author | haftmann |
Thu, 06 May 2010 18:16:07 +0200 | |
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parent 36714 | ae84ddf03c58 |
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permissions | -rw-r--r-- |
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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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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(1-2)})); |
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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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271 |
lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0" |
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272 |
proof- |
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273 |
have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel) |
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274 |
thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0) |
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|
275 |
qed |
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276 |
|
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277 |
declare normalizing_semiring_axioms' [normalizer del] |
23252 | 278 |
|
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279 |
lemmas normalizing_semiring_cancel_axioms' = |
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280 |
normalizing_semiring_cancel_axioms [normalizer |
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|
281 |
semiring ops: semiring_ops |
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|
282 |
semiring rules: semiring_rules |
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283 |
idom rules: noteq_reduce add_scale_eq_noteq] |
23252 | 284 |
|
285 |
end |
|
286 |
||
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|
287 |
locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + |
25250
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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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288 |
assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y" |
23252 | 289 |
begin |
290 |
||
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291 |
declare normalizing_ring_axioms' [normalizer del] |
23252 | 292 |
|
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293 |
lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer |
23252 | 294 |
semiring ops: semiring_ops |
295 |
semiring rules: semiring_rules |
|
296 |
ring ops: ring_ops |
|
297 |
ring rules: ring_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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298 |
idom rules: noteq_reduce add_scale_eq_noteq |
26314 | 299 |
ideal rules: subr0_iff add_r0_iff] |
23252 | 300 |
|
301 |
end |
|
302 |
||
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303 |
lemma (in no_zero_divisors) prod_eq_zero_eq_zero: |
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304 |
assumes "a * b = 0" and "a \<noteq> 0" |
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|
305 |
shows "b = 0" |
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|
306 |
proof (rule classical) |
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|
307 |
assume "b \<noteq> 0" with `a \<noteq> 0` no_zero_divisors have "a * b \<noteq> 0" by blast |
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|
308 |
with `a * b = 0` show ?thesis by simp |
23252 | 309 |
qed |
310 |
||
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|
311 |
(*FIXME introduce class*) |
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|
312 |
interpretation normalizing!: normalizing_ring_cancel |
31017 | 313 |
"op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus" |
35216 | 314 |
proof(unfold_locales, simp add: algebra_simps, auto) |
31017 | 315 |
fix w x y z ::"'a::{idom,number_ring}" |
23252 | 316 |
assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" |
317 |
hence ynz': "y - z \<noteq> 0" by simp |
|
318 |
from p have "w * y + x* z - w*z - x*y = 0" by simp |
|
29667 | 319 |
hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps) |
320 |
hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps) |
|
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|
321 |
with prod_eq_zero_eq_zero [OF _ ynz'] |
23252 | 322 |
have "w - x = 0" by blast |
323 |
thus "w = x" by simp |
|
324 |
qed |
|
325 |
||
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|
326 |
declaration {* normalizer_funs' @{thm normalizing.normalizing_ring_cancel_axioms'} *} |
23252 | 327 |
|
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|
328 |
interpretation normalizing_nat!: normalizing_semiring_cancel |
29223 | 329 |
"op +" "op *" "op ^" "0::nat" "1" |
35216 | 330 |
proof (unfold_locales, simp add: algebra_simps) |
23252 | 331 |
fix w x y z ::"nat" |
332 |
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" |
|
333 |
hence "y < z \<or> y > z" by arith |
|
334 |
moreover { |
|
335 |
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto) |
|
336 |
then obtain k where kp: "k>0" and yz:"z = y + k" by blast |
|
29667 | 337 |
from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps) |
23252 | 338 |
hence "x*k = w*k" by simp |
35216 | 339 |
hence "w = x" using kp by simp } |
23252 | 340 |
moreover { |
341 |
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto) |
|
342 |
then obtain k where kp: "k>0" and yz:"y = z + k" by blast |
|
29667 | 343 |
from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) |
23252 | 344 |
hence "w*k = x*k" by simp |
35216 | 345 |
hence "w = x" using kp by simp } |
23252 | 346 |
ultimately have "w=x" by blast } |
347 |
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto |
|
348 |
qed |
|
349 |
||
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|
350 |
declaration {* normalizer_funs' @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *} |
23252 | 351 |
|
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|
352 |
locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field |
23327 | 353 |
begin |
354 |
||
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|
355 |
declare normalizing_field_axioms' [normalizer del] |
23327 | 356 |
|
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|
357 |
lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer |
23327 | 358 |
semiring ops: semiring_ops |
359 |
semiring rules: semiring_rules |
|
360 |
ring ops: ring_ops |
|
361 |
ring rules: ring_rules |
|
30866 | 362 |
field ops: field_ops |
363 |
field rules: field_rules |
|
25250
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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)
chaieb
parents:
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changeset
|
364 |
idom rules: noteq_reduce add_scale_eq_noteq |
26314 | 365 |
ideal rules: subr0_iff add_r0_iff] |
