src/HOL/Groebner_Basis.thy
 changeset 36751 7f1da69cacb3 parent 36720 41da7025e59c child 36752 cf558aeb35b0
```     1.1 --- a/src/HOL/Groebner_Basis.thy	Fri May 07 10:00:24 2010 +0200
1.2 +++ b/src/HOL/Groebner_Basis.thy	Fri May 07 15:05:52 2010 +0200
1.3 @@ -2,341 +2,14 @@
1.4      Author:     Amine Chaieb, TU Muenchen
1.5  *)
1.6
1.7 -header {* Semiring normalization and Groebner Bases *}
1.8 +header {* Groebner bases *}
1.9
1.10  theory Groebner_Basis
1.11 -imports Numeral_Simprocs Nat_Transfer
1.12 +imports Semiring_Normalization
1.13  uses
1.14 -  "Tools/Groebner_Basis/normalizer.ML"
1.15    ("Tools/Groebner_Basis/groebner.ML")
1.16  begin
1.17
1.18 -subsection {* Semiring normalization *}
1.19 -
1.20 -setup Normalizer.setup
1.21 -
1.22 -locale normalizing_semiring =
1.23 -  fixes add mul pwr r0 r1
1.26 -    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
1.27 -    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
1.28 -    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
1.29 -    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
1.30 -begin
1.31 -
1.32 -lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
1.33 -proof (induct p)
1.34 -  case 0
1.35 -  then show ?case by (auto simp add: pwr_0 mul_1)
1.36 -next
1.37 -  case Suc
1.38 -  from this [symmetric] show ?case
1.39 -    by (auto simp add: pwr_Suc mul_1 mul_a)
1.40 -qed
1.41 -
1.42 -lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
1.43 -proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
1.44 -  fix q x y
1.45 -  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
1.46 -  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
1.47 -    by (simp add: mul_a)
1.48 -  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
1.49 -  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
1.50 -  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
1.51 -    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
1.52 -qed
1.53 -
1.54 -lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
1.55 -proof (induct p arbitrary: q)
1.56 -  case 0
1.57 -  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
1.58 -next
1.59 -  case Suc
1.60 -  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
1.61 -qed
1.62 -
1.63 -lemma semiring_ops:
1.64 -  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
1.65 -    and "TERM r0" and "TERM r1" .
1.66 -
1.67 -lemma semiring_rules:
1.68 -  "add (mul a m) (mul b m) = mul (add a b) m"
1.69 -  "add (mul a m) m = mul (add a r1) m"
1.70 -  "add m (mul a m) = mul (add a r1) m"
1.72 -  "add r0 a = a"
1.73 -  "add a r0 = a"
1.74 -  "mul a b = mul b a"
1.75 -  "mul (add a b) c = add (mul a c) (mul b c)"
1.76 -  "mul r0 a = r0"
1.77 -  "mul a r0 = r0"
1.78 -  "mul r1 a = a"
1.79 -  "mul a r1 = a"
1.80 -  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
1.81 -  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
1.82 -  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
1.83 -  "mul (mul lx ly) rx = mul (mul lx rx) ly"
1.84 -  "mul (mul lx ly) rx = mul lx (mul ly rx)"
1.85 -  "mul lx (mul rx ry) = mul (mul lx rx) ry"
1.86 -  "mul lx (mul rx ry) = mul rx (mul lx ry)"
1.93 -  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
1.94 -  "mul x (pwr x q) = pwr x (Suc q)"
1.95 -  "mul (pwr x q) x = pwr x (Suc q)"
1.96 -  "mul x x = pwr x 2"
1.97 -  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
1.98 -  "pwr (pwr x p) q = pwr x (p * q)"
1.99 -  "pwr x 0 = r1"
1.100 -  "pwr x 1 = x"
1.101 -  "mul x (add y z) = add (mul x y) (mul x z)"
1.102 -  "pwr x (Suc q) = mul x (pwr x q)"
1.103 -  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
1.104 -  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
1.105 -proof -
1.106 -  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
1.107 -next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
1.108 -next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
1.109 -next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
1.110 -next show "add r0 a = a" using add_0 by simp
1.112 -next show "mul a b = mul b a" using mul_c by simp
1.113 -next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
1.114 -next show "mul r0 a = r0" using mul_0 by simp
1.115 -next show "mul a r0 = r0" using mul_0 mul_c by simp
1.116 -next show "mul r1 a = a" using mul_1 by simp
1.117 -next show "mul a r1 = a" using mul_1 mul_c by simp
1.118 -next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
1.119 -    using mul_c mul_a by simp
1.120 -next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
1.121 -    using mul_a by simp
1.122 -next
1.123 -  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
1.124 -  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
1.125 -  finally
1.126 -  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
1.127 -    using mul_c by simp
1.128 -next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
1.129 -next
1.130 -  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
1.131 -next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
1.132 -next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
1.141 -next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
1.142 -next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
1.143 -next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
1.144 -next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
1.145 -next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
1.146 -next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
1.147 -next show "pwr x 0 = r1" using pwr_0 .
