author  haftmann 
Thu, 06 May 2010 16:40:02 +0200  
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parent 36700  9b85b9d74b83 
child 36712  2f4c318861b3 
permissions  rwrr 
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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 gb_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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subsubsection {* Declaring the abstract theory *} 

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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 gb_semiring_axioms' = 
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gb_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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interpretation class_semiring: gb_semiring 
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"op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1" 
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proof qed (auto simp add: algebra_simps) 
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lemmas nat_arith = 

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add_nat_number_of 
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diff_nat_number_of 

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mult_nat_number_of 

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eq_nat_number_of 

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less_nat_number_of 

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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 comp_arith = 
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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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lemmas semiring_norm = comp_arith 

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ML {* 

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local 
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open Conv; 
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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 class_semiring.gb_semiring_axioms'} *} 
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locale gb_ring = gb_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 gb_ring_axioms' = 
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gb_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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interpretation class_ring: gb_ring "op +" "op *" "op ^" 
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"0::'a::{comm_semiring_1,number_ring}" 1 "op " "uminus" 
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proof qed simp_all 
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declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *} 
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locale gb_field = gb_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 gb_field_axioms' = 
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gb_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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23266  270 
subsection {* Groebner Bases *} 
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locale semiringb = gb_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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26462  304 
declare gb_semiring_axioms' [normalizer del] 
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26462  306 
lemmas semiringb_axioms' = semiringb_axioms [normalizer 
23252  307 
semiring ops: semiring_ops 
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semiring rules: semiring_rules 

26314  309 
idom rules: noteq_reduce add_scale_eq_noteq] 
23252  310 

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end 

312 

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locale ringb = semiringb + gb_ring + 
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assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y" 
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begin 
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26462  317 
declare gb_ring_axioms' [normalizer del] 
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26462  319 
lemmas ringb_axioms' = ringb_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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idom rules: noteq_reduce add_scale_eq_noteq 
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ideal rules: subr0_iff add_r0_iff] 
23252  326 

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end 

328 

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lemma no_zero_divirors_neq0: 
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assumes az: "(a::'a::no_zero_divisors) \<noteq> 0" 

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and ab: "a*b = 0" shows "b = 0" 

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proof  

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{ assume bz: "b \<noteq> 0" 

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from no_zero_divisors [OF az bz] ab have False by blast } 

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thus "b = 0" by blast 

337 
qed 

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interpretation class_ringb: ringb 
31017  340 
"op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op " "uminus" 
35216  341 
proof(unfold_locales, simp add: algebra_simps, auto) 
31017  342 
fix w x y z ::"'a::{idom,number_ring}" 
23252  343 
assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 
344 
hence ynz': "y  z \<noteq> 0" by simp 

345 
from p have "w * y + x* z  w*z  x*y = 0" by simp 

29667  346 
hence "w* (y  z)  x * (y  z) = 0" by (simp add: algebra_simps) 
347 
hence "(y  z) * (w  x) = 0" by (simp add: algebra_simps) 

23252  348 
with no_zero_divirors_neq0 [OF ynz'] 
349 
have "w  x = 0" by blast 

350 
thus "w = x" by simp 

351 
qed 

352 

26462  353 
declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *} 
23252  354 

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355 
interpretation natgb: semiringb 
29223  356 
"op +" "op *" "op ^" "0::nat" "1" 
35216  357 
proof (unfold_locales, simp add: algebra_simps) 
23252  358 
fix w x y z ::"nat" 
359 
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 

360 
hence "y < z \<or> y > z" by arith 

361 
moreover { 

362 
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z  y" in exI, auto) 

363 
then obtain k where kp: "k>0" and yz:"z = y + k" by blast 

29667  364 
from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps) 
23252  365 
hence "x*k = w*k" by simp 
35216  366 
hence "w = x" using kp by simp } 
23252  367 
moreover { 
368 
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y  z" in exI, auto) 

369 
then obtain k where kp: "k>0" and yz:"y = z + k" by blast 

29667  370 
from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) 
23252  371 
hence "w*k = x*k" by simp 
35216  372 
hence "w = x" using kp by simp } 
23252  373 
ultimately have "w=x" by blast } 
374 
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto 

375 
qed 

376 

26462  377 
declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *} 
23252  378 

23327  379 
locale fieldgb = ringb + gb_field 
380 
begin 

381 

26462  382 
declare gb_field_axioms' [normalizer del] 
23327  383 

26462  384 
lemmas fieldgb_axioms' = fieldgb_axioms [normalizer 
23327  385 
semiring ops: semiring_ops 
386 
semiring rules: semiring_rules 

