src/HOL/Groebner_Basis.thy
 author haftmann Thu May 06 16:32:20 2010 +0200 (2010-05-06) changeset 36700 9b85b9d74b83 parent 36699 816da1023508 child 36702 b455ebd63799 permissions -rw-r--r--
dropped auxiliary method sring_norm; integrated normalizer.ML and normalizer_data.ML
```     1 (*  Title:      HOL/Groebner_Basis.thy
```
```     2     Author:     Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Semiring normalization and Groebner Bases *}
```
```     6
```
```     7 theory Groebner_Basis
```
```     8 imports Numeral_Simprocs Nat_Transfer
```
```     9 uses
```
```    10   "Tools/Groebner_Basis/normalizer.ML"
```
```    11   ("Tools/Groebner_Basis/groebner.ML")
```
```    12 begin
```
```    13
```
```    14 subsection {* Semiring normalization *}
```
```    15
```
```    16 setup Normalizer.setup
```
```    17
```
```    18 locale gb_semiring =
```
```    19   fixes add mul pwr r0 r1
```
```    20   assumes add_a:"(add x (add y z) = add (add x y) z)"
```
```    21     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
```
```    22     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
```
```    23     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
```
```    24     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
```
```    25     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
```
```    26 begin
```
```    27
```
```    28 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    29 proof (induct p)
```
```    30   case 0
```
```    31   then show ?case by (auto simp add: pwr_0 mul_1)
```
```    32 next
```
```    33   case Suc
```
```    34   from this [symmetric] show ?case
```
```    35     by (auto simp add: pwr_Suc mul_1 mul_a)
```
```    36 qed
```
```    37
```
```    38 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    39 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
```
```    40   fix q x y
```
```    41   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    42   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
```
```    43     by (simp add: mul_a)
```
```    44   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
```
```    45   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
```
```    46   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
```
```    47     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
```
```    48 qed
```
```    49
```
```    50 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
```
```    51 proof (induct p arbitrary: q)
```
```    52   case 0
```
```    53   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
```
```    54 next
```
```    55   case Suc
```
```    56   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
```
```    57 qed
```
```    58
```
```    59
```
```    60 subsubsection {* Declaring the abstract theory *}
```
```    61
```
```    62 lemma semiring_ops:
```
```    63   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    64     and "TERM r0" and "TERM r1" .
```
```    65
```
```    66 lemma semiring_rules:
```
```    67   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    68   "add (mul a m) m = mul (add a r1) m"
```
```    69   "add m (mul a m) = mul (add a r1) m"
```
```    70   "add m m = mul (add r1 r1) m"
```
```    71   "add r0 a = a"
```
```    72   "add a r0 = a"
```
```    73   "mul a b = mul b a"
```
```    74   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    75   "mul r0 a = r0"
```
```    76   "mul a r0 = r0"
```
```    77   "mul r1 a = a"
```
```    78   "mul a r1 = a"
```
```    79   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    80   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    81   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    82   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    83   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    84   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    85   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    86   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    87   "add (add a b) c = add a (add b c)"
```
```    88   "add a (add c d) = add c (add a d)"
```
```    89   "add (add a b) c = add (add a c) b"
```
```    90   "add a c = add c a"
```
```    91   "add a (add c d) = add (add a c) d"
```
```    92   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    93   "mul x (pwr x q) = pwr x (Suc q)"
```
```    94   "mul (pwr x q) x = pwr x (Suc q)"
```
```    95   "mul x x = pwr x 2"
```
```    96   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    97   "pwr (pwr x p) q = pwr x (p * q)"
```
```    98   "pwr x 0 = r1"
```
```    99   "pwr x 1 = x"
```
```   100   "mul x (add y z) = add (mul x y) (mul x z)"
```
```   101   "pwr x (Suc q) = mul x (pwr x q)"
```
```   102   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   103   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   104 proof -
```
```   105   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   106 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   107 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   108 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   109 next show "add r0 a = a" using add_0 by simp
```
```   110 next show "add a r0 = a" using add_0 add_c by simp
```
```   111 next show "mul a b = mul b a" using mul_c by simp
```
```   112 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   113 next show "mul r0 a = r0" using mul_0 by simp
```
```   114 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   115 next show "mul r1 a = a" using mul_1 by simp
```
```   116 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   117 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   118     using mul_c mul_a by simp
```
```   119 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   120     using mul_a by simp
```
```   121 next
```
```   122   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   123   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   124   finally
```
```   125   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   126     using mul_c by simp
```
```   127 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   128 next
```
```   129   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   130 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   131 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   132 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   133     using add_c add_a by simp
```
```   134 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   135 next show "add a (add c d) = add c (add a d)"
```
```   136     apply (simp add: add_a) by (simp only: add_c)
```
```   137 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   138 next show "add a c = add c a" by (rule add_c)
```
```   139 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   140 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   141 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   142 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   143 next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   144 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   145 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   146 next show "pwr x 0 = r1" using pwr_0 .