1.148 -next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
1.149 -next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
1.150 -next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
1.151 -next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
1.152 -next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
1.153 -    by (simp add: nat_number' pwr_Suc mul_pwr)
1.154 -qed
1.155 -
1.156 -
1.157 -lemmas normalizing_semiring_axioms' =
1.158 -  normalizing_semiring_axioms [normalizer
1.159 -    semiring ops: semiring_ops
1.160 -    semiring rules: semiring_rules]
1.161 -
1.162 -end
1.163 -
1.164 -sublocale comm_semiring_1
1.165 -  < normalizing!: normalizing_semiring plus times power zero one
1.166 -proof
1.168 -
1.169 -declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
1.170 -
1.171 -locale normalizing_ring = normalizing_semiring +
1.172 -  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.173 -    and neg :: "'a \<Rightarrow> 'a"
1.174 -  assumes neg_mul: "neg x = mul (neg r1) x"
1.175 -    and sub_add: "sub x y = add x (neg y)"
1.176 -begin
1.177 -
1.178 -lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
1.179 -
1.180 -lemmas ring_rules = neg_mul sub_add
1.181 -
1.182 -lemmas normalizing_ring_axioms' =
1.183 -  normalizing_ring_axioms [normalizer
1.184 -    semiring ops: semiring_ops
1.185 -    semiring rules: semiring_rules
1.186 -    ring ops: ring_ops
1.187 -    ring rules: ring_rules]
1.188 -
1.189 -end
1.190 -
1.191 -sublocale comm_ring_1
1.192 -  < normalizing!: normalizing_ring plus times power zero one minus uminus
1.193 -proof
1.195 -
1.196 -declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
1.197 -
1.198 -locale normalizing_field = normalizing_ring +
1.199 -  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.200 -    and inverse:: "'a \<Rightarrow> 'a"
1.201 -  assumes divide_inverse: "divide x y = mul x (inverse y)"
1.202 -     and inverse_divide: "inverse x = divide r1 x"
1.203 -begin
1.204 -
1.205 -lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
1.206 -
1.207 -lemmas field_rules = divide_inverse inverse_divide
1.208 -
1.209 -lemmas normalizing_field_axioms' =
1.210 -  normalizing_field_axioms [normalizer
1.211 -    semiring ops: semiring_ops
1.212 -    semiring rules: semiring_rules
1.213 -    ring ops: ring_ops
1.214 -    ring rules: ring_rules
1.215 -    field ops: field_ops
1.216 -    field rules: field_rules]
1.217 -
1.218 -end
1.219 -
1.220 -locale normalizing_semiring_cancel = normalizing_semiring +
1.222 -  and add_mul_solve: "add (mul w y) (mul x z) =
1.223 -    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
1.224 -begin
1.225 -
1.226 -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)"
1.227 -proof-
1.228 -  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
1.229 -  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
1.230 -    using add_mul_solve by blast
1.231 -  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)"
1.232 -    by simp
1.233 -qed
1.234 -
1.235 -lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
1.236 -  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
1.237 -proof(clarify)
1.238 -  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
1.239 -    and eq: "add b (mul r c) = add b (mul r d)"
1.240 -  hence "mul r c = mul r d" using cnd add_cancel by simp
1.241 -  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
1.242 -    using mul_0 add_cancel by simp
1.243 -  thus "False" using add_mul_solve nz cnd by simp
1.244 -qed
1.245 -
1.246 -lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
1.247 -proof-
1.250 -qed
1.251 -
1.252 -declare normalizing_semiring_axioms' [normalizer del]
1.253 -
1.254 -lemmas normalizing_semiring_cancel_axioms' =
1.255 -  normalizing_semiring_cancel_axioms [normalizer
1.256 -    semiring ops: semiring_ops
1.257 -    semiring rules: semiring_rules
