src/HOL/Groebner_Basis.thy
 author nipkow Fri Aug 28 18:52:41 2009 +0200 (2009-08-28) changeset 32436 10cd49e0c067 parent 31790 05c92381363c child 33296 a3924d1069e5 permissions -rw-r--r--
Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
```     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 Nat_Numeral
```
```     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 subsection {* Semiring normalization *}
```
```    17
```
```    18 setup NormalizerData.setup
```
```    19
```
```    20
```
```    21 locale gb_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   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    67     and "TERM r0" and "TERM r1" .
```
```    68
```
```    69 lemma semiring_rules:
```
```    70   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    71   "add (mul a m) m = mul (add a r1) m"
```
```    72   "add m (mul a m) = mul (add a r1) m"
```
```    73   "add m m = mul (add r1 r1) m"
```
```    74   "add r0 a = a"
```
```    75   "add a r0 = a"
```
```    76   "mul a b = mul b a"
```
```    77   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    78   "mul r0 a = r0"
```
```    79   "mul a r0 = r0"
```
```    80   "mul r1 a = a"
```
```    81   "mul a r1 = a"
```
```    82   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    83   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    84   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    85   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    86   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    87   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    88   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    89   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    90   "add (add a b) c = add a (add b c)"
```
```    91   "add a (add c d) = add c (add a d)"
```
```    92   "add (add a b) c = add (add a c) b"
```
```    93   "add a c = add c a"
```
```    94   "add a (add c d) = add (add a c) d"
```
```    95   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    96   "mul x (pwr x q) = pwr x (Suc q)"
```
```    97   "mul (pwr x q) x = pwr x (Suc q)"
```
```    98   "mul x x = pwr x 2"
```
```    99   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```   100   "pwr (pwr x p) q = pwr x (p * q)"
```
```   101   "pwr x 0 = r1"
```
```   102   "pwr x 1 = x"
```
```   103   "mul x (add y z) = add (mul x y) (mul x z)"
```
```   104   "pwr x (Suc q) = mul x (pwr x q)"
```
```   105   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   106   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   107 proof -
```
```   108   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   109 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   110 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   111 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   112 next show "add r0 a = a" using add_0 by simp
```
```   113 next show "add a r0 = a" using add_0 add_c by simp
```
```   114 next show "mul a b = mul b a" using mul_c by simp
```
```   115 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   116 next show "mul r0 a = r0" using mul_0 by simp
```
```   117 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   118 next show "mul r1 a = a" using mul_1 by simp
```
```   119 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   120 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   121     using mul_c mul_a by simp
```
```   122 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   123     using mul_a by simp
```
```   124 next
```
```   125   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   126   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   127   finally
```
```   128   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   129     using mul_c by simp
```
```   130 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   131 next
```
```   132   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   133 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   134 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   135 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   136     using add_c add_a by simp
```
```   137 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   138 next show "add a (add c d) = add c (add a d)"
```
```   139     apply (simp add: add_a) by (simp only: add_c)
```
```   140 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   141 next show "add a c = add c a" by (rule add_c)
```
```   142 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   143 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   144 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   145 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   146 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   147 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   148 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   149 next show "pwr x 0 = r1" using pwr_0 .
