src/HOL/Groebner_Basis.thy
 author chaieb Mon Jun 11 16:23:17 2007 +0200 (2007-06-11) changeset 23327 1654013ec97c parent 23312 6e32a5bfc30f child 23330 01c09922ce59 permissions -rw-r--r--
Added instantiation of algebra method to fields
```     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 NatBin
```
```    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   includes meta_term_syntax
```
```    68   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    69     and "TERM r0" and "TERM r1"
```
```    70   by rule+
```
```    71
```
```    72 lemma semiring_rules:
```
```    73   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    74   "add (mul a m) m = mul (add a r1) m"
```
```    75   "add m (mul a m) = mul (add a r1) m"
```
```    76   "add m m = mul (add r1 r1) m"
```
```    77   "add r0 a = a"
```
```    78   "add a r0 = a"
```
```    79   "mul a b = mul b a"
```
```    80   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    81   "mul r0 a = r0"
```
```    82   "mul a r0 = r0"
```
```    83   "mul r1 a = a"
```
```    84   "mul a r1 = a"
```
```    85   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    86   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    87   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    88   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    89   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    90   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    91   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    92   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    93   "add (add a b) c = add a (add b c)"
```
```    94   "add a (add c d) = add c (add a d)"
```
```    95   "add (add a b) c = add (add a c) b"
```
```    96   "add a c = add c a"
```
```    97   "add a (add c d) = add (add a c) d"
```
```    98   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    99   "mul x (pwr x q) = pwr x (Suc q)"
```
```   100   "mul (pwr x q) x = pwr x (Suc q)"
```
```   101   "mul x x = pwr x 2"
```
```   102   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```   103   "pwr (pwr x p) q = pwr x (p * q)"
```
```   104   "pwr x 0 = r1"
```
```   105   "pwr x 1 = x"
```
```   106   "mul x (add y z) = add (mul x y) (mul x z)"
```
```   107   "pwr x (Suc q) = mul x (pwr x q)"
```
```   108   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   109   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   110 proof -
```
```   111   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   112 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   113 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   114 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   115 next show "add r0 a = a" using add_0 by simp
```
```   116 next show "add a r0 = a" using add_0 add_c by simp
```
```   117 next show "mul a b = mul b a" using mul_c by simp
```
```   118 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   119 next show "mul r0 a = r0" using mul_0 by simp
```
```   120 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   121 next show "mul r1 a = a" using mul_1 by simp
```
```   122 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   123 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   124     using mul_c mul_a by simp
```
```   125 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   126     using mul_a by simp
```
```   127 next
```
```   128   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   129   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   130   finally
```
```   131   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   132     using mul_c by simp
```
```   133 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   134 next
```
```   135   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   136 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   137 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   138 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   139     using add_c add_a by simp
```
```   140 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   141 next show "add a (add c d) = add c (add a d)"
```
```   142     apply (simp add: add_a) by (simp only: add_c)
```
```   143 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   144 next show "add a c = add c a" by (rule add_c)
```
```   145 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   146 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   147 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   148 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   149 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   150 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   151 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   152 next show "pwr x 0 = r1" using pwr_0 .
```
```   153 next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   154 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   155 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   156 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
```
```   157 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   158     by (simp add: nat_number pwr_Suc mul_pwr)
```
```   159 qed
```
```   160
```
```   161
```
```   162 lemma "axioms" [normalizer
```
```   163     semiring ops: semiring_ops
```
```   164     semiring rules: semiring_rules]:
```
```   165   "gb_semiring add mul pwr r0 r1" .
