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(* Title: HOL/Groebner_Basis.thy 
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Author: Amine Chaieb, TU Muenchen 

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*) 

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header {* Semiring normalization and Groebner Bases *} 

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theory Groebner_Basis 
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imports Numeral_Simprocs Nat_Transfer 
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uses 
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"Tools/Groebner_Basis/normalizer.ML" 
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("Tools/Groebner_Basis/groebner.ML") 
23252  12 
begin 
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subsection {* Semiring normalization *} 

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setup Normalizer.setup 
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locale normalizing_semiring = 
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fixes add mul pwr r0 r1 
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assumes add_a:"(add x (add y z) = add (add x y) z)" 

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and add_c: "add x y = add y x" and add_0:"add r0 x = x" 

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and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x" 

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and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0" 

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and mul_d:"mul x (add y z) = add (mul x y) (mul x z)" 

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and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)" 

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begin 

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lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)" 

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proof (induct p) 

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case 0 

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then show ?case by (auto simp add: pwr_0 mul_1) 

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next 

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case Suc 

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from this [symmetric] show ?case 

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by (auto simp add: pwr_Suc mul_1 mul_a) 

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qed 

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lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)" 

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proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1) 

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fix q x y 

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assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)" 

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have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))" 

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by (simp add: mul_a) 

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also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c) 

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also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a) 

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finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) = 

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mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c) 

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qed 

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lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)" 

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proof (induct p arbitrary: q) 

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case 0 

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show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto 

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next 

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case Suc 

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thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc) 

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qed 

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lemma semiring_ops: 

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shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)" 

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and "TERM r0" and "TERM r1" . 
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lemma semiring_rules: 

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"add (mul a m) (mul b m) = mul (add a b) m" 

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"add (mul a m) m = mul (add a r1) m" 

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"add m (mul a m) = mul (add a r1) m" 

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"add m m = mul (add r1 r1) m" 

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"add r0 a = a" 

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"add a r0 = a" 

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"mul a b = mul b a" 

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"mul (add a b) c = add (mul a c) (mul b c)" 

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"mul r0 a = r0" 

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"mul a r0 = r0" 

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"mul r1 a = a" 

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"mul a r1 = a" 

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"mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" 

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"mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" 

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"mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" 

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"mul (mul lx ly) rx = mul (mul lx rx) ly" 

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"mul (mul lx ly) rx = mul lx (mul ly rx)" 

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"mul lx (mul rx ry) = mul (mul lx rx) ry" 

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"mul lx (mul rx ry) = mul rx (mul lx ry)" 

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"add (add a b) (add c d) = add (add a c) (add b d)" 

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"add (add a b) c = add a (add b c)" 

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"add a (add c d) = add c (add a d)" 

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"add (add a b) c = add (add a c) b" 

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"add a c = add c a" 

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"add a (add c d) = add (add a c) d" 

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"mul (pwr x p) (pwr x q) = pwr x (p + q)" 

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"mul x (pwr x q) = pwr x (Suc q)" 

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"mul (pwr x q) x = pwr x (Suc q)" 

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"mul x x = pwr x 2" 

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"pwr (mul x y) q = mul (pwr x q) (pwr y q)" 

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"pwr (pwr x p) q = pwr x (p * q)" 

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"pwr x 0 = r1" 

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"pwr x 1 = x" 

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"mul x (add y z) = add (mul x y) (mul x z)" 

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"pwr x (Suc q) = mul x (pwr x q)" 

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"pwr x (2*n) = mul (pwr x n) (pwr x n)" 

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"pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))" 

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proof  

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show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp 

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next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp 

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next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp 

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next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp 

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next show "add r0 a = a" using add_0 by simp 

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next show "add a r0 = a" using add_0 add_c by simp 

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next show "mul a b = mul b a" using mul_c by simp 

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next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp 

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next show "mul r0 a = r0" using mul_0 by simp 

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next show "mul a r0 = r0" using mul_0 mul_c by simp 

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next show "mul r1 a = a" using mul_1 by simp 

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next show "mul a r1 = a" using mul_1 mul_c by simp 

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next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" 

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using mul_c mul_a by simp 

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next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" 

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using mul_a by simp 

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next 

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have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c) 

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also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp 

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finally 

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show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" 

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using mul_c by simp 

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next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp 

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next 

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show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a) 

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next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a ) 

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next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c) 

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next show "add (add a b) (add c d) = add (add a c) (add b d)" 

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using add_c add_a by simp 

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next show "add (add a b) c = add a (add b c)" using add_a by simp 

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next show "add a (add c d) = add c (add a d)" 

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apply (simp add: add_a) by (simp only: add_c) 

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next show "add (add a b) c = add (add a c) b" using add_a add_c by simp 

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next show "add a c = add c a" by (rule add_c) 

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next show "add a (add c d) = add (add a c) d" using add_a by simp 

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next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr) 

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next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp 

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next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp 

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next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c) 
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next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul) 
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next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr) 

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next show "pwr x 0 = r1" using pwr_0 . 

