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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") 
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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) 
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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] 
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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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lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)" 

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by simp 
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lemmas semiring_norm = 
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Let_def arith_simps nat_arith rel_simps neg_simps if_False 
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if_True add_0 add_Suc add_number_of_left mult_number_of_left 
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numeral_1_eq_1[symmetric] Suc_eq_plus1 
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numeral_0_eq_0[symmetric] numerals[symmetric] 
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iszero_simps not_iszero_Numeral1 

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ML {* 

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local 
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fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct); 
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fun int_of_rat x = 
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(case Rat.quotient_of_rat x of (i, 1) => i 

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 _ => error "int_of_rat: bad int"); 

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val numeral_conv = 
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Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv 

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Simplifier.rewrite (HOL_basic_ss addsimps 

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(@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(12)})); 

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in 

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fun normalizer_funs' key = 
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Normalizer.funs key 
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{is_const = fn phi => numeral_is_const, 
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dest_const = fn phi => fn ct => 

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Rat.rat_of_int (snd 

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(HOLogic.dest_number (Thm.term_of ct) 

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handle TERM _ => error "ring_dest_const")), 

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mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x), 
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conv = fn phi => K numeral_conv} 
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end 

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*} 
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declaration {* normalizer_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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(*FIXME add class*) 
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interpretation normalizing!: normalizing_ring plus times power 
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"0::'a::{comm_semiring_1,number_ring}" 1 minus uminus proof 
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qed simp_all 
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declaration {* normalizer_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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293 
idom rules: noteq_reduce add_scale_eq_noteq] 
23252  294 

295 
end 

296 

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297 
locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
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298 
assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y" 
23252  299 
begin 
300 

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301 
declare normalizing_ring_axioms' [normalizer del] 
23252  302 

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303 
lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer 
23252  304 
semiring ops: semiring_ops 
305 
semiring rules: semiring_rules 

306 
ring ops: ring_ops 

307 
ring rules: ring_rules 

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308 
idom rules: noteq_reduce add_scale_eq_noteq 
26314  309 
ideal rules: subr0_iff add_r0_iff] 
23252  310 

311 
end 

312 

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313 
lemma (in no_zero_divisors) prod_eq_zero_eq_zero: 
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314 
assumes "a * b = 0" and "a \<noteq> 0" 
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315 
shows "b = 0" 
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316 
proof (rule classical) 
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317 
assume "b \<noteq> 0" with `a \<noteq> 0` no_zero_divisors have "a * b \<noteq> 0" by blast 
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318 
with `a * b = 0` show ?thesis by simp 
23252  319 
qed 
320 

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321 
(*FIXME introduce class*) 
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322 
interpretation normalizing!: normalizing_ring_cancel 
31017  323 
"op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op " "uminus" 
35216  324 
proof(unfold_locales, simp add: algebra_simps, auto) 
31017  325 
fix w x y z ::"'a::{idom,number_ring}" 
23252  326 
assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 
327 
hence ynz': "y  z \<noteq> 0" by simp 

328 
from p have "w * y + x* z  w*z  x*y = 0" by simp 

29667  329 
hence "w* (y  z)  x * (y  z) = 0" by (simp add: algebra_simps) 
330 
hence "(y  z) * (w  x) = 0" by (simp add: algebra_simps) 

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331 
with prod_eq_zero_eq_zero [OF _ ynz'] 
23252  332 
have "w  x = 0" by blast 
333 
thus "w = x" by simp 

334 
qed 

335 

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336 
declaration {* normalizer_funs' @{thm normalizing.normalizing_ring_cancel_axioms'} *} 
23252  337 

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338 
interpretation normalizing_nat!: normalizing_semiring_cancel 
29223  339 
"op +" "op *" "op ^" "0::nat" "1" 
35216  340 
proof (unfold_locales, simp add: algebra_simps) 
23252  341 
fix w x y z ::"nat" 
342 
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 

343 
hence "y < z \<or> y > z" by arith 

344 
moreover { 

345 
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z  y" in exI, auto) 

346 
then obtain k where kp: "k>0" and yz:"z = y + k" by blast 

29667  347 
from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps) 
23252  348 
hence "x*k = w*k" by simp 
35216  349 
hence "w = x" using kp by simp } 
23252  350 
moreover { 
351 
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y  z" in exI, auto) 

352 
then obtain k where kp: "k>0" and yz:"y = z + k" by blast 

29667  353 
from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) 
23252  354 
hence "w*k = x*k" by simp 
35216  355 
hence "w = x" using kp by simp } 
23252  356 
ultimately have "w=x" by blast } 
357 
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto 

