src/HOL/Groebner_Basis.thy
 author wenzelm Tue Jun 05 16:26:04 2007 +0200 (2007-06-05) changeset 23252 67268bb40b21 child 23258 9062e98fdab1 permissions -rw-r--r--
Semiring normalization and Groebner Bases.
```     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 begin
```
```    15
```
```    16 subsection {* Semiring normalization *}
```
```    17
```
```    18 setup NormalizerData.setup
```
```    19
```
```    20
```
```    21 locale semiring =
```
```    22   fixes add mul pwr r0 r1
```
```    23   assumes add_a:"(add x (add y z) = add (add x y) z)"
```
```    24     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
```
```    25     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
```
```    26     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
```
```    27     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
```
```    28     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
```
```    29 begin
```
```    30
```
```    31 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    32 proof (induct p)
```
```    33   case 0
```
```    34   then show ?case by (auto simp add: pwr_0 mul_1)
```
```    35 next
```
```    36   case Suc
```
```    37   from this [symmetric] show ?case
```
```    38     by (auto simp add: pwr_Suc mul_1 mul_a)
```
```    39 qed
```
```    40
```
```    41 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    42 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
```
```    43   fix q x y
```
```    44   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    45   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
```
```    46     by (simp add: mul_a)
```
```    47   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
```
```    48   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
```
```    49   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
```
```    50     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
```
```    51 qed
```
```    52
```
```    53 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
```
```    54 proof (induct p arbitrary: q)
```
```    55   case 0
```
```    56   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
```
```    57 next
```
```    58   case Suc
```
```    59   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
```
```    60 qed
```
```    61
```
```    62
```
```    63 subsubsection {* Declaring the abstract theory *}
```
```    64
```
```    65 lemma semiring_ops:
```
```    66   includes meta_term_syntax
```
```    67   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    68     and "TERM r0" and "TERM r1"
```
```    69   by rule+
```
```    70
```
```    71 lemma semiring_rules:
```
```    72   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    73   "add (mul a m) m = mul (add a r1) m"
```
```    74   "add m (mul a m) = mul (add a r1) m"
```
```    75   "add m m = mul (add r1 r1) m"
```
```    76   "add r0 a = a"
```
```    77   "add a r0 = a"
```
```    78   "mul a b = mul b a"
```
```    79   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    80   "mul r0 a = r0"
```
```    81   "mul a r0 = r0"
```
```    82   "mul r1 a = a"
```
```    83   "mul a r1 = a"
```
```    84   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    85   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    86   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    87   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    88   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    89   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    90   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    91   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    92   "add (add a b) c = add a (add b c)"
```
```    93   "add a (add c d) = add c (add a d)"
```
```    94   "add (add a b) c = add (add a c) b"
```
```    95   "add a c = add c a"
```
```    96   "add a (add c d) = add (add a c) d"
```
```    97   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    98   "mul x (pwr x q) = pwr x (Suc q)"
```
```    99   "mul (pwr x q) x = pwr x (Suc q)"
```
```   100   "mul x x = pwr x 2"
```
```   101   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```   102   "pwr (pwr x p) q = pwr x (p * q)"
```
```   103   "pwr x 0 = r1"
```
```   104   "pwr x 1 = x"
```
```   105   "mul x (add y z) = add (mul x y) (mul x z)"
```
```   106   "pwr x (Suc q) = mul x (pwr x q)"
```
```   107   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   108   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   109 proof -
```
```   110   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   111 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   112 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   113 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   114 next show "add r0 a = a" using add_0 by simp
```
```   115 next show "add a r0 = a" using add_0 add_c by simp
```
```   116 next show "mul a b = mul b a" using mul_c by simp
```
```   117 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   118 next show "mul r0 a = r0" using mul_0 by simp
```
```   119 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   120 next show "mul r1 a = a" using mul_1 by simp
```
```   121 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   122 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   123     using mul_c mul_a by simp
```
```   124 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   125     using mul_a by simp
```
```   126 next
```
```   127   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   128   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   129   finally
```
```   130   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   131     using mul_c by simp
```
```   132 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   133 next
```
```   134   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   135 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   136 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   137 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   138     using add_c add_a by simp
```
```   139 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   140 next show "add a (add c d) = add c (add a d)"
```
```   141     apply (simp add: add_a) by (simp only: add_c)
```
```   142 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   143 next show "add a c = add c a" by (rule add_c)
```
```   144 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   145 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   146 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   147 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   148 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   149 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   150 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   151 next show "pwr x 0 = r1" using pwr_0 .
```
```   152 next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   153 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   154 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   155 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
```
```   156 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   157     by (simp add: nat_number pwr_Suc mul_pwr)
```
```   158 qed
```
```   159
```
```   160
```
```   161 lemma "axioms" [normalizer
```
```   162     semiring ops: semiring_ops
```
```   163     semiring rules: semiring_rules]:
```
```   164   "semiring add mul pwr r0 r1" .