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next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c) 
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next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp 
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next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp 

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next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr) 
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next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))" 
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by (simp add: nat_number' pwr_Suc mul_pwr) 
23252  150 
qed 
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lemmas normalizing_semiring_axioms' = 
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normalizing_semiring_axioms [normalizer 
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semiring ops: semiring_ops 
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semiring rules: semiring_rules] 
23252  157 

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end 

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sublocale comm_semiring_1 
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< normalizing!: normalizing_semiring plus times power zero one 
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proof 
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qed (simp_all add: algebra_simps) 
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declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *} 
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locale normalizing_ring = normalizing_semiring + 
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fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 
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and neg :: "'a \<Rightarrow> 'a" 

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assumes neg_mul: "neg x = mul (neg r1) x" 

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and sub_add: "sub x y = add x (neg y)" 

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begin 

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lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" . 
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lemmas ring_rules = neg_mul sub_add 

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lemmas normalizing_ring_axioms' = 
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normalizing_ring_axioms [normalizer 
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semiring ops: semiring_ops 
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semiring rules: semiring_rules 

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ring ops: ring_ops 

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ring rules: ring_rules] 

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end 

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sublocale comm_ring_1 
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< normalizing!: normalizing_ring plus times power zero one minus uminus 

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proof 

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qed (simp_all add: diff_minus) 

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declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *} 
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locale normalizing_field = normalizing_ring + 
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fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 
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and inverse:: "'a \<Rightarrow> 'a" 

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assumes divide_inverse: "divide x y = mul x (inverse y)" 
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and inverse_divide: "inverse x = divide r1 x" 

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begin 
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lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" . 
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lemmas field_rules = divide_inverse inverse_divide 

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lemmas normalizing_field_axioms' = 
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normalizing_field_axioms [normalizer 
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semiring ops: semiring_ops 
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semiring rules: semiring_rules 

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ring ops: ring_ops 

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ring rules: ring_rules 
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field ops: field_ops 

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field rules: field_rules] 

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end 

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locale normalizing_semiring_cancel = normalizing_semiring + 
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assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z" 
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and add_mul_solve: "add (mul w y) (mul x z) = 

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add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z" 

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begin 

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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)" 

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proof 

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have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp 

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also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)" 

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using add_mul_solve by blast 

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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)" 

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by simp 

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qed 

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lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk> 

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\<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)" 

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proof(clarify) 

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assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d" 

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and eq: "add b (mul r c) = add b (mul r d)" 

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hence "mul r c = mul r d" using cnd add_cancel by simp 

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hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)" 

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using mul_0 add_cancel by simp 

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thus "False" using add_mul_solve nz cnd by simp 

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qed 

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lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0" 
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proof 
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have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel) 
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thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0) 
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qed 
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declare normalizing_semiring_axioms' [normalizer del] 
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lemmas normalizing_semiring_cancel_axioms' = 
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normalizing_semiring_cancel_axioms [normalizer 
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semiring ops: semiring_ops 
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semiring rules: semiring_rules 
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idom rules: noteq_reduce add_scale_eq_noteq] 
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end 

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locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
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assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y" 
23252  260 
begin 
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declare normalizing_ring_axioms' [normalizer del] 
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lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer 
23252  265 
semiring ops: semiring_ops 
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semiring rules: semiring_rules 

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ring ops: ring_ops 

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ring rules: ring_rules 

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idom rules: noteq_reduce add_scale_eq_noteq 
26314  270 
ideal rules: subr0_iff add_r0_iff] 
23252  271 

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end 

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36720  274 
sublocale idom 
275 
< normalizing!: normalizing_ring_cancel plus times power zero one minus uminus 

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proof 

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fix w x y z 

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show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" 

279 
proof 

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assume "w * y + x * z = w * z + x * y" 

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then have "w * y + x * z  w * z  x * y = 0" by (simp add: algebra_simps) 

282 
then have "w * (y  z)  x * (y  z) = 0" by (simp add: algebra_simps) 

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then have "(y  z) * (w  x) = 0" by (simp add: algebra_simps) 

