src/HOL/Groebner_Basis.thy
 author wenzelm Thu Jun 14 18:33:31 2007 +0200 (2007-06-14) changeset 23389 aaca6a8e5414 parent 23332 b91295432e6d child 23458 b2267a9e9e28 permissions -rw-r--r--
tuned proofs: avoid implicit prems;
```     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 theory Groebner_Basis
```
```     8 imports NatBin
```
```     9 uses
```
```    10   "Tools/Groebner_Basis/misc.ML"
```
```    11   "Tools/Groebner_Basis/normalizer_data.ML"
```
```    12   ("Tools/Groebner_Basis/normalizer.ML")
```
```    13   ("Tools/Groebner_Basis/groebner.ML")
```
```    14 begin
```
```    15
```
```    16
```
```    17
```
```    18 subsection {* Semiring normalization *}
```
```    19
```
```    20 setup NormalizerData.setup
```
```    21
```
```    22
```
```    23 locale gb_semiring =
```
```    24   fixes add mul pwr r0 r1
```
```    25   assumes add_a:"(add x (add y z) = add (add x y) z)"
```
```    26     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
```
```    27     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
```
```    28     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
```
```    29     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
```
```    30     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
```
```    31 begin
```
```    32
```
```    33 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    34 proof (induct p)
```
```    35   case 0
```
```    36   then show ?case by (auto simp add: pwr_0 mul_1)
```
```    37 next
```
```    38   case Suc
```
```    39   from this [symmetric] show ?case
```
```    40     by (auto simp add: pwr_Suc mul_1 mul_a)
```
```    41 qed
```
```    42
```
```    43 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    44 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
```
```    45   fix q x y
```
```    46   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    47   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
```
```    48     by (simp add: mul_a)
```
```    49   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
```
```    50   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
```
```    51   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
```
```    52     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
```
```    53 qed
```
```    54
```
```    55 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
```
```    56 proof (induct p arbitrary: q)
```
```    57   case 0
```
```    58   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
```
```    59 next
```
```    60   case Suc
```
```    61   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
```
```    62 qed
```
```    63
```
```    64
```
```    65 subsubsection {* Declaring the abstract theory *}
```
```    66
```
```    67 lemma semiring_ops:
```
```    68   includes meta_term_syntax
```
```    69   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    70     and "TERM r0" and "TERM r1"
```
```    71   by rule+
```
```    72
```
```    73 lemma semiring_rules:
```
```    74   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    75   "add (mul a m) m = mul (add a r1) m"
```
```    76   "add m (mul a m) = mul (add a r1) m"
```
```    77   "add m m = mul (add r1 r1) m"
```
```    78   "add r0 a = a"
```
```    79   "add a r0 = a"
```
```    80   "mul a b = mul b a"
```
```    81   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    82   "mul r0 a = r0"
```
```    83   "mul a r0 = r0"
```
```    84   "mul r1 a = a"
```
```    85   "mul a r1 = a"
```
```    86   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    87   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    88   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    89   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    90   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    91   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    92   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    93   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    94   "add (add a b) c = add a (add b c)"
```
```    95   "add a (add c d) = add c (add a d)"
```
```    96   "add (add a b) c = add (add a c) b"
```
```    97   "add a c = add c a"
```
```    98   "add a (add c d) = add (add a c) d"
```
```    99   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```   100   "mul x (pwr x q) = pwr x (Suc q)"
```
```   101   "mul (pwr x q) x = pwr x (Suc q)"
```
```   102   "mul x x = pwr x 2"
```
```   103   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```   104   "pwr (pwr x p) q = pwr x (p * q)"
```
```   105   "pwr x 0 = r1"
```
```   106   "pwr x 1 = x"
```
```   107   "mul x (add y z) = add (mul x y) (mul x z)"
```
```   108   "pwr x (Suc q) = mul x (pwr x q)"
```
```   109   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   110   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   111 proof -
```
```   112   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   113 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   114 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   115 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   116 next show "add r0 a = a" using add_0 by simp
```
```   117 next show "add a r0 = a" using add_0 add_c by simp
```
```   118 next show "mul a b = mul b a" using mul_c by simp
```
```   119 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   120 next show "mul r0 a = r0" using mul_0 by simp
```
```   121 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   122 next show "mul r1 a = a" using mul_1 by simp
```
```   123 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   124 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   125     using mul_c mul_a by simp
```
```   126 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   127     using mul_a by simp
```
```   128 next
```
```   129   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   130   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   131   finally
```
```   132   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   133     using mul_c by simp
```
```   134 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   135 next
```
```   136   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   137 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   138 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   139 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   140     using add_c add_a by simp
```
```   141 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   142 next show "add a (add c d) = add c (add a d)"
```
```   143     apply (simp add: add_a) by (simp only: add_c)
```
```   144 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   145 next show "add a c = add c a" by (rule add_c)
```
```   146 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   147 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   148 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   149 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   150 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   151 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   152 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   153 next show "pwr x 0 = r1" using pwr_0 .