387 
ring ops: ring_ops 

388 
ring rules: ring_rules 

30866  389 
field ops: field_ops 
390 
field rules: field_rules 

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(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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391 
idom rules: noteq_reduce add_scale_eq_noteq 
26314  392 
ideal rules: subr0_iff add_r0_iff] 
393 

23327  394 
end 
395 

396 

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397 
lemmas bool_simps = simp_thms(134) 
23252  398 
lemma dnf: 
399 
"(P & (Q  R)) = ((P&Q)  (P&R))" "((Q  R) & P) = ((Q&P)  (R&P))" 

400 
"(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)" 

401 
by blast+ 

402 

403 
lemmas weak_dnf_simps = dnf bool_simps 

404 

405 
lemma nnf_simps: 

406 
"(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)" 

407 
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P" 

408 
by blast+ 

409 

410 
lemma PFalse: 

411 
"P \<equiv> False \<Longrightarrow> \<not> P" 

412 
"\<not> P \<Longrightarrow> (P \<equiv> False)" 

413 
by auto 

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414 

36702
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415 
ML {* 
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416 
structure Algebra_Simplification = Named_Thms( 
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417 
val name = "algebra" 
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418 
val description = "presimplification rules for algebraic methods" 
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419 
) 
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420 
*} 
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421 

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422 
setup Algebra_Simplification.setup 
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423 

23252  424 
use "Tools/Groebner_Basis/groebner.ML" 
425 

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426 
method_setup algebra = 
23458  427 
{* 
23332
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changeset

428 
let 
b91295432e6d
algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
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429 
fun keyword k = Scan.lift (Args.$$$ k  Args.colon) >> K () 
b91295432e6d
algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
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changeset

430 
val addN = "add" 
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algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
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431 
val delN = "del" 
b91295432e6d
algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
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432 
val any_keyword = keyword addN  keyword delN 
b91295432e6d
algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
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433 
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; 
b91295432e6d
algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
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434 
in 
30549  435 
((Scan.optional (keyword addN  thms) [])  
436 
(Scan.optional (keyword delN  thms) [])) >> 

437 
(fn (add_ths, del_ths) => fn ctxt => 

30510
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unified type Proof.method and pervasive METHOD combinators;
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438 
SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt)) 
23332
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439 
end 
25250
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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440 
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases" 
27666  441 
declare dvd_def[algebra] 
442 
declare dvd_eq_mod_eq_0[symmetric, algebra] 

30027  443 
declare mod_div_trivial[algebra] 
444 
declare mod_mod_trivial[algebra] 

27666  445 
declare conjunct1[OF DIVISION_BY_ZERO, algebra] 
446 
declare conjunct2[OF DIVISION_BY_ZERO, algebra] 

447 
declare zmod_zdiv_equality[symmetric,algebra] 

448 
declare zdiv_zmod_equality[symmetric, algebra] 

449 
declare zdiv_zminus_zminus[algebra] 

450 
declare zmod_zminus_zminus[algebra] 

451 
declare zdiv_zminus2[algebra] 

452 
declare zmod_zminus2[algebra] 

453 
declare zdiv_zero[algebra] 

454 
declare zmod_zero[algebra] 

30031  455 
declare mod_by_1[algebra] 
456 
declare div_by_1[algebra] 

27666  457 
declare zmod_minus1_right[algebra] 
458 
declare zdiv_minus1_right[algebra] 

459 
declare mod_div_trivial[algebra] 

460 
declare mod_mod_trivial[algebra] 

30034  461 
declare mod_mult_self2_is_0[algebra] 
462 
declare mod_mult_self1_is_0[algebra] 

27666  463 
declare zmod_eq_0_iff[algebra] 
30042  464 
declare dvd_0_left_iff[algebra] 
27666  465 
declare zdvd1_eq[algebra] 
466 
declare zmod_eq_dvd_iff[algebra] 

467 
declare nat_mod_eq_iff[algebra] 

23252  468 

28402  469 
subsection{* Groebner Bases for fields *} 
470 

30729
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471 
interpretation class_fieldgb: 
31017  472 
fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op " "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse) 
28402  473 

36409  474 
lemma divide_Numeral1: "(x::'a::{field, number_ring}) / Numeral1 = x" by simp 
475 
lemma divide_Numeral0: "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0" 

28402  476 
by simp 
36409  477 
lemma mult_frac_frac: "((x::'a::field_inverse_zero) / y) * (z / w) = (x*z) / (y*w)" 
28402  478 
by simp 
36409  479 
lemma mult_frac_num: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y" 
28402  480 
by simp 
36409  481 
lemma mult_num_frac: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y" 
28402  482 
by simp 
483 