```
```   150 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
```
```   151 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   152 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   153 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
```
```   154 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   155     by (simp add: nat_number pwr_Suc mul_pwr)
```
```   156 qed
```
```   157
```
```   158
```
```   159 lemmas gb_semiring_axioms' =
```
```   160   gb_semiring_axioms [normalizer
```
```   161     semiring ops: semiring_ops
```
```   162     semiring rules: semiring_rules]
```
```   163
```
```   164 end
```
```   165
```
```   166 interpretation class_semiring: gb_semiring
```
```   167     "op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1"
```
```   168   proof qed (auto simp add: algebra_simps power_Suc)
```
```   169
```
```   170 lemmas nat_arith =
```
```   171   add_nat_number_of
```
```   172   diff_nat_number_of
```
```   173   mult_nat_number_of
```
```   174   eq_nat_number_of
```
```   175   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
```
```   180 lemmas comp_arith =
```
```   181   Let_def arith_simps nat_arith rel_simps neg_simps if_False
```
```   182   if_True add_0 add_Suc add_number_of_left mult_number_of_left
```
```   183   numeral_1_eq_1[symmetric] Suc_eq_plus1
```
```   184   numeral_0_eq_0[symmetric] numerals[symmetric]
```
```   185   iszero_simps not_iszero_Numeral1
```
```   186
```
```   187 lemmas semiring_norm = comp_arith
```
```   188
```
```   189 ML {*
```
```   190 local
```
```   191
```
```   192 open Conv;
```
```   193
```
```   194 fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);
```
```   195
```
```   196 fun int_of_rat x =
```
```   197   (case Rat.quotient_of_rat x of (i, 1) => i
```
```   198   | _ => error "int_of_rat: bad int");
```
```   199
```
```   200 val numeral_conv =
```
```   201   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
```
```   202   Simplifier.rewrite (HOL_basic_ss addsimps
```
```   203     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
```
```   204
```
```   205 in
```
```   206
```
```   207 fun normalizer_funs key =
```
```   208   NormalizerData.funs key
```
```   209    {is_const = fn phi => numeral_is_const,
```
```   210     dest_const = fn phi => fn ct =>
```
```   211       Rat.rat_of_int (snd
```
```   212         (HOLogic.dest_number (Thm.term_of ct)
```
```   213           handle TERM _ => error "ring_dest_const")),
```
```   214     mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
```
```   215     conv = fn phi => K numeral_conv}
```
```   216
```
```   217 end
```
```   218 *}
```
```   219
```
```   220 declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
```
```   221
```
```   222
```
```   223 locale gb_ring = gb_semiring +
```
```   224   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   225     and neg :: "'a \<Rightarrow> 'a"
```
```   226   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   227     and sub_add: "sub x y = add x (neg y)"
```
```   228 begin
```
```   229
```
```   230 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   231
```
```   232 lemmas ring_rules = neg_mul sub_add
```
```   233
```
```   234 lemmas gb_ring_axioms' =
```
```   235   gb_ring_axioms [normalizer
```
```   236     semiring ops: semiring_ops
```
```   237     semiring rules: semiring_rules
```
```   238     ring ops: ring_ops
```
```   239     ring rules: ring_rules]
```
```   240
```
```   241 end
```
```   242
```
```   243
```
```   244 interpretation class_ring: gb_ring "op +" "op *" "op ^"
```
```   245     "0::'a::{comm_semiring_1,number_ring}" 1 "op -" "uminus"
```
```   246   proof qed simp_all
```
```   247
```
```   248
```
```   249 declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
```
```   250
```
```   251 use "Tools/Groebner_Basis/normalizer.ML"
```
```   252
```
```   253
```
```   254 method_setup sring_norm = {*
```
```   255   Scan.succeed (SIMPLE_METHOD' o Normalizer.semiring_normalize_tac)
```
```   256 *} "semiring normalizer"
```
```   257
```
```   258
```
```   259 locale gb_field = gb_ring +
```
```   260   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   261     and inverse:: "'a \<Rightarrow> 'a"
```
```   262   assumes divide_inverse: "divide x y = mul x (inverse y)"
```
```   263      and inverse_divide: "inverse x = divide r1 x"
```
```   264 begin
```
```   265
```
```   266 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
```
```   267
```
```   268 lemmas field_rules = divide_inverse inverse_divide
```
```   269
```
```   270 lemmas gb_field_axioms' =
```
```   271   gb_field_axioms [normalizer
```
```   272     semiring ops: semiring_ops
```
```   273     semiring rules: semiring_rules
```
```   274     ring ops: ring_ops
```
```   275     ring rules: ring_rules
```
```   276     field ops: field_ops
```