```
```   154 next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   155 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   156 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   157 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
```
```   158 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   159     by (simp add: nat_number pwr_Suc mul_pwr)
```
```   160 qed
```
```   161
```
```   162
```
```   163 lemma "axioms" [normalizer
```
```   164     semiring ops: semiring_ops
```
```   165     semiring rules: semiring_rules]:
```
```   166   "gb_semiring add mul pwr r0 r1" by fact
```
```   167
```
```   168 end
```
```   169
```
```   170 interpretation class_semiring: gb_semiring
```
```   171     ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
```
```   172   by unfold_locales (auto simp add: ring_eq_simps power_Suc)
```
```   173
```
```   174 lemmas nat_arith =
```
```   175   add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
```
```   176
```
```   177 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
```
```   178   by (simp add: numeral_1_eq_1)
```
```   179 lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
```
```   180   if_True add_0 add_Suc add_number_of_left mult_number_of_left
```
```   181   numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
```
```   182   numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
```
```   183   iszero_number_of_1 iszero_number_of_0 nonzero_number_of_Min
```
```   184   iszero_number_of_Pls iszero_0 not_iszero_Numeral1
```
```   185
```
```   186 lemmas semiring_norm = comp_arith
```
```   187
```
```   188 ML {*
```
```   189   fun numeral_is_const ct =
```
```   190     can HOLogic.dest_number (Thm.term_of ct);
```
```   191
```
```   192   val numeral_conv =
```
```   193     Conv.then_conv (Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}),
```
```   194    Simplifier.rewrite (HOL_basic_ss addsimps
```
```   195   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}));
```
```   196 *}
```
```   197
```
```   198 ML {*
```
```   199   fun int_of_rat x =
```
```   200     (case Rat.quotient_of_rat x of (i, 1) => i
```
```   201     | _ => error "int_of_rat: bad int")
```
```   202 *}
```
```   203
```
```   204 declaration {*
```
```   205   NormalizerData.funs @{thm class_semiring.axioms}
```
```   206    {is_const = fn phi => numeral_is_const,
```
```   207     dest_const = fn phi => fn ct =>
```
```   208       Rat.rat_of_int (snd
```
```   209         (HOLogic.dest_number (Thm.term_of ct)
```
```   210           handle TERM _ => error "ring_dest_const")),
```
```   211     mk_const = fn phi => fn cT => fn x =>
```
```   212       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   213     conv = fn phi => K numeral_conv}
```
```   214 *}
```
```   215
```
```   216
```
```   217 locale gb_ring = gb_semiring +
```
```   218   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   219     and neg :: "'a \<Rightarrow> 'a"
```
```   220   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   221     and sub_add: "sub x y = add x (neg y)"
```
```   222 begin
```
```   223
```
```   224 lemma ring_ops:
```
```   225   includes meta_term_syntax
```
```   226   shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   227
```
```   228 lemmas ring_rules = neg_mul sub_add
```
```   229
```
```   230 lemma "axioms" [normalizer
```
```   231   semiring ops: semiring_ops
```
```   232   semiring rules: semiring_rules
```
```   233   ring ops: ring_ops
```
```   234   ring rules: ring_rules]:
```
```   235   "gb_ring add mul pwr r0 r1 sub neg" by fact
```
```   236
```
```   237 end
```
```   238
```
```   239
```
```   240 interpretation class_ring: gb_ring ["op +" "op *" "op ^"
```
```   241     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
```
```   242   by unfold_locales simp_all
```
```   243
```
```   244
```
```   245 declaration {*
```
```   246   NormalizerData.funs @{thm class_ring.axioms}
```
```   247    {is_const = fn phi => numeral_is_const,
```
```   248     dest_const = fn phi => fn ct =>
```
```   249       Rat.rat_of_int (snd
```
```   250         (HOLogic.dest_number (Thm.term_of ct)
```
```   251           handle TERM _ => error "ring_dest_const")),
```
```   252     mk_const = fn phi => fn cT => fn x =>
```