```
```   147 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   148 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   149 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   150 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
```
```   151 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   152     by (simp add: nat_number' pwr_Suc mul_pwr)
```
```   153 qed
```
```   154
```
```   155
```
```   156 lemmas gb_semiring_axioms' =
```
```   157   gb_semiring_axioms [normalizer
```
```   158     semiring ops: semiring_ops
```
```   159     semiring rules: semiring_rules]
```
```   160
```
```   161 end
```
```   162
```
```   163 interpretation class_semiring: gb_semiring
```
```   164     "op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1"
```
```   165   proof qed (auto simp add: algebra_simps)
```
```   166
```
```   167 lemmas nat_arith =
```
```   168   add_nat_number_of
```
```   169   diff_nat_number_of
```
```   170   mult_nat_number_of
```
```   171   eq_nat_number_of
```
```   172   less_nat_number_of
```
```   173
```
```   174 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
```
```   175   by simp
```
```   176
```
```   177 lemmas comp_arith =
```
```   178   Let_def arith_simps nat_arith rel_simps neg_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_plus1
```
```   181   numeral_0_eq_0[symmetric] numerals[symmetric]
```
```   182   iszero_simps not_iszero_Numeral1
```
```   183
```
```   184 lemmas semiring_norm = comp_arith
```
```   185
```
```   186 ML {*
```
```   187 local
```
```   188
```
```   189 open Conv;
```
```   190
```
```   191 fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);
```
```   192
```
```   193 fun int_of_rat x =
```
```   194   (case Rat.quotient_of_rat x of (i, 1) => i
```
```   195   | _ => error "int_of_rat: bad int");
```
```   196
```
```   197 val numeral_conv =
```
```   198   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
```
```   199   Simplifier.rewrite (HOL_basic_ss addsimps
```
```   200     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
```
```   201
```
```   202 in
```
```   203
```
```   204 fun normalizer_funs key =
```
```   205   Normalizer.funs key
```
```   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 => Numeral.mk_cnumber cT (int_of_rat x),
```
```   212     conv = fn phi => K numeral_conv}
```
```   213
```
```   214 end
```
```   215 *}
```
```   216
```
```   217 declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
```
```   218
```
```   219 locale gb_ring = gb_semiring +
```
```   220   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   221     and neg :: "'a \<Rightarrow> 'a"
```
```   222   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   223     and sub_add: "sub x y = add x (neg y)"
```
```   224 begin
```
```   225
```
```   226 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   227
```
```   228 lemmas ring_rules = neg_mul sub_add
```
```   229
```
```   230 lemmas gb_ring_axioms' =
```
```   231   gb_ring_axioms [normalizer
```
```   232     semiring ops: semiring_ops
```
```   233     semiring rules: semiring_rules
```
```   234     ring ops: ring_ops
```
```   235     ring rules: ring_rules]
```
```   236
```
```   237 end
```
```   238
```
```   239
```
```   240 interpretation class_ring: gb_ring "op +" "op *" "op ^"
```
```   241     "0::'a::{comm_semiring_1,number_ring}" 1 "op -" "uminus"
```
```   242   proof qed simp_all
```
```   243
```
```   244
```
```   245 declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
```
```   246
```
```   247 locale gb_field = gb_ring +
```
```   248   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   249     and inverse:: "'a \<Rightarrow> 'a"
```
```   250   assumes divide_inverse: "divide x y = mul x (inverse y)"
```
```   251      and inverse_divide: "inverse x = divide r1 x"
```
```   252 begin
```
```   253
```
```   254 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
```
```   255
```
```   256 lemmas field_rules = divide_inverse inverse_divide
```
```   257
```
```   258 lemmas gb_field_axioms' =
```
```   259   gb_field_axioms [normalizer
```
```   260     semiring ops: semiring_ops
```
```   261     semiring rules: semiring_rules
```
```   262     ring ops: ring_ops
```
```   263     ring rules: ring_rules
```
```   264     field ops: field_ops
```
```   265     field rules: field_rules]
```
```   266
```
```   267 end
```
```   268
```
```   269
```
```   270 subsection {* Groebner Bases *}
```
```   271
```
```   272 locale semiringb = gb_semiring +
```