358 
qed 

359 

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360 
declaration {* normalizer_funs' @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *} 
23252  361 

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362 
locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field 
23327  363 
begin 
364 

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365 
declare normalizing_field_axioms' [normalizer del] 
23327  366 

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367 
lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer 
23327  368 
semiring ops: semiring_ops 
369 
semiring rules: semiring_rules 

370 
ring ops: ring_ops 

371 
ring rules: ring_rules 

30866  372 
field ops: field_ops 
373 
field rules: field_rules 

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374 
idom rules: noteq_reduce add_scale_eq_noteq 
26314  375 
ideal rules: subr0_iff add_r0_iff] 
376 

23327  377 
end 
378 

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379 
(*FIXME introduce class*) 
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380 
interpretation normalizing!: normalizing_field_cancel "op +" "op *" "op ^" 
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381 
"0::'a::{field,number_ring}" "1" "op " "uminus" "op /" "inverse" 
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382 
apply (unfold_locales) by (simp_all add: divide_inverse) 
28402  383 

36409  384 
lemma divide_Numeral1: "(x::'a::{field, number_ring}) / Numeral1 = x" by simp 
385 
lemma divide_Numeral0: "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0" 

28402  386 
by simp 
36409  387 
lemma mult_frac_frac: "((x::'a::field_inverse_zero) / y) * (z / w) = (x*z) / (y*w)" 
28402  388 
by simp 
36409  389 
lemma mult_frac_num: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y" 
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390 
by (fact times_divide_eq_left) 
36409  391 
lemma mult_num_frac: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y" 
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392 
by (fact times_divide_eq_left) 
28402  393 

394 
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp 

395 

36409  396 
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::field_inverse_zero) / y + z = (x + z*y) / y" 
28402  397 
by (simp add: add_divide_distrib) 
36409  398 
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::field_inverse_zero) / y = (x + z*y) / y" 
28402  399 
by (simp add: add_divide_distrib) 
35084  400 

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401 
ML {* 
28402  402 
local 
403 
val zr = @{cpat "0"} 

404 
val zT = ctyp_of_term zr 

405 
val geq = @{cpat "op ="} 

406 
val eqT = Thm.dest_ctyp (ctyp_of_term geq) > hd 

407 
val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} 

408 
val add_frac_num = mk_meta_eq @{thm "add_frac_num"} 

409 
val add_num_frac = mk_meta_eq @{thm "add_num_frac"} 

410 

411 
fun prove_nz ss T t = 

412 
let 

413 
val z = instantiate_cterm ([(zT,T)],[]) zr 

414 
val eq = instantiate_cterm ([(eqT,T)],[]) geq 

35410  415 
val th = Simplifier.rewrite (ss addsimps @{thms simp_thms}) 
28402  416 
(Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} 
417 
(Thm.capply (Thm.capply eq t) z))) 

418 
in equal_elim (symmetric th) TrueI 

419 
end 

420 

421 
fun proc phi ss ct = 

422 
let 

423 
val ((x,y),(w,z)) = 

424 
(Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct 

425 
val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] 

426 
val T = ctyp_of_term x 

427 
val [y_nz, z_nz] = map (prove_nz ss T) [y, z] 

428 
val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq 

429 
in SOME (implies_elim (implies_elim th y_nz) z_nz) 

430 
end 

431 
handle CTERM _ => NONE  TERM _ => NONE  THM _ => NONE 

432 

433 
fun proc2 phi ss ct = 

434 
let 

435 
val (l,r) = Thm.dest_binop ct 

436 
val T = ctyp_of_term l 

437 
in (case (term_of l, term_of r) of 

35084  438 
(Const(@{const_name Rings.divide},_)$_$_, _) => 
28402  439 
let val (x,y) = Thm.dest_binop l val z = r 
440 
val _ = map (HOLogic.dest_number o term_of) [x,y,z] 

441 
val ynz = prove_nz ss T y 

442 
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz) 

443 
end 

35084  444 
 (_, Const (@{const_name Rings.divide},_)$_$_) => 
28402  445 
let val (x,y) = Thm.dest_binop r val z = l 
446 
val _ = map (HOLogic.dest_number o term_of) [x,y,z] 

447 
val ynz = prove_nz ss T y 

448 
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz) 

449 
end 

450 
 _ => NONE) 

451 
end 

452 
handle CTERM _ => NONE  TERM _ => NONE  THM _ => NONE 

453 

35084  454 
fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b 
28402  455 
 is_number t = can HOLogic.dest_number t 
456 