```
```   166
```
```   167 end
```
```   168
```
```   169 interpretation class_semiring: gb_semiring
```
```   170     ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
```
```   171   by unfold_locales (auto simp add: ring_eq_simps power_Suc)
```
```   172
```
```   173 lemmas nat_arith =
```
```   174   add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
```
```   175
```
```   176 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
```
```   177   by (simp add: numeral_1_eq_1)
```
```   178 lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
```
```   179   if_True add_0 add_Suc add_number_of_left mult_number_of_left
```
```   180   numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
```
```   181   numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
```
```   182   iszero_number_of_1 iszero_number_of_0 nonzero_number_of_Min
```
```   183   iszero_number_of_Pls iszero_0 not_iszero_Numeral1
```
```   184
```
```   185 lemmas semiring_norm = comp_arith
```
```   186
```
```   187 ML {*
```
```   188   fun numeral_is_const ct =
```
```   189     can HOLogic.dest_number (Thm.term_of ct);
```
```   190
```
```   191   val numeral_conv =
```
```   192     Conv.then_conv (Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}),
```
```   193    Simplifier.rewrite (HOL_basic_ss addsimps
```
```   194   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}));
```
```   195 *}
```
```   196
```
```   197 ML {*
```
```   198   fun int_of_rat x =
```
```   199     (case Rat.quotient_of_rat x of (i, 1) => i
```
```   200     | _ => error "int_of_rat: bad int")
```
```   201 *}
```
```   202
```
```   203 declaration {*
```
```   204   NormalizerData.funs @{thm class_semiring.axioms}
```
```   205    {is_const = fn phi => numeral_is_const,
```
```   206     dest_const = fn phi => fn ct =>
```
```   207       Rat.rat_of_int (snd
```
```   208         (HOLogic.dest_number (Thm.term_of ct)
```
```   209           handle TERM _ => error "ring_dest_const")),
```
```   210     mk_const = fn phi => fn cT => fn x =>
```
```   211       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   212     conv = fn phi => numeral_conv}
```
```   213 *}
```
```   214
```
```   215
```
```   216 locale gb_ring = gb_semiring +
```
```   217   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   218     and neg :: "'a \<Rightarrow> 'a"
```
```   219   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   220     and sub_add: "sub x y = add x (neg y)"
```
```   221 begin
```
```   222
```
```   223 lemma ring_ops:
```
```   224   includes meta_term_syntax
```
```   225   shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   226
```
```   227 lemmas ring_rules = neg_mul sub_add
```
```   228
```
```   229 lemma "axioms" [normalizer
```
```   230   semiring ops: semiring_ops
```
```   231   semiring rules: semiring_rules
```
```   232   ring ops: ring_ops
```
```   233   ring rules: ring_rules]:
```
```   234   "gb_ring add mul pwr r0 r1 sub neg" .
```
```   235
```
```   236 end
```
```   237
```
```   238
```
```   239 interpretation class_ring: gb_ring ["op +" "op *" "op ^"
```
```   240     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
```
```   241   by unfold_locales simp_all
```
```   242
```
```   243
```
```   244 declaration {*
```
```   245   NormalizerData.funs @{thm class_ring.axioms}
```
```   246    {is_const = fn phi => numeral_is_const,
```
```   247     dest_const = fn phi => fn ct =>
```
```   248       Rat.rat_of_int (snd
```
```   249         (HOLogic.dest_number (Thm.term_of ct)
```
```   250           handle TERM _ => error "ring_dest_const")),
```
```   251     mk_const = fn phi => fn cT => fn x =>
```
```   252       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   253     conv = fn phi => numeral_conv}
```
```   254 *}
```
```   255
```
```   256 use "Tools/Groebner_Basis/normalizer.ML"
```
```   257
```
```   258 method_setup sring_norm = {*
```
```   259   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
```
```   260 *} "Semiring_normalizer"
```
```   261
```
```   262
```
```   263 locale gb_field = gb_ring +
```
```   264   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   265     and inverse:: "'a \<Rightarrow> 'a"
```
```   266   assumes divide: "divide x y = mul x (inverse y)"
```
```   267      and inverse: "inverse x = divide r1 x"
```
```   268 begin
```
```   269
```
```   270 lemma "axioms" [normalizer
```
```   271   semiring ops: semiring_ops
```
```   272   semiring rules: semiring_rules
```
```   273   ring ops: ring_ops
```
```   274   ring rules: ring_rules]:
```
```   275   "gb_field add mul pwr r0 r1 sub neg divide inverse" .
```
```   276
```
```   277 end
```
```   278
```
```   279 subsection {* Groebner Bases *}
```
```   280
```
```   281 locale semiringb = gb_semiring +
```
```   282   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   283   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   284     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   285 begin
```
```   286
```
```   287 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)"
```
```   288 proof-
```
```   289   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   290   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   291     using add_mul_solve by blast
```
```   292   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)"
```
```   293     by simp
```
```   294 qed
```
```   295
```
```   296 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   297   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   298 proof(clarify)
```
```   299   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   300     and eq: "add b (mul r c) = add b (mul r d)"
```
```   301   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   302   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   303     using mul_0 add_cancel by simp
```
```   304   thus "False" using add_mul_solve nz cnd by simp
```
```   305 qed
```
```   306
```
```   307 declare "axioms" [normalizer del]
```
```   308
```
```   309 lemma "axioms" [normalizer
```
```   310   semiring ops: semiring_ops
```
```   311   semiring rules: semiring_rules
```
```   312   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   313   "semiringb add mul pwr r0 r1" .
```
```   314
```
```   315 end
```
```   316
```
```   317 locale ringb = semiringb + gb_ring
```
```   318 begin
```
```   319
```
```   320 declare "axioms" [normalizer del]
```
```   321
```
```   322 lemma "axioms" [normalizer
```
```   323   semiring ops: semiring_ops
```
```   324   semiring rules: semiring_rules
```
```   325   ring ops: ring_ops
```
```   326   ring rules: ring_rules
```
```   327   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   328   "ringb add mul pwr r0 r1 sub neg" .