```   253       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   254     conv = fn phi => K numeral_conv}
```
```   255 *}
```
```   256
```
```   257 use "Tools/Groebner_Basis/normalizer.ML"
```
```   258
```
```   259 method_setup sring_norm = {*
```
```   260   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
```
```   261 *} "Semiring_normalizer"
```
```   262
```
```   263
```
```   264 locale gb_field = gb_ring +
```
```   265   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   266     and inverse:: "'a \<Rightarrow> 'a"
```
```   267   assumes divide: "divide x y = mul x (inverse y)"
```
```   268      and inverse: "inverse x = divide r1 x"
```
```   269 begin
```
```   270
```
```   271 lemma "axioms" [normalizer
```
```   272   semiring ops: semiring_ops
```
```   273   semiring rules: semiring_rules
```
```   274   ring ops: ring_ops
```
```   275   ring rules: ring_rules]:
```
```   276   "gb_field add mul pwr r0 r1 sub neg divide inverse" by fact
```
```   277
```
```   278 end
```
```   279
```
```   280 subsection {* Groebner Bases *}
```
```   281
```
```   282 locale semiringb = gb_semiring +
```
```   283   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   284   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   285     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   286 begin
```
```   287
```
```   288 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)"
```
```   289 proof-
```
```   290   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   291   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   292     using add_mul_solve by blast
```
```   293   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)"
```
```   294     by simp
```
```   295 qed
```
```   296
```
```   297 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   298   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   299 proof(clarify)
```
```   300   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   301     and eq: "add b (mul r c) = add b (mul r d)"
```
```   302   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   303   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   304     using mul_0 add_cancel by simp
```
```   305   thus "False" using add_mul_solve nz cnd by simp
```
```   306 qed
```
```   307
```
```   308 declare "axioms" [normalizer del]
```
```   309
```
```   310 lemma "axioms" [normalizer
```
```   311   semiring ops: semiring_ops
```
```   312   semiring rules: semiring_rules
```
```   313   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   314   "semiringb add mul pwr r0 r1" by fact
```
```   315
```
```   316 end
```
```   317
```
```   318 locale ringb = semiringb + gb_ring
```
```   319 begin
```
```   320
```
```   321 declare "axioms" [normalizer del]
```
```   322
```
```   323 lemma "axioms" [normalizer
```
```   324   semiring ops: semiring_ops
```
```   325   semiring rules: semiring_rules
```
```   326   ring ops: ring_ops
```
```   327   ring rules: ring_rules
```
```   328   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   329   "ringb add mul pwr r0 r1 sub neg" by fact
```
```   330
```
```   331 end
```
```   332
```
```   333 lemma no_zero_divirors_neq0:
```
```   334   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
```
```   335     and ab: "a*b = 0" shows "b = 0"
```
```   336 proof -
```
```   337   { assume bz: "b \<noteq> 0"
```
```   338     from no_zero_divisors [OF az bz] ab have False by blast }
```
```   339   thus "b = 0" by blast
```
```   340 qed
```
```   341
```
```   342 interpretation class_ringb: ringb
```
```   343   ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
```
```   344 proof(unfold_locales, simp add: ring_eq_simps power_Suc, auto)
```
```   345   fix w x y z ::"'a::{idom,recpower,number_ring}"
```
```   346   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   347   hence ynz': "y - z \<noteq> 0" by simp
```
```   348   from p have "w * y + x* z - w*z - x*y = 0" by simp
```
```   349   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_eq_simps)
```
```   350   hence "(y - z) * (w - x) = 0" by (simp add: ring_eq_simps)
```
```   351   with  no_zero_divirors_neq0 [OF ynz']
```
```   352   have "w - x = 0" by blast
```
```   353   thus "w = x"  by simp
```
```   354 qed
```
```   355
```
```   356
```
```   357 declaration {*
```
```   358   NormalizerData.funs @{thm class_ringb.axioms}