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then have "y  z = 0 \<or> w  x = 0" by (rule divisors_zero) 

285 
then show "w = x \<or> y = z" by auto 

286 
qed (auto simp add: add_ac) 

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qed (simp_all add: algebra_simps) 

23252  288 

36720  289 
declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *} 
23252  290 

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interpretation normalizing_nat!: normalizing_semiring_cancel 
29223  292 
"op +" "op *" "op ^" "0::nat" "1" 
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proof (unfold_locales, simp add: algebra_simps) 
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fix w x y z ::"nat" 
295 
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 

296 
hence "y < z \<or> y > z" by arith 

297 
moreover { 

298 
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z  y" in exI, auto) 

299 
then obtain k where kp: "k>0" and yz:"z = y + k" by blast 

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from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps) 
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hence "x*k = w*k" by simp 
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hence "w = x" using kp by simp } 
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moreover { 
304 
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y  z" in exI, auto) 

305 
then obtain k where kp: "k>0" and yz:"y = z + k" by blast 

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from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) 
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hence "w*k = x*k" by simp 
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hence "w = x" using kp by simp } 
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ultimately have "w=x" by blast } 
310 
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto 

311 
qed 

312 

36720  313 
declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *} 
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locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field 
23327  316 
begin 
317 

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declare normalizing_field_axioms' [normalizer del] 
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lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer 
23327  321 
semiring ops: semiring_ops 
322 
semiring rules: semiring_rules 

323 
ring ops: ring_ops 

324 
ring rules: ring_rules 

30866  325 
field ops: field_ops 
326 
field rules: field_rules 

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b3a485b98963
(1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
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idom rules: noteq_reduce add_scale_eq_noteq 
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ideal rules: subr0_iff add_r0_iff] 
329 

23327  330 
end 
331 

36720  332 
sublocale field 
333 
< normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse 

334 
proof 

335 
qed (simp_all add: divide_inverse) 

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36720  337 
declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *} 
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338 

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339 

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subsection {* Groebner Bases *} 
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341 

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lemmas bool_simps = simp_thms(134) 
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343 

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lemma dnf: 
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"(P & (Q  R)) = ((P&Q)  (P&R))" "((Q  R) & P) = ((Q&P)  (R&P))" 
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"(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)" 
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by blast+ 
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348 

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lemmas weak_dnf_simps = dnf bool_simps 
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350 

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lemma nnf_simps: 
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"(\<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)" 
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"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P" 
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354 
by blast+ 
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355 

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lemma PFalse: 
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"P \<equiv> False \<Longrightarrow> \<not> P" 
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"\<not> P \<Longrightarrow> (P \<equiv> False)" 
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359 
by auto 
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360 

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361 
ML {* 
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structure Algebra_Simplification = Named_Thms( 
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val name = "algebra" 
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val description = "presimplification rules for algebraic methods" 
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) 
28402  366 
*} 
367 

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setup Algebra_Simplification.setup 
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369 

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declare dvd_def[algebra] 
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declare dvd_eq_mod_eq_0[symmetric, algebra] 
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declare mod_div_trivial[algebra] 
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declare mod_mod_trivial[algebra] 
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declare conjunct1[OF DIVISION_BY_ZERO, algebra] 
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declare conjunct2[OF DIVISION_BY_ZERO, algebra] 
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declare zmod_zdiv_equality[symmetric,algebra] 
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declare zdiv_zmod_equality[symmetric, algebra] 
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declare zdiv_zminus_zminus[algebra] 
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declare zmod_zminus_zminus[algebra] 
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declare zdiv_zminus2[algebra] 
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declare zmod_zminus2[algebra] 
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declare zdiv_zero[algebra] 
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declare zmod_zero[algebra] 
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declare mod_by_1[algebra] 
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declare div_by_1[algebra] 
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declare zmod_minus1_right[algebra] 
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declare zdiv_minus1_right[algebra] 
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declare mod_div_trivial[algebra] 
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declare mod_mod_trivial[algebra] 
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declare mod_mult_self2_is_0[algebra] 
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declare mod_mult_self1_is_0[algebra] 
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declare zmod_eq_0_iff[algebra] 
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declare dvd_0_left_iff[algebra] 
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declare zdvd1_eq[algebra] 
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declare zmod_eq_dvd_iff[algebra] 
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declare nat_mod_eq_iff[algebra] 
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397 

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use "Tools/Groebner_Basis/groebner.ML" 
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399 

36720  400 
method_setup algebra = Groebner.algebra_method 
401 
"solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases" 

28402  402 

403 
end 