```
```   329
```
```   330 end
```
```   331
```
```   332 lemma no_zero_divirors_neq0:
```
```   333   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
```
```   334     and ab: "a*b = 0" shows "b = 0"
```
```   335 proof -
```
```   336   { assume bz: "b \<noteq> 0"
```
```   337     from no_zero_divisors [OF az bz] ab have False by blast }
```
```   338   thus "b = 0" by blast
```
```   339 qed
```
```   340
```
```   341 interpretation class_ringb: ringb
```
```   342   ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
```
```   343 proof(unfold_locales, simp add: ring_eq_simps power_Suc, auto)
```
```   344   fix w x y z ::"'a::{idom,recpower,number_ring}"
```
```   345   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   346   hence ynz': "y - z \<noteq> 0" by simp
```
```   347   from p have "w * y + x* z - w*z - x*y = 0" by simp
```
```   348   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_eq_simps)
```
```   349   hence "(y - z) * (w - x) = 0" by (simp add: ring_eq_simps)
```
```   350   with  no_zero_divirors_neq0 [OF ynz']
```
```   351   have "w - x = 0" by blast
```
```   352   thus "w = x"  by simp
```
```   353 qed
```
```   354
```
```   355
```
```   356 declaration {*
```
```   357   NormalizerData.funs @{thm class_ringb.axioms}
```
```   358    {is_const = fn phi => numeral_is_const,
```
```   359     dest_const = fn phi => fn ct =>
```
```   360       Rat.rat_of_int (snd
```
```   361         (HOLogic.dest_number (Thm.term_of ct)
```
```   362           handle TERM _ => error "ring_dest_const")),
```
```   363     mk_const = fn phi => fn cT => fn x =>
```
```   364       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   365     conv = fn phi => numeral_conv}
```
```   366 *}
```
```   367
```
```   368
```
```   369 interpretation natgb: semiringb
```
```   370   ["op +" "op *" "op ^" "0::nat" "1"]
```
```   371 proof (unfold_locales, simp add: ring_eq_simps power_Suc)
```
```   372   fix w x y z ::"nat"
```
```   373   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   374     hence "y < z \<or> y > z" by arith
```
```   375     moreover {
```
```   376       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   377       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   378       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_eq_simps)
```
```   379       hence "x*k = w*k" by simp
```
```   380       hence "w = x" using kp by (simp add: mult_cancel2) }
```
```   381     moreover {
```
```   382       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   383       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   384       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_eq_simps)
```
```   385       hence "w*k = x*k" by simp
```
```   386       hence "w = x" using kp by (simp add: mult_cancel2)}
```
```   387     ultimately have "w=x" by blast }
```
```   388   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   389 qed
```
```   390
```
```   391 declaration {*
```
```   392   NormalizerData.funs @{thm natgb.axioms}
```
```   393    {is_const = fn phi => numeral_is_const,
```
```   394     dest_const = fn phi => fn ct =>
```
```   395       Rat.rat_of_int (snd
```
```   396         (HOLogic.dest_number (Thm.term_of ct)
```
```   397           handle TERM _ => error "ring_dest_const")),
```
```   398     mk_const = fn phi => fn cT => fn x =>
```
```   399       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   400     conv = fn phi => numeral_conv}
```
```   401 *}
```
```   402
```
```   403 locale fieldgb = ringb + gb_field
```
```   404 begin
```
```   405
```
```   406 declare "axioms" [normalizer del]
```
```   407
```
```   408 lemma "axioms" [normalizer
```
```   409   semiring ops: semiring_ops
```
```   410   semiring rules: semiring_rules
```
```   411   ring ops: ring_ops
```
```   412   ring rules: ring_rules
```
```   413   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   414   "fieldgb add mul pwr r0 r1 sub neg divide inverse" by unfold_locales
```
```   415 end
```
```   416
```
```   417
```
```   418
```
```   419 lemmas bool_simps = simp_thms(1-34)
```
```   420 lemma dnf:
```
```   421     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   422     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
```
```   423   by blast+
```
```   424
```
```   425 lemmas weak_dnf_simps = dnf bool_simps
```
```   426
```
```   427 lemma nnf_simps:
```
```   428     "(\<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)"
```
```   429     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   430   by blast+
```
```   431
```
```   432 lemma PFalse:
```
```   433     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```   434     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```   435   by auto
```
```   436
```
```   437 use "Tools/Groebner_Basis/groebner.ML"
```
```   438
```
```   439 ML {*
```
```   440   fun algebra_tac ctxt i = ObjectLogic.full_atomize_tac i THEN (fn st =>
```
```   441   rtac (Groebner.ring_conv ctxt (Thm.dest_arg (nth (cprems_of st) (i - 1)))) i st);
```
```   442 *}
```
```   443
```
```   444 method_setup algebra = {*
```
```   445   Method.ctxt_args (Method.SIMPLE_METHOD' o algebra_tac)
```
```   446 *} ""
```
```   447
```
```   448
```
```   449
```
```   450 end
```