1.258 -    idom rules: noteq_reduce add_scale_eq_noteq]
1.259 -
1.260 -end
1.261 -
1.262 -locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
1.263 -  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
1.264 -begin
1.265 -
1.266 -declare normalizing_ring_axioms' [normalizer del]
1.267 -
1.268 -lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
1.269 -  semiring ops: semiring_ops
1.270 -  semiring rules: semiring_rules
1.271 -  ring ops: ring_ops
1.272 -  ring rules: ring_rules
1.273 -  idom rules: noteq_reduce add_scale_eq_noteq
1.274 -  ideal rules: subr0_iff add_r0_iff]
1.275 -
1.276 -end
1.277 -
1.278 -sublocale idom
1.279 -  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
1.280 -proof
1.281 -  fix w x y z
1.282 -  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
1.283 -  proof
1.284 -    assume "w * y + x * z = w * z + x * y"
1.285 -    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
1.286 -    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
1.287 -    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
1.288 -    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
1.289 -    then show "w = x \<or> y = z" by auto
1.292 -
1.293 -declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
1.294 -
1.295 -interpretation normalizing_nat!: normalizing_semiring_cancel
1.296 -  "op +" "op *" "op ^" "0::nat" "1"
1.297 -proof (unfold_locales, simp add: algebra_simps)
1.298 -  fix w x y z ::"nat"
1.299 -  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
1.300 -    hence "y < z \<or> y > z" by arith
1.301 -    moreover {
1.302 -      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
1.303 -      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
1.304 -      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
1.305 -      hence "x*k = w*k" by simp
1.306 -      hence "w = x" using kp by simp }
1.307 -    moreover {
1.308 -      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
1.309 -      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
1.310 -      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
1.311 -      hence "w*k = x*k" by simp
1.312 -      hence "w = x" using kp by simp }
1.313 -    ultimately have "w=x" by blast }
1.314 -  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
1.315 -qed
1.316 -
1.317 -declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
1.318 -
1.319 -locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
1.320 -begin
1.321 -
1.322 -declare normalizing_field_axioms' [normalizer del]
1.323 -
1.324 -lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
1.325 -  semiring ops: semiring_ops
1.326 -  semiring rules: semiring_rules
1.327 -  ring ops: ring_ops
1.328 -  ring rules: ring_rules
1.329 -  field ops: field_ops
1.330 -  field rules: field_rules
1.331 -  idom rules: noteq_reduce add_scale_eq_noteq
1.332 -  ideal rules: subr0_iff add_r0_iff]
1.333 -
1.334 -end
1.335 -
1.336 -sublocale field
1.337 -  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
1.338 -proof
1.340 -
1.341 -declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
1.342 -
1.343 -
1.344  subsection {* Groebner Bases *}
1.345
1.346  lemmas bool_simps = simp_thms(1-34)
1.347 @@ -367,6 +40,11 @@
1.348
1.349  setup Algebra_Simplification.setup
1.350
1.351 +use "Tools/Groebner_Basis/groebner.ML"
1.352 +
1.353 +method_setup algebra = Groebner.algebra_method
1.354 +  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
1.355 +
1.356  declare dvd_def[algebra]
1.357  declare dvd_eq_mod_eq_0[symmetric, algebra]
1.358  declare mod_div_trivial[algebra]
1.359 @@ -395,9 +73,4 @@
1.360  declare zmod_eq_dvd_iff[algebra]
1.361  declare nat_mod_eq_iff[algebra]
1.362
1.363 -use "Tools/Groebner_Basis/groebner.ML"
1.364 -
1.365 -method_setup algebra = Groebner.algebra_method
1.366 -  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
1.367 -
1.368  end
```