457 
val is_number = is_number o term_of 

458 

459 
fun proc3 phi ss ct = 

460 
(case term_of ct of 

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461 
Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => 
28402  462 
let 
463 
val ((a,b),c) = Thm.dest_binop ct >> Thm.dest_binop 

464 
val _ = map is_number [a,b,c] 

465 
val T = ctyp_of_term c 

466 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"} 

467 
in SOME (mk_meta_eq th) end 

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468 
 Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => 
28402  469 
let 
470 
val ((a,b),c) = Thm.dest_binop ct >> Thm.dest_binop 

471 
val _ = map is_number [a,b,c] 

472 
val T = ctyp_of_term c 

473 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"} 

474 
in SOME (mk_meta_eq th) end 

35084  475 
 Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => 
28402  476 
let 
477 
val ((a,b),c) = Thm.dest_binop ct >> Thm.dest_binop 

478 
val _ = map is_number [a,b,c] 

479 
val T = ctyp_of_term c 

480 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"} 

481 
in SOME (mk_meta_eq th) end 

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482 
 Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) => 
28402  483 
let 
484 
val (a,(b,c)) = Thm.dest_binop ct > Thm.dest_binop 

485 
val _ = map is_number [a,b,c] 

486 
val T = ctyp_of_term c 

487 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"} 

488 
in SOME (mk_meta_eq th) end 

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489 
 Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) => 
28402  490 
let 
491 
val (a,(b,c)) = Thm.dest_binop ct > Thm.dest_binop 

492 
val _ = map is_number [a,b,c] 

493 
val T = ctyp_of_term c 

494 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"} 

495 
in SOME (mk_meta_eq th) end 

35084  496 
 Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) => 
28402  497 
let 
498 
val (a,(b,c)) = Thm.dest_binop ct > Thm.dest_binop 

499 
val _ = map is_number [a,b,c] 

500 
val T = ctyp_of_term c 

501 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"} 

502 
in SOME (mk_meta_eq th) end 

503 
 _ => NONE) 

504 
handle TERM _ => NONE  CTERM _ => NONE  THM _ => NONE 

505 

506 
val add_frac_frac_simproc = 

507 
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], 

508 
name = "add_frac_frac_simproc", 

509 
proc = proc, identifier = []} 

510 

511 
val add_frac_num_simproc = 

512 
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], 

513 
name = "add_frac_num_simproc", 

514 
proc = proc2, identifier = []} 

515 

516 
val ord_frac_simproc = 

517 
make_simproc 

518 
{lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, 

519 
@{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"}, 

520 
@{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, 

521 
@{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"}, 

522 
@{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"}, 

523 
@{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}], 

524 
name = "ord_frac_simproc", proc = proc3, identifier = []} 

525 

526 
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 

527 
@{thm "divide_Numeral1"}, 

36305  528 
@{thm "divide_zero"}, @{thm "divide_Numeral0"}, 
28402  529 
@{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"}, 
530 
@{thm "mult_num_frac"}, @{thm "mult_frac_num"}, 

531 
@{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, 

532 
@{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, 

533 
@{thm "diff_def"}, @{thm "minus_divide_left"}, 

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534 
@{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym, 
35084  535 
@{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
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536 
Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute})))) 
35084  537 
(@{thm field_divide_inverse} RS sym)] 
28402  538 

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539 
in 
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540 

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541 
val field_comp_conv = (Simplifier.rewrite 
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542 
(HOL_basic_ss addsimps @{thms "semiring_norm"} 
35410  543 
addsimps ths addsimps @{thms simp_thms} 
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544 
addsimprocs Numeral_Simprocs.field_cancel_numeral_factors 
28402  545 
addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, 
546 
ord_frac_simproc] 

547 
addcongs [@{thm "if_weak_cong"}])) 

548 
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps 

549 
[@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(12)})) 

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550 

23252  551 
end 
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552 
*} 
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553 

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554 
declaration {* 
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555 
let 
28402  556 

557 
fun numeral_is_const ct = 

558 
case term_of ct of 

35084  559 
Const (@{const_name Rings.divide},_) $ a $ b => 
30866  560 
can HOLogic.dest_number a andalso can HOLogic.dest_number b 
35084  561 
 Const (@{const_name Rings.inverse},_)$t => can HOLogic.dest_number t 
28402  562 
 t => can HOLogic.dest_number t 
563 