```   277     field rules: field_rules]
```
```   278
```
```   279 end
```
```   280
```
```   281
```
```   282 subsection {* Groebner Bases *}
```
```   283
```
```   284 locale semiringb = gb_semiring +
```
```   285   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   286   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   287     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   288 begin
```
```   289
```
```   290 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)"
```
```   291 proof-
```
```   292   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   293   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   294     using add_mul_solve by blast
```
```   295   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)"
```
```   296     by simp
```
```   297 qed
```
```   298
```
```   299 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   300   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   301 proof(clarify)
```
```   302   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   303     and eq: "add b (mul r c) = add b (mul r d)"
```
```   304   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   305   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   306     using mul_0 add_cancel by simp
```
```   307   thus "False" using add_mul_solve nz cnd by simp
```
```   308 qed
```
```   309
```
```   310 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
```
```   311 proof-
```
```   312   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
```
```   313   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
```
```   314 qed
```
```   315
```
```   316 declare gb_semiring_axioms' [normalizer del]
```
```   317
```
```   318 lemmas semiringb_axioms' = semiringb_axioms [normalizer
```
```   319   semiring ops: semiring_ops
```
```   320   semiring rules: semiring_rules
```
```   321   idom rules: noteq_reduce add_scale_eq_noteq]
```
```   322
```
```   323 end
```
```   324
```
```   325 locale ringb = semiringb + gb_ring +
```
```   326   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
```
```   327 begin
```
```   328
```
```   329 declare gb_ring_axioms' [normalizer del]
```
```   330
```
```   331 lemmas ringb_axioms' = ringb_axioms [normalizer
```
```   332   semiring ops: semiring_ops
```
```   333   semiring rules: semiring_rules
```
```   334   ring ops: ring_ops
```
```   335   ring rules: ring_rules
```
```   336   idom rules: noteq_reduce add_scale_eq_noteq
```
```   337   ideal rules: subr0_iff add_r0_iff]
```
```   338
```
```   339 end
```
```   340
```
```   341
```
```   342 lemma no_zero_divirors_neq0:
```
```   343   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
```
```   344     and ab: "a*b = 0" shows "b = 0"
```
```   345 proof -
```
```   346   { assume bz: "b \<noteq> 0"
```
```   347     from no_zero_divisors [OF az bz] ab have False by blast }
```
```   348   thus "b = 0" by blast
```
```   349 qed
```
```   350
```
```   351 interpretation class_ringb: ringb
```
```   352   "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
```
```   353 proof(unfold_locales, simp add: algebra_simps power_Suc, auto)
```
```   354   fix w x y z ::"'a::{idom,number_ring}"
```
```   355   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   356   hence ynz': "y - z \<noteq> 0" by simp
```
```   357   from p have "w * y + x* z - w*z - x*y = 0" by simp
```
```   358   hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
```
```   359   hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
```
```   360   with  no_zero_divirors_neq0 [OF ynz']
```
```   361   have "w - x = 0" by blast
```
```   362   thus "w = x"  by simp
```
```   363 qed
```
```   364
```
```   365 declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
```
```   366
```
```   367 interpretation natgb: semiringb
```
```   368   "op +" "op *" "op ^" "0::nat" "1"
```
```   369 proof (unfold_locales, simp add: algebra_simps power_Suc)
```
```   370   fix w x y z ::"nat"
```
```   371   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   372     hence "y < z \<or> y > z" by arith
```
```   373     moreover {
```
```   374       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   375       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   376       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
```
```   377       hence "x*k = w*k" by simp
```
```   378       hence "w = x" using kp by (simp add: mult_cancel2) }
```
```   379     moreover {
```
```   380       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   381       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   382       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
```
```   383       hence "w*k = x*k" by simp
```
```   384       hence "w = x" using kp by (simp add: mult_cancel2)}
```
```   385     ultimately have "w=x" by blast }
```
```   386   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   387 qed