484 
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp 

485 

36409  486 
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::field_inverse_zero) / y + z = (x + z*y) / y" 
28402  487 
by (simp add: add_divide_distrib) 
36409  488 
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::field_inverse_zero) / y = (x + z*y) / y" 
28402  489 
by (simp add: add_divide_distrib) 
35084  490 

491 
ML {* 

492 
let open Conv 

493 
in fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute})))) (@{thm field_divide_inverse} RS sym) 

494 
end 

495 
*} 

496 

28402  497 
ML{* 
498 
local 

499 
val zr = @{cpat "0"} 

500 
val zT = ctyp_of_term zr 

501 
val geq = @{cpat "op ="} 

502 
val eqT = Thm.dest_ctyp (ctyp_of_term geq) > hd 

503 
val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} 

504 
val add_frac_num = mk_meta_eq @{thm "add_frac_num"} 

505 
val add_num_frac = mk_meta_eq @{thm "add_num_frac"} 

506 

507 
fun prove_nz ss T t = 

508 
let 

509 
val z = instantiate_cterm ([(zT,T)],[]) zr 

510 
val eq = instantiate_cterm ([(eqT,T)],[]) geq 

35410  511 
val th = Simplifier.rewrite (ss addsimps @{thms simp_thms}) 
28402  512 
(Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} 
513 
(Thm.capply (Thm.capply eq t) z))) 

514 
in equal_elim (symmetric th) TrueI 

515 
end 

516 

517 
fun proc phi ss ct = 

518 
let 

519 
val ((x,y),(w,z)) = 

520 
(Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct 

521 
val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] 

522 
val T = ctyp_of_term x 

523 
val [y_nz, z_nz] = map (prove_nz ss T) [y, z] 

524 
val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq 

525 
in SOME (implies_elim (implies_elim th y_nz) z_nz) 

526 
end 

527 
handle CTERM _ => NONE  TERM _ => NONE  THM _ => NONE 

528 

529 
fun proc2 phi ss ct = 

530 
let 

531 
val (l,r) = Thm.dest_binop ct 

532 
val T = ctyp_of_term l 

533 
in (case (term_of l, term_of r) of 

35084  534 
(Const(@{const_name Rings.divide},_)$_$_, _) => 
28402  535 
let val (x,y) = Thm.dest_binop l val z = r 
536 
val _ = map (HOLogic.dest_number o term_of) [x,y,z] 

537 
val ynz = prove_nz ss T y 

538 
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz) 

539 
end 

35084  540 
 (_, Const (@{const_name Rings.divide},_)$_$_) => 
28402  541 
let val (x,y) = Thm.dest_binop r val z = l 
542 
val _ = map (HOLogic.dest_number o term_of) [x,y,z] 

543 
val ynz = prove_nz ss T y 

544 
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz) 

545 
end 

546 
 _ => NONE) 

547 
end 

548 
handle CTERM _ => NONE  TERM _ => NONE  THM _ => NONE 

549 

35084  550 
fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b 
28402  551 
 is_number t = can HOLogic.dest_number t 
552 

553 
val is_number = is_number o term_of 

554 

555 
fun proc3 phi ss ct = 

556 
(case term_of ct of 

35092
cfe605c54e50
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haftmann
parents:
35084
diff
changeset

557 
Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => 
28402  558 
let 
559 
val ((a,b),c) = Thm.dest_binop ct >> Thm.dest_binop 

560 
val _ = map is_number [a,b,c] 

561 
val T = ctyp_of_term c 

562 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"} 

563 
in SOME (mk_meta_eq th) end 

35092
cfe605c54e50
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haftmann
parents:
35084
diff
changeset

564 
 Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => 
28402  565 
let 
566 
val ((a,b),c) = Thm.dest_binop ct >> Thm.dest_binop 

567 
val _ = map is_number [a,b,c] 

568 
val T = ctyp_of_term c 

569 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"} 

570 
in SOME (mk_meta_eq th) end 

35084  571 
 Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => 
28402  572 
let 
573 
val ((a,b),c) = Thm.dest_binop ct >> Thm.dest_binop 

574 
val _ = map is_number [a,b,c] 

575 
val T = ctyp_of_term c 

576 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"} 

577 
in SOME (mk_meta_eq th) end 

35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35084
diff
changeset

578 
 Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) => 
28402  579 
let 
580 
val (a,(b,c)) = Thm.dest_binop ct > Thm.dest_binop 

581 
val _ = map is_number [a,b,c] 