564 
fun dest_const ct = ((case term_of ct of 

35084  565 
Const (@{const_name Rings.divide},_) $ a $ b=> 
28402  566 
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) 
35084  567 
 Const (@{const_name Rings.inverse},_)$t => 
30869
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568 
Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t))) 
28402  569 
 t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
570 
handle TERM _ => error "ring_dest_const") 

571 

572 
fun mk_const phi cT x = 

573 
let val (a, b) = Rat.quotient_of_rat x 

574 
in if b = 1 then Numeral.mk_cnumber cT a 

575 
else Thm.capply 

576 
(Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) 

577 
(Numeral.mk_cnumber cT a)) 

578 
(Numeral.mk_cnumber cT b) 

579 
end 

580 

581 
in 

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582 

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583 
Normalizer.funs @{thm normalizing.normalizing_field_cancel_axioms'} 
28402  584 
{is_const = K numeral_is_const, 
585 
dest_const = K dest_const, 

586 
mk_const = mk_const, 

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587 
conv = K (K field_comp_conv)} 
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588 

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589 
end 
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590 
*} 
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591 

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592 
lemmas comp_arith = semiring_norm (*FIXME*) 
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593 

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594 

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595 
subsection {* Groebner Bases *} 
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596 

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597 
lemmas bool_simps = simp_thms(134) 
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598 

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599 
lemma dnf: 
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600 
"(P & (Q  R)) = ((P&Q)  (P&R))" "((Q  R) & P) = ((Q&P)  (R&P))" 
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601 
"(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)" 
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602 
by blast+ 
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603 

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604 
lemmas weak_dnf_simps = dnf bool_simps 
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605 

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606 
lemma nnf_simps: 
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607 
"(\<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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608 
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P" 
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609 
by blast+ 
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610 

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611 
lemma PFalse: 
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612 
"P \<equiv> False \<Longrightarrow> \<not> P" 
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613 
"\<not> P \<Longrightarrow> (P \<equiv> False)" 
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614 
by auto 
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615 

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616 
ML {* 
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617 
structure Algebra_Simplification = Named_Thms( 
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618 
val name = "algebra" 
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619 
val description = "presimplification rules for algebraic methods" 
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620 
) 
28402  621 
*} 
622 

36712
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623 
setup Algebra_Simplification.setup 
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624 

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625 
declare dvd_def[algebra] 
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626 
declare dvd_eq_mod_eq_0[symmetric, algebra] 
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627 
declare mod_div_trivial[algebra] 
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628 
declare mod_mod_trivial[algebra] 
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629 
declare conjunct1[OF DIVISION_BY_ZERO, algebra] 
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630 
declare conjunct2[OF DIVISION_BY_ZERO, algebra] 
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631 
declare zmod_zdiv_equality[symmetric,algebra] 
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632 
declare zdiv_zmod_equality[symmetric, algebra] 
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633 
declare zdiv_zminus_zminus[algebra] 
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634 
declare zmod_zminus_zminus[algebra] 
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635 
declare zdiv_zminus2[algebra] 
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636 
declare zmod_zminus2[algebra] 
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637 
declare zdiv_zero[algebra] 
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638 
declare zmod_zero[algebra] 
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639 
declare mod_by_1[algebra] 
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640 
declare div_by_1[algebra] 
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641 
declare zmod_minus1_right[algebra] 
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642 
declare zdiv_minus1_right[algebra] 
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643 
declare mod_div_trivial[algebra] 
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644 
declare mod_mod_trivial[algebra] 
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645 
declare mod_mult_self2_is_0[algebra] 
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646 
declare mod_mult_self1_is_0[algebra] 
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647 
declare zmod_eq_0_iff[algebra] 
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648 
declare dvd_0_left_iff[algebra] 
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649 
declare zdvd1_eq[algebra] 
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650 
declare zmod_eq_dvd_iff[algebra] 
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651 
declare nat_mod_eq_iff[algebra] 
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652 

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653 
use "Tools/Groebner_Basis/groebner.ML" 
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654 

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655 
method_setup algebra = 
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656 
{* 
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657 
let 
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658 
fun keyword k = Scan.lift (Args.$$$ k  Args.colon) >> K () 
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659 
val addN = "add" 
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660 
val delN = "del" 
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661 
val any_keyword = keyword addN  keyword delN 
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662 
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; 
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663 
in 
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664 
((Scan.optional (keyword addN  thms) [])  
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665 
(Scan.optional (keyword delN  thms) [])) >> 
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666 
(fn (add_ths, del_ths) => fn ctxt => 
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667 
SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt)) 
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668 
end 
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669 
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases" 
28402  670 

671 
end 