```   273   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   274   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   275     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   276 begin
```
```   277
```
```   278 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)"
```
```   279 proof-
```
```   280   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   281   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   282     using add_mul_solve by blast
```
```   283   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)"
```
```   284     by simp
```
```   285 qed
```
```   286
```
```   287 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   288   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   289 proof(clarify)
```
```   290   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   291     and eq: "add b (mul r c) = add b (mul r d)"
```
```   292   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   293   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   294     using mul_0 add_cancel by simp
```
```   295   thus "False" using add_mul_solve nz cnd by simp
```
```   296 qed
```
```   297
```
```   298 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
```
```   299 proof-
```
```   300   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
```
```   301   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
```
```   302 qed
```
```   303
```
```   304 declare gb_semiring_axioms' [normalizer del]
```
```   305
```
```   306 lemmas semiringb_axioms' = semiringb_axioms [normalizer
```
```   307   semiring ops: semiring_ops
```
```   308   semiring rules: semiring_rules
```
```   309   idom rules: noteq_reduce add_scale_eq_noteq]
```
```   310
```
```   311 end
```
```   312
```
```   313 locale ringb = semiringb + gb_ring +
```
```   314   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
```
```   315 begin
```
```   316
```
```   317 declare gb_ring_axioms' [normalizer del]
```
```   318
```
```   319 lemmas ringb_axioms' = ringb_axioms [normalizer
```
```   320   semiring ops: semiring_ops
```
```   321   semiring rules: semiring_rules
```
```   322   ring ops: ring_ops
```
```   323   ring rules: ring_rules
```
```   324   idom rules: noteq_reduce add_scale_eq_noteq
```
```   325   ideal rules: subr0_iff add_r0_iff]
```
```   326
```
```   327 end
```
```   328
```
```   329
```
```   330 lemma no_zero_divirors_neq0:
```
```   331   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
```
```   332     and ab: "a*b = 0" shows "b = 0"
```
```   333 proof -
```
```   334   { assume bz: "b \<noteq> 0"
```
```   335     from no_zero_divisors [OF az bz] ab have False by blast }
```
```   336   thus "b = 0" by blast
```
```   337 qed
```
```   338
```
```   339 interpretation class_ringb: ringb
```
```   340   "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
```
```   341 proof(unfold_locales, simp add: algebra_simps, auto)
```
```   342   fix w x y z ::"'a::{idom,number_ring}"
```
```   343   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   344   hence ynz': "y - z \<noteq> 0" by simp
```
```   345   from p have "w * y + x* z - w*z - x*y = 0" by simp
```
```   346   hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
```
```   347   hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
```
```   348   with  no_zero_divirors_neq0 [OF ynz']
```
```   349   have "w - x = 0" by blast
```
```   350   thus "w = x"  by simp
```
```   351 qed
```
```   352
```
```   353 declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
```
```   354
```
```   355 interpretation natgb: semiringb
```
```   356   "op +" "op *" "op ^" "0::nat" "1"
```
```   357 proof (unfold_locales, simp add: algebra_simps)
```
```   358   fix w x y z ::"nat"
```
```   359   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   360     hence "y < z \<or> y > z" by arith
```
```   361     moreover {
```
```   362       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   363       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   364       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
```
```   365       hence "x*k = w*k" by simp
```
```   366       hence "w = x" using kp by simp }
```
```   367     moreover {
```
```   368       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   369       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   370       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
```
```   371       hence "w*k = x*k" by simp
```
```   372       hence "w = x" using kp by simp }
```
```   373     ultimately have "w=x" by blast }
```
