author  haftmann 
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parent 36720  src/HOL/Groebner_Basis.thy@41da7025e59c 
child 36753  5cf4e9128f22 
permissions  rwrr 
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(* Title: HOL/Semiring_Normalization.thy 
23252  2 
Author: Amine Chaieb, TU Muenchen 
3 
*) 

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header {* Semiring normalization *} 
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theory Semiring_Normalization 
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imports Numeral_Simprocs Nat_Transfer 
23252  9 
uses 
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"Tools/Groebner_Basis/normalizer.ML" 
23252  11 
begin 
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setup Normalizer.setup 
23252  14 

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locale normalizing_semiring = 
23252  16 
fixes add mul pwr r0 r1 
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assumes add_a:"(add x (add y z) = add (add x y) z)" 

18 
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 

24 

25 
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 

28 
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 

34 

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

36 
proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1) 

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fix q x y 

38 
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" . 
23252  59 

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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)" 

89 
"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" 

93 
"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))" 

98 
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 

100 
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 

119 
show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" 

120 
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 

35216  137 
next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c) 
23252  138 
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 . 

35216  141 
next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c) 
23252  142 
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 

35216  144 
next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr) 
23252  145 
next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))" 
35216  146 
by (simp add: nat_number' pwr_Suc mul_pwr) 
23252  147 
qed 
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lemmas normalizing_semiring_axioms' = 
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normalizing_semiring_axioms [normalizer 
23252  152 
semiring ops: semiring_ops 
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semiring rules: semiring_rules] 
23252  154 

155 
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) 
23252  161 

36720  162 
declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *} 
23573  163 

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locale normalizing_ring = normalizing_semiring + 
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fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 
166 
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] 

23252  181 

182 
end 

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sublocale comm_ring_1 
185 
< normalizing!: normalizing_ring plus times power zero one minus uminus 

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proof 

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qed (simp_all add: diff_minus) 

23252  188 

36720  189 
declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *} 
23252  190 

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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" 

30866  194 
assumes divide_inverse: "divide x y = mul x (inverse y)" 
195 
and inverse_divide: "inverse x = divide r1 x" 

23327  196 
begin 
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30866  198 
lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" . 
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200 
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 

30866  207 
ring rules: ring_rules 
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field ops: field_ops 

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field rules: field_rules] 

23327  210 

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end 

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locale normalizing_semiring_cancel = normalizing_semiring + 
23252  214 
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 

218 

219 
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> 

229 
\<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" 

232 
and eq: "add b (mul r c) = add b (mul r d)" 

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

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

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using mul_0 add_cancel by simp 

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thus "False" using add_mul_solve nz cnd by simp 

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qed 

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lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0" 
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proof 
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have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel) 
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thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0) 
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qed 
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declare normalizing_semiring_axioms' [normalizer del] 
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lemmas normalizing_semiring_cancel_axioms' = 
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normalizing_semiring_cancel_axioms [normalizer 
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semiring ops: semiring_ops 
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semiring rules: semiring_rules 
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idom rules: noteq_reduce add_scale_eq_noteq] 
23252  252 

253 
end 

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locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
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assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y" 
23252  257 
begin 
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declare normalizing_ring_axioms' [normalizer del] 
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lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer 
23252  262 
semiring ops: semiring_ops 
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semiring rules: semiring_rules 

264 
ring ops: ring_ops 

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ring rules: ring_rules 

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idom rules: noteq_reduce add_scale_eq_noteq 
26314  267 
ideal rules: subr0_iff add_r0_iff] 
23252  268 

269 
end 

270 

36720  271 
sublocale idom 
272 
< normalizing!: normalizing_ring_cancel plus times power zero one minus uminus 

273 
proof 

274 
fix w x y z 

275 
show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" 

276 
proof 

277 
assume "w * y + x * z = w * z + x * y" 

278 
then have "w * y + x * z  w * z  x * y = 0" by (simp add: algebra_simps) 

279 
then have "w * (y  z)  x * (y  z) = 0" by (simp add: algebra_simps) 

280 
then have "(y  z) * (w  x) = 0" by (simp add: algebra_simps) 

281 
then have "y  z = 0 \<or> w  x = 0" by (rule divisors_zero) 

282 
then show "w = x \<or> y = z" by auto 

283 
qed (auto simp add: add_ac) 

284 
qed (simp_all add: algebra_simps) 

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declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *} 
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interpretation normalizing_nat!: normalizing_semiring_cancel 
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"op +" "op *" "op ^" "0::nat" "1" 
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proof (unfold_locales, simp add: algebra_simps) 
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fix w x y z ::"nat" 
292 
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 

293 
hence "y < z \<or> y > z" by arith 

294 
moreover { 

295 
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z  y" in exI, auto) 

296 
then obtain k where kp: "k>0" and yz:"z = y + k" by blast 

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

302 
then obtain k where kp: "k>0" and yz:"y = z + k" by blast 

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from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) 
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hence "w*k = x*k" by simp 
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hence "w = x" using kp by simp } 
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ultimately have "w=x" by blast } 
307 
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto 

308 
qed 

309 

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declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *} 
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locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field 
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begin 
314 

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

320 
ring ops: ring_ops 

321 
ring rules: ring_rules 

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field ops: field_ops 
323 
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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idom rules: noteq_reduce add_scale_eq_noteq 
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ideal rules: subr0_iff add_r0_iff] 
326 

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end 
328 

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sublocale field 
330 
< normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse 

331 
proof 

332 
qed (simp_all add: divide_inverse) 

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declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *} 
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336 
end 