```
```   388
```
```   389 declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
```
```   390
```
```   391 locale fieldgb = ringb + gb_field
```
```   392 begin
```
```   393
```
```   394 declare gb_field_axioms' [normalizer del]
```
```   395
```
```   396 lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
```
```   397   semiring ops: semiring_ops
```
```   398   semiring rules: semiring_rules
```
```   399   ring ops: ring_ops
```
```   400   ring rules: ring_rules
```
```   401   field ops: field_ops
```
```   402   field rules: field_rules
```
```   403   idom rules: noteq_reduce add_scale_eq_noteq
```
```   404   ideal rules: subr0_iff add_r0_iff]
```
```   405
```
```   406 end
```
```   407
```
```   408
```
```   409 lemmas bool_simps = simp_thms(1-34)
```
```   410 lemma dnf:
```
```   411     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   412     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
```
```   413   by blast+
```
```   414
```
```   415 lemmas weak_dnf_simps = dnf bool_simps
```
```   416
```
```   417 lemma nnf_simps:
```
```   418     "(\<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)"
```
```   419     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   420   by blast+
```
```   421
```
```   422 lemma PFalse:
```
```   423     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```   424     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```   425   by auto
```
```   426 use "Tools/Groebner_Basis/groebner.ML"
```
```   427
```
```   428 method_setup algebra =
```
```   429 {*
```
```   430 let
```
```   431  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```   432  val addN = "add"
```
```   433  val delN = "del"
```
```   434  val any_keyword = keyword addN || keyword delN
```
```   435  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
```
```   436 in
```
```   437   ((Scan.optional (keyword addN |-- thms) []) --
```
```   438    (Scan.optional (keyword delN |-- thms) [])) >>
```
```   439   (fn (add_ths, del_ths) => fn ctxt =>
```
```   440        SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
```
```   441 end
```
```   442 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
```
```   443 declare dvd_def[algebra]
```
```   444 declare dvd_eq_mod_eq_0[symmetric, algebra]
```
```   445 declare mod_div_trivial[algebra]
```
```   446 declare mod_mod_trivial[algebra]
```
```   447 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
```
```   448 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
```
```   449 declare zmod_zdiv_equality[symmetric,algebra]
```
```   450 declare zdiv_zmod_equality[symmetric, algebra]
```
```   451 declare zdiv_zminus_zminus[algebra]
```
```   452 declare zmod_zminus_zminus[algebra]
```
```   453 declare zdiv_zminus2[algebra]
```
```   454 declare zmod_zminus2[algebra]
```
```   455 declare zdiv_zero[algebra]
```
```   456 declare zmod_zero[algebra]
```
```   457 declare mod_by_1[algebra]
```
```   458 declare div_by_1[algebra]
```
```   459 declare zmod_minus1_right[algebra]
```
```   460 declare zdiv_minus1_right[algebra]
```
```   461 declare mod_div_trivial[algebra]
```
```   462 declare mod_mod_trivial[algebra]
```
```   463 declare mod_mult_self2_is_0[algebra]
```
```   464 declare mod_mult_self1_is_0[algebra]
```
```   465 declare zmod_eq_0_iff[algebra]
```
```   466 declare dvd_0_left_iff[algebra]
```
```   467 declare zdvd1_eq[algebra]
```
```   468 declare zmod_eq_dvd_iff[algebra]
```
```   469 declare nat_mod_eq_iff[algebra]
```
```   470
```
```   471 subsection{* Groebner Bases for fields *}
```
```   472
```
```   473 interpretation class_fieldgb:
```
```   474   fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
```
```   475
```
```   476 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
```
```   477 lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
```
```   478   by simp
```
```   479 lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
```
```   480   by simp
```
```   481 lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
```
```   482   by simp
```
```   483 lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
```
```   484   by simp
```
```   485
```
```   486 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
```
```   487
```
```   488 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
```
```   489   by (simp add: add_divide_distrib)
```
```   490 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
```
```   491   by (simp add: add_divide_distrib)
```
```   492 ML{* let open Conv in fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))   (@{thm divide_inverse} RS sym)end*}
```
```   493 ML{*
```
```   494 local
```
```   495  val zr = @{cpat "0"}
```
```   496  val zT = ctyp_of_term zr
```
```   497  val geq = @{cpat "op ="}
```
```   498  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
```