582 
val T = ctyp_of_term c 

583 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"} 

584 
in SOME (mk_meta_eq th) end 

35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35084
diff
changeset

585 
 Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) => 
28402  586 
let 
587 
val (a,(b,c)) = Thm.dest_binop ct > Thm.dest_binop 

588 
val _ = map is_number [a,b,c] 

589 
val T = ctyp_of_term c 

590 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"} 

591 
in SOME (mk_meta_eq th) end 

35084  592 
 Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) => 
28402  593 
let 
594 
val (a,(b,c)) = Thm.dest_binop ct > Thm.dest_binop 

595 
val _ = map is_number [a,b,c] 

596 
val T = ctyp_of_term c 

597 
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"} 

598 
in SOME (mk_meta_eq th) end 

599 
 _ => NONE) 

600 
handle TERM _ => NONE  CTERM _ => NONE  THM _ => NONE 

601 

602 
val add_frac_frac_simproc = 

603 
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], 

604 
name = "add_frac_frac_simproc", 

605 
proc = proc, identifier = []} 

606 

607 
val add_frac_num_simproc = 

608 
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], 

609 
name = "add_frac_num_simproc", 

610 
proc = proc2, identifier = []} 

611 

612 
val ord_frac_simproc = 

613 
make_simproc 

614 
{lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, 

615 
@{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"}, 

616 
@{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, 

617 
@{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"}, 

618 
@{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"}, 

619 
@{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}], 

620 
name = "ord_frac_simproc", proc = proc3, identifier = []} 

621 

30869
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

622 
local 
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

623 
open Conv 
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

624 
in 
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

625 

28402  626 
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 
627 
@{thm "divide_Numeral1"}, 

36305  628 
@{thm "divide_zero"}, @{thm "divide_Numeral0"}, 
28402  629 
@{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"}, 
630 
@{thm "mult_num_frac"}, @{thm "mult_frac_num"}, 

631 
@{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, 

632 
@{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, 

633 
@{thm "diff_def"}, @{thm "minus_divide_left"}, 

30869
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

634 
@{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym, 
35084  635 
@{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
30869
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

636 
fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute})))) 
35084  637 
(@{thm field_divide_inverse} RS sym)] 
28402  638 

639 
val comp_conv = (Simplifier.rewrite 

640 
(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"} 

35410  641 
addsimps ths addsimps @{thms simp_thms} 
31068
f591144b0f17
modules numeral_simprocs, nat_numeral_simprocs; proper structures for numeral simprocs
haftmann
parents:
31017
diff
changeset

642 
addsimprocs Numeral_Simprocs.field_cancel_numeral_factors 
28402  643 
addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, 
644 
ord_frac_simproc] 

645 
addcongs [@{thm "if_weak_cong"}])) 

646 
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps 

647 
[@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(12)})) 

23252  648 
end 
28402  649 

650 
fun numeral_is_const ct = 

651 
case term_of ct of 

35084  652 
Const (@{const_name Rings.divide},_) $ a $ b => 
30866  653 
can HOLogic.dest_number a andalso can HOLogic.dest_number b 
35084  654 
 Const (@{const_name Rings.inverse},_)$t => can HOLogic.dest_number t 
28402  655 
 t => can HOLogic.dest_number t 
656 

657 
fun dest_const ct = ((case term_of ct of 

35084  658 
Const (@{const_name Rings.divide},_) $ a $ b=> 
28402  659 
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) 
35084  660 
 Const (@{const_name Rings.inverse},_)$t => 
30869
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset

661 
Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t))) 
28402  662 
 t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
663 
handle TERM _ => error "ring_dest_const") 

664 

665 
fun mk_const phi cT x = 

666 
let val (a, b) = Rat.quotient_of_rat x 

667 
in if b = 1 then Numeral.mk_cnumber cT a 

668 
else Thm.capply 

669 
(Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) 

670 
(Numeral.mk_cnumber cT a)) 

671 
(Numeral.mk_cnumber cT b) 

672 
end 

673 

674 
in 

675 
val field_comp_conv = comp_conv; 

676 
val fieldgb_declaration = 

36700
9b85b9d74b83
dropped auxiliary method sring_norm; integrated normalizer.ML and normalizer_data.ML
haftmann
parents:
36699
diff
changeset

677 
Normalizer.funs @{thm class_fieldgb.fieldgb_axioms'} 
28402  678 
{is_const = K numeral_is_const, 
679 
dest_const = K dest_const, 

680 
mk_const = mk_const, 

681 
conv = K (K comp_conv)} 

682 
end; 

683 
*} 

684 

685 
declaration fieldgb_declaration 

686 

687 
end 