "divide x y = mul x (inverse y)"
   264      and inverse: "inverse x = divide r1 x"
   265 begin
   266 
   267 lemmas gb_field_axioms' =
   268   gb_field_axioms [normalizer
   269     semiring ops: semiring_ops
   270     semiring rules: semiring_rules
   271     ring ops: ring_ops
   272     ring rules: ring_rules]
   273 
   274 end
   275 
   276 
   277 subsection {* Groebner Bases *}
   278 
   279 locale semiringb = gb_semiring +
   280   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
   281   and add_mul_solve: "add (mul w y) (mul x z) =
   282     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
   283 begin
   284 
   285 lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   286 proof-
   287   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
   288   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   289     using add_mul_solve by blast
   290   finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   291     by simp
   292 qed
   293 
   294 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
   295   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
   296 proof(clarify)
   297   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
   298     and eq: "add b (mul r c) = add b (mul r d)"
   299   hence "mul r c = mul r d" using cnd add_cancel by simp
   300   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
   301     using mul_0 add_cancel by simp
   302   thus "False" using add_mul_solve nz cnd by simp
   303 qed
   304 
   305 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
   306 proof-
   307   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
   308   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
   309 qed
   310 
   311 declare gb_semiring_axioms' [normalizer del]
   312 
   313 lemmas semiringb_axioms' = semiringb_axioms [normalizer
   314   semiring ops: semiring_ops
   315   semiring rules: semiring_rules
   316   idom rules: noteq_reduce add_scale_eq_noteq]
   317 
   318 end
   319 
   320 locale ringb = semiringb + gb_ring + 
   321   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
   322 begin
   323 
   324 declare gb_ring_axioms' [normalizer del]
   325 
   326 lemmas ringb_axioms' = ringb_axioms [normalizer
   327   semiring ops: semiring_ops
   328   semiring rules: semiring_rules
   329   ring ops: ring_ops
   330   ring rules: ring_rules
   331   idom rules: noteq_reduce add_scale_eq_noteq
   332   ideal rules: subr0_iff add_r0_iff]
   333 
   334 end
   335 
   336 
   337 lemma no_zero_divirors_neq0:
   338   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
   339     and ab: "a*b = 0" shows "b = 0"
   340 proof -
   341   { assume bz: "b \<noteq> 0"
   342     from no_zero_divisors [OF az bz] ab have False by blast }
   343   thus "b = 0" by blast
   344 qed
   345 
   346 interpretation class_ringb: ringb
   347   ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
   348 proof(unfold_locales, simp add: ring_simps power_Suc, auto)
   349   fix w x y z ::"'a::{idom,recpower,number_ring}"
   350   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   351   hence ynz': "y - z \<noteq> 0" by simp
   352   from p have "w * y + x* z - w*z - x*y = 0" by simp
   353   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
   354   hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
   355   with  no_zero_divirors_neq0 [OF ynz']
   356   have "w - x = 0" by blast
   357   thus "w = x"  by simp
   358 qed
   359 
   360 declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
   361 
   362 interpretation natgb: semiringb
   363   ["op +" "op *" "op ^" "0::nat" "1"]
   364 proof (unfold_locales, simp add: ring_simps power_Suc)
   365   fix w x y z ::"nat"
   366   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   367     hence "y < z \<or> y > z" by arith
   368     moreover {
   369       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
   370       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
   371       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
   372       hence "x*k = w*k" by simp
   373       hence "w = x" using kp by (simp add: mult_cancel2) }
   374     moreover {
   375       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
   376       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
   377       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
   378       hence "w*k = x*k" by simp
   379       hence "w = x" using kp by (simp add: mult_cancel2)}
   380     ultimately have "w=x" by blast }
   381   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
   382 qed
   383 
   384 declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
   385 
   386 locale fieldgb = ringb + gb_field
   387 begin
   388 
   389 declare gb_field_axioms' [normalizer del]
   390 
   391 lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
   392   semiring ops: semiring_ops
   393   semiring rules: semiring_rules
   394   ring ops: ring_ops
   395   ring rules: ring_rules
   396   idom rules: noteq_reduce add_scale_eq_noteq
   397   ideal rules: subr0_iff add_r0_iff]
   398 
   399 end
   400 
   401 
   402 lemmas bool_simps = simp_thms(1-34)
   403 lemma dnf:
   404     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
   405     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
   406   by blast+
   407 
   408 lemmas weak_dnf_simps = dnf bool_simps
   409 
   410 lemma nnf_simps:
   411     "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