```   499  val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
```
```   500  val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
```
```   501  val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
```
```   502
```
```   503  fun prove_nz ss T t =
```
```   504     let
```
```   505       val z = instantiate_cterm ([(zT,T)],[]) zr
```
```   506       val eq = instantiate_cterm ([(eqT,T)],[]) geq
```
```   507       val th = Simplifier.rewrite (ss addsimps simp_thms)
```
```   508            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
```
```   509                   (Thm.capply (Thm.capply eq t) z)))
```
```   510     in equal_elim (symmetric th) TrueI
```
```   511     end
```
```   512
```
```   513  fun proc phi ss ct =
```
```   514   let
```
```   515     val ((x,y),(w,z)) =
```
```   516          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
```
```   517     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
```
```   518     val T = ctyp_of_term x
```
```   519     val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
```
```   520     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
```
```   521   in SOME (implies_elim (implies_elim th y_nz) z_nz)
```
```   522   end
```
```   523   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
```
```   524
```
```   525  fun proc2 phi ss ct =
```
```   526   let
```
```   527     val (l,r) = Thm.dest_binop ct
```
```   528     val T = ctyp_of_term l
```
```   529   in (case (term_of l, term_of r) of
```
```   530       (Const(@{const_name "HOL.divide"},_)\$_\$_, _) =>
```
```   531         let val (x,y) = Thm.dest_binop l val z = r
```
```   532             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
```
```   533             val ynz = prove_nz ss T y
```
```   534         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
```
```   535         end
```
```   536      | (_, Const (@{const_name "HOL.divide"},_)\$_\$_) =>
```
```   537         let val (x,y) = Thm.dest_binop r val z = l
```
```   538             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
```
```   539             val ynz = prove_nz ss T y
```
```   540         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
```
```   541         end
```
```   542      | _ => NONE)
```
```   543   end
```
```   544   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
```
```   545
```
```   546  fun is_number (Const(@{const_name "HOL.divide"},_)\$a\$b) = is_number a andalso is_number b
```
```   547    | is_number t = can HOLogic.dest_number t
```
```   548
```
```   549  val is_number = is_number o term_of
```
```   550
```
```   551  fun proc3 phi ss ct =
```
```   552   (case term_of ct of
```
```   553     Const(@{const_name HOL.less},_)\$(Const(@{const_name "HOL.divide"},_)\$_\$_)\$_ =>
```
```   554       let
```
```   555         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
```
```   556         val _ = map is_number [a,b,c]
```
```   557         val T = ctyp_of_term c
```
```   558         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
```
```   559       in SOME (mk_meta_eq th) end
```
```   560   | Const(@{const_name HOL.less_eq},_)\$(Const(@{const_name "HOL.divide"},_)\$_\$_)\$_ =>
```
```   561       let
```
```   562         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
```
```   563         val _ = map is_number [a,b,c]
```
```   564         val T = ctyp_of_term c
```
```   565         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
```
```   566       in SOME (mk_meta_eq th) end
```
```   567   | Const("op =",_)\$(Const(@{const_name "HOL.divide"},_)\$_\$_)\$_ =>
```
```   568       let
```
```   569         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
```
```   570         val _ = map is_number [a,b,c]
```
```   571         val T = ctyp_of_term c
```
```   572         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
```
```   573       in SOME (mk_meta_eq th) end
```
```   574   | Const(@{const_name HOL.less},_)\$_\$(Const(@{const_name "HOL.divide"},_)\$_\$_) =>
```
```   575     let
```
```   576       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
```
```   577         val _ = map is_number [a,b,c]
```
```   578         val T = ctyp_of_term c
```
```   579         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
```
```   580       in SOME (mk_meta_eq th) end
```
```   581   | Const(@{const_name HOL.less_eq},_)\$_\$(Const(@{const_name "HOL.divide"},_)\$_\$_) =>
```
```   582     let
```
```   583       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
```
```   584         val _ = map is_number [a,b,c]
```
```   585         val T = ctyp_of_term c
```
```   586         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
```
```   587       in SOME (mk_meta_eq th) end
```
```   588   | Const("op =",_)\$_\$(Const(@{const_name "HOL.divide"},_)\$_\$_) =>
```
```   589     let
```