```
```   165
```
```   166 end
```
```   167
```
```   168 interpretation class_semiring: semiring
```
```   169     ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
```
```   170   by unfold_locales (auto simp add: ring_eq_simps power_Suc)
```
```   171
```
```   172 lemmas nat_arith =
```
```   173   add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
```
```   174
```
```   175 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
```
```   176   by (simp add: numeral_1_eq_1)
```
```   177 lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
```
```   178   if_True add_0 add_Suc add_number_of_left mult_number_of_left
```
```   179   numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
```
```   180   numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
```
```   181   iszero_number_of_1 iszero_number_of_0 nonzero_number_of_Min
```
```   182   iszero_number_of_Pls iszero_0 not_iszero_Numeral1
```
```   183
```
```   184 lemmas semiring_norm = comp_arith
```
```   185
```
```   186 ML {*
```
```   187   fun numeral_is_const ct =
```
```   188     can HOLogic.dest_number (Thm.term_of ct);
```
```   189
```
```   190   val numeral_conv =
```
```   191     Conv.then_conv (Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}),
```
```   192    Simplifier.rewrite (HOL_basic_ss addsimps
```
```   193   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}));
```
```   194 *}
```
```   195
```
```   196 ML {*
```
```   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
```
```   202 declaration {*
```
```   203   NormalizerData.funs @{thm class_semiring.axioms}
```
```   204    {is_const = fn phi => numeral_is_const,
```
```   205     dest_const = fn phi => fn ct =>
```
```   206       Rat.rat_of_int (snd
```
```   207         (HOLogic.dest_number (Thm.term_of ct)
```
```   208           handle TERM _ => error "ring_dest_const")),
```
```   209     mk_const = fn phi => fn cT => fn x =>
```
```   210       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   211     conv = fn phi => numeral_conv}
```
```   212 *}
```
```   213
```
```   214
```
```   215 locale ring = semiring +
```
```   216   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   217     and neg :: "'a \<Rightarrow> 'a"
```
```   218   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   219     and sub_add: "sub x y = add x (neg y)"
```
```   220 begin
```
```   221
```
```   222 lemma ring_ops:
```
```   223   includes meta_term_syntax
```
```   224   shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   225
```
```   226 lemmas ring_rules = neg_mul sub_add
```
```   227
```
```   228 lemma "axioms" [normalizer
```
```   229   semiring ops: semiring_ops
```
```   230   semiring rules: semiring_rules
```
```   231   ring ops: ring_ops
```
```   232   ring rules: ring_rules]:
```
```   233   "ring add mul pwr r0 r1 sub neg" .
```
```   234
```
```   235 end
```
```   236
```
```   237
```
```   238 interpretation class_ring: ring ["op +" "op *" "op ^"
```
```   239     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
```
```   240   by unfold_locales simp_all
```
```   241
```
```   242
```
```   243 declaration {*
```
```   244   NormalizerData.funs @{thm class_ring.axioms}
```
```   245    {is_const = fn phi => numeral_is_const,
```
```   246     dest_const = fn phi => fn ct =>
```
```   247       Rat.rat_of_int (snd
```
```   248         (HOLogic.dest_number (Thm.term_of ct)
```
```   249           handle TERM _ => error "ring_dest_const")),
```
```   250     mk_const = fn phi => fn cT => fn x =>
```
```   251       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   252     conv = fn phi => numeral_conv}
```
```   253 *}
```
```   254
```
```   255 use "Tools/Groebner_Basis/normalizer.ML"
```
```   256
```
```   257 method_setup sring_norm = {*
```
```   258   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
```
```   259 *} "Semiring_normalizer"
```
```   260
```
```   261
```
```   262 subsection {* Gr�bner Bases *}
```
```   263
```
```   264 locale semiringb = semiring +
```
```   265   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   266   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   267     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   268 begin
```
```   269
```
```   270 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)"
```
```   271 proof-
```
```   272   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   273   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   274     using add_mul_solve by blast
```
```   275   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)"
```
```   276     by simp
```
```   277 qed
```
```   278
```
```   279 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   280   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   281 proof(clarify)
```
```   282   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   283     and eq: "add b (mul r c) = add b (mul r d)"
```
```   284   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   285   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   286     using mul_0 add_cancel by simp
```
```   287   thus "False" using add_mul_solve nz cnd by simp
```
```   288 qed
```
```   289
```
```   290 declare "axioms" [normalizer del]
```
```   291
```
```   292 lemma "axioms" [normalizer
```
```   293   semiring ops: semiring_ops
```
```   294   semiring rules: semiring_rules
```
```   295   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   296   "semiringb add mul pwr r0 r1" .