   412     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   413   by blast+
   414 
   415 lemma PFalse:
   416     "P \<equiv> False \<Longrightarrow> \<not> P"
   417     "\<not> P \<Longrightarrow> (P \<equiv> False)"
   418   by auto
   419 use "Tools/Groebner_Basis/groebner.ML"
   420 
   421 method_setup algebra =
   422 {*
   423 let
   424  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
   425  val addN = "add"
   426  val delN = "del"
   427  val any_keyword = keyword addN || keyword delN
   428  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
   429 in
   430 fn src => Method.syntax 
   431     ((Scan.optional (keyword addN |-- thms) []) -- 
   432     (Scan.optional (keyword delN |-- thms) [])) src 
   433  #> (fn ((add_ths, del_ths), ctxt) => 
   434        Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
   435 end
   436 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
   437 declare dvd_def[algebra]
   438 declare dvd_eq_mod_eq_0[symmetric, algebra]
   439 declare nat_mod_div_trivial[algebra]
   440 declare nat_mod_mod_trivial[algebra]
   441 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
   442 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
   443 declare zmod_zdiv_equality[symmetric,algebra]
   444 declare zdiv_zmod_equality[symmetric, algebra]
   445 declare zdiv_zminus_zminus[algebra]
   446 declare zmod_zminus_zminus[algebra]
   447 declare zdiv_zminus2[algebra]
   448 declare zmod_zminus2[algebra]
   449 declare zdiv_zero[algebra]
   450 declare zmod_zero[algebra]
   451 declare zmod_1[algebra]
   452 declare zdiv_1[algebra]
   453 declare zmod_minus1_right[algebra]
   454 declare zdiv_minus1_right[algebra]
   455 declare mod_div_trivial[algebra]
   456 declare mod_mod_trivial[algebra]
   457 declare zmod_zmult_self1[algebra]
   458 declare zmod_zmult_self2[algebra]
   459 declare zmod_eq_0_iff[algebra]
   460 declare zdvd_0_left[algebra]
   461 declare zdvd1_eq[algebra]
   462 declare zmod_eq_dvd_iff[algebra]
   463 declare nat_mod_eq_iff[algebra]
   464 
   465 subsection{* Groebner Bases for fields *}
   466 
   467 interpretation class_fieldgb:
   468   fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
   469 
   470 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
   471 lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
   472   by simp
   473 lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
   474   by simp
   475 lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   476   by simp
   477 lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   478   by simp
   479 
   480 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
   481 
   482 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
   483   by (simp add: add_divide_distrib)
   484 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
   485   by (simp add: add_divide_distrib)
   486 
   487 
   488 ML{* 
   489 local
   490  val zr = @{cpat "0"}
   491  val zT = ctyp_of_term zr
   492  val geq = @{cpat "op ="}
   493  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
   494  val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
   495  val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
   496  val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
   497 
   498  fun prove_nz ss T t =
   499     let
   500       val z = instantiate_cterm ([(zT,T)],[]) zr
   501       val eq = instantiate_cterm ([(eqT,T)],[]) geq
   502       val th = Simplifier.rewrite (ss addsimps simp_thms)
   503            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
   504                   (Thm.capply (Thm.capply eq t) z)))
   505     in equal_elim (symmetric th) TrueI
   506     end
   507 
   508  fun proc phi ss ct =
   509   let
   510     val ((x,y),(w,z)) =
   511          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
   512     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
   513     val T = ctyp_of_term x
   514     val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
   515     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
   516   in SOME (implies_elim (implies_elim th y_nz) z_nz)
   517   end
   518   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
   519 
   520  fun proc2 phi ss ct =
   521   let
   522     val (l,r) = Thm.dest_binop ct
   523     val T = ctyp_of_term l
   524   in (case (term_of l, term_of r) of
   525       (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
   526         let val (x,y) = Thm.dest_binop l val z = r
   527             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
   528             val ynz = prove_nz ss T y
   529         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
   530         end
   531      | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
   532         let val (x,y) = Thm.dest_binop r val z = l
   533             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
   534             val ynz = prove_nz ss T y
   535         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
   536         end
   537      | _ => NONE)
   538   end
   539   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
   540 
   541  fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
   542    | is_number t = can HOLogic.dest_number t
   543 
   544  val is_number = is_number o term_of
   545 
   546  fun proc3 phi ss ct =
   547   (case term_of ct of
   548     Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   549       let
   550         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   551         val _ = map is_number [a,b,c]