```
```   359    {is_const = fn phi => numeral_is_const,
```
```   360     dest_const = fn phi => fn ct =>
```
```   361       Rat.rat_of_int (snd
```
```   362         (HOLogic.dest_number (Thm.term_of ct)
```
```   363           handle TERM _ => error "ring_dest_const")),
```
```   364     mk_const = fn phi => fn cT => fn x =>
```
```   365       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   366     conv = fn phi => K numeral_conv}
```
```   367 *}
```
```   368
```
```   369
```
```   370 interpretation natgb: semiringb
```
```   371   ["op +" "op *" "op ^" "0::nat" "1"]
```
```   372 proof (unfold_locales, simp add: ring_eq_simps power_Suc)
```
```   373   fix w x y z ::"nat"
```
```   374   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   375     hence "y < z \<or> y > z" by arith
```
```   376     moreover {
```
```   377       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   378       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   379       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_eq_simps)
```
```   380       hence "x*k = w*k" by simp
```
```   381       hence "w = x" using kp by (simp add: mult_cancel2) }
```
```   382     moreover {
```
```   383       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   384       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   385       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_eq_simps)
```
```   386       hence "w*k = x*k" by simp
```
```   387       hence "w = x" using kp by (simp add: mult_cancel2)}
```
```   388     ultimately have "w=x" by blast }
```
```   389   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   390 qed
```
```   391
```
```   392 declaration {*
```
```   393   NormalizerData.funs @{thm natgb.axioms}
```
```   394    {is_const = fn phi => numeral_is_const,
```
```   395     dest_const = fn phi => fn ct =>
```
```   396       Rat.rat_of_int (snd
```
```   397         (HOLogic.dest_number (Thm.term_of ct)
```
```   398           handle TERM _ => error "ring_dest_const")),
```
```   399     mk_const = fn phi => fn cT => fn x =>
```
```   400       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   401     conv = fn phi => K numeral_conv}
```
```   402 *}
```
```   403
```
```   404 locale fieldgb = ringb + gb_field
```
```   405 begin
```
```   406
```
```   407 declare "axioms" [normalizer del]
```
```   408
```
```   409 lemma "axioms" [normalizer
```
```   410   semiring ops: semiring_ops
```
```   411   semiring rules: semiring_rules
```
```   412   ring ops: ring_ops
```
```   413   ring rules: ring_rules
```
```   414   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   415   "fieldgb add mul pwr r0 r1 sub neg divide inverse" by unfold_locales
```
```   416 end
```
```   417
```
```   418
```
```   419
```
```   420 lemmas bool_simps = simp_thms(1-34)
```
```   421 lemma dnf:
```
```   422     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   423     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
```
```   424   by blast+
```
```   425
```
```   426 lemmas weak_dnf_simps = dnf bool_simps
```
```   427
```
```   428 lemma nnf_simps:
```
```   429     "(\<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)"
```
```   430     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   431   by blast+
```
```   432
```
```   433 lemma PFalse:
```
```   434     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```   435     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```   436   by auto
```
```   437
```
```   438 use "Tools/Groebner_Basis/groebner.ML"
```
```   439
```
```   440 method_setup algebra =
```
```   441 {*
```
```   442 let
```
```   443  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```   444  val addN = "add"
```
```   445  val delN = "del"
```
```   446  val any_keyword = keyword addN || keyword delN
```
```   447  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
```
```   448 in
```
```   449 fn src => Method.syntax
```
```   450     ((Scan.optional (keyword addN |-- thms) []) --
```
```   451     (Scan.optional (keyword delN |-- thms) [])) src
```
```   452  #> (fn ((add_ths, del_ths), ctxt) =>
```
```   453        Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
```
```   454 end
```
```   455
```
```   456 *} ""
```
```   457
```
```   458 end
```