```   374   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   375 qed
```
```   376
```
```   377 declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
```
```   378
```
```   379 locale fieldgb = ringb + gb_field
```
```   380 begin
```
```   381
```
```   382 declare gb_field_axioms' [normalizer del]
```
```   383
```
```   384 lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
```
```   385   semiring ops: semiring_ops
```
```   386   semiring rules: semiring_rules
```
```   387   ring ops: ring_ops
```
```   388   ring rules: ring_rules
```
```   389   field ops: field_ops
```
```   390   field rules: field_rules
```
```   391   idom rules: noteq_reduce add_scale_eq_noteq
```
```   392   ideal rules: subr0_iff add_r0_iff]
```
```   393
```
```   394 end
```
```   395
```
```   396
```
```   397 lemmas bool_simps = simp_thms(1-34)
```
```   398 lemma dnf:
```
```   399     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   400     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
```
```   401   by blast+
```
```   402
```
```   403 lemmas weak_dnf_simps = dnf bool_simps
```
```   404
```
```   405 lemma nnf_simps:
```
```   406     "(\<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)"
```
```   407     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   408   by blast+
```
```   409
```
```   410 lemma PFalse:
```
```   411     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```   412     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```   413   by auto
```
```   414
```
```   415 use "Tools/Groebner_Basis/groebner.ML"
```
```   416
```
```   417 method_setup algebra =
```
```   418 {*
```
```   419 let
```
```   420  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```   421  val addN = "add"
```
```   422  val delN = "del"
```
```   423  val any_keyword = keyword addN || keyword delN
```
```   424  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
```
```   425 in
```
```   426   ((Scan.optional (keyword addN |-- thms) []) --
```
```   427    (Scan.optional (keyword delN |-- thms) [])) >>
```
```   428   (fn (add_ths, del_ths) => fn ctxt =>
```
```   429        SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
```
```   430 end
```
```   431 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
```
```   432 declare dvd_def[algebra]
```
```   433 declare dvd_eq_mod_eq_0[symmetric, algebra]
```
```   434 declare mod_div_trivial[algebra]
```
```   435 declare mod_mod_trivial[algebra]
```
```   436 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
```
```   437 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
```
```   438 declare zmod_zdiv_equality[symmetric,algebra]
```
```   439 declare zdiv_zmod_equality[symmetric, algebra]
```
```   440 declare zdiv_zminus_zminus[algebra]
```
```   441 declare zmod_zminus_zminus[algebra]
```
```   442 declare zdiv_zminus2[algebra]
```
```   443 declare zmod_zminus2[algebra]
```
```   444 declare zdiv_zero[algebra]
```
```   445 declare zmod_zero[algebra]
```
```   446 declare mod_by_1[algebra]
```
```   447 declare div_by_1[algebra]
```
```   448 declare zmod_minus1_right[algebra]
```
```   449 declare zdiv_minus1_right[algebra]
```
```   450 declare mod_div_trivial[algebra]
```
```   451 declare mod_mod_trivial[algebra]
```
```   452 declare mod_mult_self2_is_0[algebra]
```
```   453 declare mod_mult_self1_is_0[algebra]
```
```   454 declare zmod_eq_0_iff[algebra]
```
```   455 declare dvd_0_left_iff[algebra]
```
```   456 declare zdvd1_eq[algebra]
```
```   457 declare zmod_eq_dvd_iff[algebra]
```
```   458 declare nat_mod_eq_iff[algebra]
```
```   459
```
```   460 subsection{* Groebner Bases for fields *}
```
```   461
```
```   462 interpretation class_fieldgb:
```
```   463   fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
```
```   464
```
```   465 lemma divide_Numeral1: "(x::'a::{field, number_ring}) / Numeral1 = x" by simp
```
```   466 lemma divide_Numeral0: "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
```
```   467   by simp
```
```   468 lemma mult_frac_frac: "((x::'a::field_inverse_zero) / y) * (z / w) = (x*z) / (y*w)"
```
```   469   by simp
```
```   470 lemma mult_frac_num: "((x::'a::field_inverse_zero) / y) * z  = (x*z) / y"
```
```   471   by simp
```
```   472 lemma mult_num_frac: "((x::'a::field_inverse_zero) / y) * z  = (x*z) / y"
```
```   473   by simp
```
```   474
```
```   475 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
```
```   476
```
```   477 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::field_inverse_zero) / y + z = (x + z*y) / y"
```