   552         val T = ctyp_of_term c
   553         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
   554       in SOME (mk_meta_eq th) end
   555   | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   556       let
   557         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   558         val _ = map is_number [a,b,c]
   559         val T = ctyp_of_term c
   560         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
   561       in SOME (mk_meta_eq th) end
   562   | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
   563       let
   564         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   565         val _ = map is_number [a,b,c]
   566         val T = ctyp_of_term c
   567         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
   568       in SOME (mk_meta_eq th) end
   569   | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   570     let
   571       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   572         val _ = map is_number [a,b,c]
   573         val T = ctyp_of_term c
   574         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
   575       in SOME (mk_meta_eq th) end
   576   | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   577     let
   578       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   579         val _ = map is_number [a,b,c]
   580         val T = ctyp_of_term c
   581         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
   582       in SOME (mk_meta_eq th) end
   583   | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
   584     let
   585       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   586         val _ = map is_number [a,b,c]
   587         val T = ctyp_of_term c
   588         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
   589       in SOME (mk_meta_eq th) end
   590   | _ => NONE)
   591   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
   592 
   593 val add_frac_frac_simproc =
   594        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
   595                      name = "add_frac_frac_simproc",
   596                      proc = proc, identifier = []}
   597 
   598 val add_frac_num_simproc =
   599        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
   600                      name = "add_frac_num_simproc",
   601                      proc = proc2, identifier = []}
   602 
   603 val ord_frac_simproc =
   604   make_simproc
   605     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
   606              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
   607              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
   608              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
   609              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
   610              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
   611              name = "ord_frac_simproc", proc = proc3, identifier = []}
   612 
   613 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
   614            @{thm "divide_Numeral1"},
   615            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
   616            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
   617            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
   618            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
   619            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
   620            @{thm "diff_def"}, @{thm "minus_divide_left"},
   621            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
   622 
   623 local
   624 open Conv
   625 in
   626 val comp_conv = (Simplifier.rewrite
   627 (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
   628               addsimps ths addsimps simp_thms
   629               addsimprocs field_cancel_numeral_factors
   630                addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
   631                             ord_frac_simproc]
   632                 addcongs [@{thm "if_weak_cong"}]))
   633 then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
   634   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
   635 end
   636 
   637 fun numeral_is_const ct =
   638   case term_of ct of
   639    Const (@{const_name "HOL.divide"},_) $ a $ b =>
   640      numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
   641  | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
   642  | t => can HOLogic.dest_number t
   643 
   644 fun dest_const ct = ((case term_of ct of
   645    Const (@{const_name "HOL.divide"},_) $ a $ b=>
   646     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
   647  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
   648    handle TERM _ => error "ring_dest_const")
   649 
   650 fun mk_const phi cT x =
   651  let val (a, b) = Rat.quotient_of_rat x
   652  in if b = 1 then Numeral.mk_cnumber cT a
   653     else Thm.capply
   654          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
   655                      (Numeral.mk_cnumber cT a))
   656          (Numeral.mk_cnumber cT b)
   657   end
   658 
   659 in
   660  val field_comp_conv = comp_conv;
   661  val fieldgb_declaration = 
   662   NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
   663    {is_const = K numeral_is_const,
   664     dest_const = K dest_const,
   665     mk_const = mk_const,
   666     conv = K (K comp_conv)}
   667 end;
   668 *}
   669 
   670 declaration fieldgb_declaration
   671 
   672 end