```
```   297
```
```   298 end
```
```   299
```
```   300 locale ringb = semiringb + ring
```
```   301 begin
```
```   302
```
```   303 declare "axioms" [normalizer del]
```
```   304
```
```   305 lemma "axioms" [normalizer
```
```   306   semiring ops: semiring_ops
```
```   307   semiring rules: semiring_rules
```
```   308   ring ops: ring_ops
```
```   309   ring rules: ring_rules
```
```   310   idom rules: noteq_reduce add_scale_eq_noteq]:
```
```   311   "ringb add mul pwr r0 r1 sub neg" .
```
```   312
```
```   313 end
```
```   314
```
```   315 lemma no_zero_divirors_neq0:
```
```   316   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
```
```   317     and ab: "a*b = 0" shows "b = 0"
```
```   318 proof -
```
```   319   { assume bz: "b \<noteq> 0"
```
```   320     from no_zero_divisors [OF az bz] ab have False by blast }
```
```   321   thus "b = 0" by blast
```
```   322 qed
```
```   323
```
```   324 interpretation class_ringb: ringb
```
```   325   ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
```
```   326 proof(unfold_locales, simp add: ring_eq_simps power_Suc, auto)
```
```   327   fix w x y z ::"'a::{idom,recpower,number_ring}"
```
```   328   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   329   hence ynz': "y - z \<noteq> 0" by simp
```
```   330   from p have "w * y + x* z - w*z - x*y = 0" by simp
```
```   331   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_eq_simps)
```
```   332   hence "(y - z) * (w - x) = 0" by (simp add: ring_eq_simps)
```
```   333   with  no_zero_divirors_neq0 [OF ynz']
```
```   334   have "w - x = 0" by blast
```
```   335   thus "w = x"  by simp
```
```   336 qed
```
```   337
```
```   338
```
```   339 declaration {*
```
```   340   NormalizerData.funs @{thm class_ringb.axioms}
```
```   341    {is_const = fn phi => numeral_is_const,
```
```   342     dest_const = fn phi => fn ct =>
```
```   343       Rat.rat_of_int (snd
```
```   344         (HOLogic.dest_number (Thm.term_of ct)
```
```   345           handle TERM _ => error "ring_dest_const")),
```
```   346     mk_const = fn phi => fn cT => fn x =>
```
```   347       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   348     conv = fn phi => numeral_conv}
```
```   349 *}
```
```   350
```
```   351
```
```   352 interpretation natgb: semiringb
```
```   353   ["op +" "op *" "op ^" "0::nat" "1"]
```
```   354 proof (unfold_locales, simp add: ring_eq_simps power_Suc)
```
```   355   fix w x y z ::"nat"
```
```   356   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   357     hence "y < z \<or> y > z" by arith
```
```   358     moreover {
```
```   359       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   360       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   361       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_eq_simps)
```
```   362       hence "x*k = w*k" by simp
```
```   363       hence "w = x" using kp by (simp add: mult_cancel2) }
```
```   364     moreover {
```
```   365       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   366       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   367       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_eq_simps)
```
```   368       hence "w*k = x*k" by simp
```
```   369       hence "w = x" using kp by (simp add: mult_cancel2)}
```
```   370     ultimately have "w=x" by blast }
```
```   371   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   372 qed
```
```   373
```
```   374 declaration {*
```
```   375   NormalizerData.funs @{thm natgb.axioms}
```
```   376    {is_const = fn phi => numeral_is_const,
```
```   377     dest_const = fn phi => fn ct =>
```
```   378       Rat.rat_of_int (snd
```
```   379         (HOLogic.dest_number (Thm.term_of ct)
```
```   380           handle TERM _ => error "ring_dest_const")),
```
```   381     mk_const = fn phi => fn cT => fn x =>
```
```   382       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
```
```   383     conv = fn phi => numeral_conv}
```
```   384 *}
```
```   385
```
```   386
```
```   387 lemmas bool_simps =  simp_thms(1-34)
```
```   388 lemma dnf:
```
```   389     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   390     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
```
```   391   by blast+
```
```   392
```
```   393 lemmas weak_dnf_simps = dnf bool_simps
```
```   394
```
```   395 lemma nnf_simps:
```
```   396     "(\<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)"
```
```   397     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   398   by blast+
```
```   399
```
```   400 lemma PFalse:
```
```   401     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```   402     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```   403   by auto
```
```   404
```
```   405 use "Tools/Groebner_Basis/groebner.ML"
```
```   406
```
```   407 ML {*
```
```   408   fun algebra_tac ctxt i = ObjectLogic.full_atomize_tac i THEN (fn st =>
```
```   409   rtac (Groebner.ring_conv ctxt (Thm.dest_arg (nth (cprems_of st) (i - 1)))) i st);
```
```   410 *}
```
```   411
```
```   412 method_setup algebra = {*
```
```   413   Method.ctxt_args (Method.SIMPLE_METHOD' o algebra_tac)
```
```   414 *} ""
```
```   415
```
```   416 end
```