```   478   by (simp add: add_divide_distrib)
```
```   479 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::field_inverse_zero) / y = (x + z*y) / y"
```
```   480   by (simp add: add_divide_distrib)
```
```   481
```
```   482 ML {*
```
```   483 let open Conv
```
```   484 in fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute})))) (@{thm field_divide_inverse} RS sym)
```
```   485 end
```
```   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 @{thms 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 Rings.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 Rings.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 Rings.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 Orderings.less},_)\$(Const(@{const_name Rings.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 Orderings.less_eq},_)\$(Const(@{const_name Rings.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 Rings.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 Orderings.less},_)\$_\$(Const(@{const_name Rings.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 Orderings.less_eq},_)\$_\$(Const(@{const_name Rings.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 Rings.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 local
```
```   614 open Conv
```
```   615 in
```
```   616
```
```   617 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
```
```   618            @{thm "divide_Numeral1"},
```
```   619            @{thm "divide_zero"}, @{thm "divide_Numeral0"},
```
```   620            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
```
```   621            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
```
```   622            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
```
```   623            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
```
```   624            @{thm "diff_def"}, @{thm "minus_divide_left"},
```
```   625            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
```
```   626            @{thm field_divide_inverse} RS sym, @{thm inverse_divide},
```
```   627            fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))
```
```   628            (@{thm field_divide_inverse} RS sym)]
```
```   629
```
```   630 val comp_conv = (Simplifier.rewrite
```
```   631 (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
```
```   632               addsimps ths addsimps @{thms simp_thms}
```
```   633               addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
```
```   634                addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
```
```   635                             ord_frac_simproc]
```
```   636                 addcongs [@{thm "if_weak_cong"}]))
```
```   637 then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
```
```   638   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
```
```   639 end
```
```   640
```
```   641 fun numeral_is_const ct =
```
```   642   case term_of ct of
```
```   643    Const (@{const_name Rings.divide},_) \$ a \$ b =>
```
```   644      can HOLogic.dest_number a andalso can HOLogic.dest_number b
```
```   645  | Const (@{const_name Rings.inverse},_)\$t => can HOLogic.dest_number t
```
```   646  | t => can HOLogic.dest_number t
```
```   647
```
```   648 fun dest_const ct = ((case term_of ct of
```
```   649    Const (@{const_name Rings.divide},_) \$ a \$ b=>
```
```   650     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
```
```   651  | Const (@{const_name Rings.inverse},_)\$t =>
```
```   652                Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
```
```   653  | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
```
```   654    handle TERM _ => error "ring_dest_const")
```
```   655
```
```   656 fun mk_const phi cT x =
```
```   657  let val (a, b) = Rat.quotient_of_rat x
```
```   658  in if b = 1 then Numeral.mk_cnumber cT a
```
```   659     else Thm.capply
```
```   660          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
```
```   661                      (Numeral.mk_cnumber cT a))
```
```   662          (Numeral.mk_cnumber cT b)
```
```   663   end
```
```   664
```
```   665 in
```
```   666  val field_comp_conv = comp_conv;
```
```   667  val fieldgb_declaration =
```
```   668   Normalizer.funs @{thm class_fieldgb.fieldgb_axioms'}
```
```   669    {is_const = K numeral_is_const,
```
```   670     dest_const = K dest_const,
```
```   671     mk_const = mk_const,
```
```   672     conv = K (K comp_conv)}
```
```   673 end;
```
```   674 *}
```
```   675
```
```   676 declaration fieldgb_declaration
```
```   677
```
```   678 end
```