```   590       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
```
```   591         val _ = map is_number [a,b,c]
```
```   592         val T = ctyp_of_term c
```
```   593         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
```
```   594       in SOME (mk_meta_eq th) end
```
```   595   | _ => NONE)
```
```   596   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
```
```   597
```
```   598 val add_frac_frac_simproc =
```
```   599        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
```
```   600                      name = "add_frac_frac_simproc",
```
```   601                      proc = proc, identifier = []}
```
```   602
```
```   603 val add_frac_num_simproc =
```
```   604        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
```
```   605                      name = "add_frac_num_simproc",
```
```   606                      proc = proc2, identifier = []}
```
```   607
```
```   608 val ord_frac_simproc =
```
```   609   make_simproc
```
```   610     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
```
```   611              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
```
```   612              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
```
```   613              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
```
```   614              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
```
```   615              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
```
```   616              name = "ord_frac_simproc", proc = proc3, identifier = []}
```
```   617
```
```   618 local
```
```   619 open Conv
```
```   620 in
```
```   621
```
```   622 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
```
```   623            @{thm "divide_Numeral1"},
```
```   624            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
```
```   625            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
```
```   626            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
```
```   627            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
```
```   628            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
```
```   629            @{thm "diff_def"}, @{thm "minus_divide_left"},
```
```   630            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
```
```   631            @{thm divide_inverse} RS sym, @{thm inverse_divide},
```
```   632            fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))
```
```   633            (@{thm divide_inverse} RS sym)]
```
```   634
```
```   635 val comp_conv = (Simplifier.rewrite
```
```   636 (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
```
```   637               addsimps ths addsimps simp_thms
```
```   638               addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
```
```   639                addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
```
```   640                             ord_frac_simproc]
```
```   641                 addcongs [@{thm "if_weak_cong"}]))
```
```   642 then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
```
```   643   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
```
```   644 end
```
```   645
```
```   646 fun numeral_is_const ct =
```
```   647   case term_of ct of
```
```   648    Const (@{const_name "HOL.divide"},_) \$ a \$ b =>
```
```   649      can HOLogic.dest_number a andalso can HOLogic.dest_number b
```
```   650  | Const (@{const_name "HOL.inverse"},_)\$t => can HOLogic.dest_number t
```
```   651  | t => can HOLogic.dest_number t
```
```   652
```
```   653 fun dest_const ct = ((case term_of ct of
```
```   654    Const (@{const_name "HOL.divide"},_) \$ a \$ b=>
```
```   655     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
```
```   656  | Const (@{const_name "HOL.inverse"},_)\$t =>
```
```   657                Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
```
```   658  | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
```
```   659    handle TERM _ => error "ring_dest_const")
```
```   660
```
```   661 fun mk_const phi cT x =
```
```   662  let val (a, b) = Rat.quotient_of_rat x
```
```   663  in if b = 1 then Numeral.mk_cnumber cT a
```
```   664     else Thm.capply
```
```   665          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
```
```   666                      (Numeral.mk_cnumber cT a))
```
```   667          (Numeral.mk_cnumber cT b)
```
```   668   end
```
```   669
```
```   670 in
```
```   671  val field_comp_conv = comp_conv;
```
```   672  val fieldgb_declaration =
```
```   673   NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
```
```   674    {is_const = K numeral_is_const,
```
```   675     dest_const = K dest_const,
```
```   676     mk_const = mk_const,
```
```   677     conv = K (K comp_conv)}
```
```   678 end;
```
```   679 *}
```
```   680
```
```   681 declaration fieldgb_declaration
```
```   682
```
```   683 end
```