366 |
||
23327 | 367 |
end |
368 |
||
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|
369 |
(*FIXME introduce class*) |
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|
370 |
interpretation normalizing!: normalizing_field_cancel "op +" "op *" "op ^" |
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|
371 |
"0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" |
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|
372 |
apply (unfold_locales) by (simp_all add: divide_inverse) |
28402 | 373 |
|
36409 | 374 |
lemma divide_Numeral1: "(x::'a::{field, number_ring}) / Numeral1 = x" by simp |
375 |
lemma divide_Numeral0: "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0" |
|
28402 | 376 |
by simp |
36409 | 377 |
lemma mult_frac_frac: "((x::'a::field_inverse_zero) / y) * (z / w) = (x*z) / (y*w)" |
28402 | 378 |
by simp |
36409 | 379 |
lemma mult_frac_num: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y" |
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|
380 |
by (fact times_divide_eq_left) |
36409 | 381 |
lemma mult_num_frac: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y" |
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|
382 |
by (fact times_divide_eq_left) |
28402 | 383 |
|
384 |
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp |
|
385 |
||
36409 | 386 |
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::field_inverse_zero) / y + z = (x + z*y) / y" |
28402 | 387 |
by (simp add: add_divide_distrib) |
36409 | 388 |
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::field_inverse_zero) / y = (x + z*y) / y" |
28402 | 389 |
by (simp add: add_divide_distrib) |
35084 | 390 |
|
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|
391 |
ML {* |
28402 | 392 |
local |
393 |
val zr = @{cpat "0"} |
|
394 |
val zT = ctyp_of_term zr |
|
395 |
val geq = @{cpat "op ="} |
|
396 |
val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd |
|
397 |
val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} |
|
398 |
val add_frac_num = mk_meta_eq @{thm "add_frac_num"} |
|
399 |
val add_num_frac = mk_meta_eq @{thm "add_num_frac"} |
|
400 |
||
401 |
fun prove_nz ss T t = |
|
402 |
let |
|
403 |
val z = instantiate_cterm ([(zT,T)],[]) zr |
|
404 |
val eq = instantiate_cterm ([(eqT,T)],[]) geq |
|
35410 | 405 |
val th = Simplifier.rewrite (ss addsimps @{thms simp_thms}) |
28402 | 406 |
(Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} |
407 |
(Thm.capply (Thm.capply eq t) z))) |
|
408 |
in equal_elim (symmetric th) TrueI |
|
409 |
end |
|
410 |
||
411 |
fun proc phi ss ct = |
|
412 |
let |
|
413 |
val ((x,y),(w,z)) = |
|
414 |
(Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct |
|
415 |
val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] |
|
416 |
val T = ctyp_of_term x |
|
417 |
val [y_nz, z_nz] = map (prove_nz ss T) [y, z] |
|
418 |
val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq |
|
419 |
in SOME (implies_elim (implies_elim th y_nz) z_nz) |
|
420 |
end |
|
421 |
handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE |
|
422 |
||
423 |
fun proc2 phi ss ct = |
|
424 |
let |
|
425 |
val (l,r) = Thm.dest_binop ct |
|
426 |
val T = ctyp_of_term l |
|
427 |
in (case (term_of l, term_of r) of |
|
35084 | 428 |
(Const(@{const_name Rings.divide},_)$_$_, _) => |
28402 | 429 |
let val (x,y) = Thm.dest_binop l val z = r |
430 |
val _ = map (HOLogic.dest_number o term_of) [x,y,z] |
|
431 |
val ynz = prove_nz ss T y |
|
432 |
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz) |
|
433 |
end |
|
35084 | 434 |
| (_, Const (@{const_name Rings.divide},_)$_$_) => |
28402 | 435 |
let val (x,y) = Thm.dest_binop r val z = l |
436 |
val _ = map (HOLogic.dest_number o term_of) [x,y,z] |
|
437 |
val ynz = prove_nz ss T y |
|
438 |
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz) |
|
439 |
end |
|
440 |
| _ => NONE) |
|
441 |
end |
|
442 |
handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE |
|
443 |
||
35084 | 444 |
fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b |
28402 | 445 |
| is_number t = can HOLogic.dest_number t |
446 |
||
447 |
val is_number = is_number o term_of |
|
448 |
||
449 |
fun proc3 phi ss ct = |
|
450 |
(case term_of ct of |
|
35092
cfe605c54e50
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haftmann
parents:
35084
diff
changeset
|
451 |
Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => |
28402 | 452 |
let |
453 |
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop |
|
454 |
val _ = map is_number [a,b,c] |
|
455 |
val T = ctyp_of_term c |
|
456 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"} |
|
457 |
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
|
458 |
| Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => |
28402 | 459 |
let |
460 |
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop |
|
461 |
val _ = map is_number [a,b,c] |
|
462 |
val T = ctyp_of_term c |
|
463 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"} |
|
464 |
in SOME (mk_meta_eq th) end |
|
35084 | 465 |
| Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ => |
28402 | 466 |
let |
467 |
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop |
|
468 |
val _ = map is_number [a,b,c] |
|
469 |
val T = ctyp_of_term c |
|
470 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"} |
|
471 |
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
|
472 |
| Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) => |
28402 | 473 |
let |
474 |
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop |
|
475 |
val _ = map is_number [a,b,c] |
|
476 |
val T = ctyp_of_term c |
|
477 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"} |
|
478 |
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
|
479 |
| Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) => |
28402 | 480 |
let |
481 |
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop |
|
482 |
val _ = map is_number [a,b,c] |
|
483 |
val T = ctyp_of_term c |
|
484 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"} |
|
485 |
in SOME (mk_meta_eq th) end |
|
35084 | 486 |
| Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) => |
28402 | 487 |
let |
488 |
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop |
|
489 |
val _ = map is_number [a,b,c] |
|
490 |
val T = ctyp_of_term c |
|
491 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"} |
|
492 |
in SOME (mk_meta_eq th) end |
|
493 |
| _ => NONE) |
|
494 |
handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE |
|
495 |
||
496 |
val add_frac_frac_simproc = |
|
497 |
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], |
|
498 |
name = "add_frac_frac_simproc", |
|
499 |
proc = proc, identifier = []} |
|
500 |
||
501 |
val add_frac_num_simproc = |
|
502 |
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], |
|
503 |
name = "add_frac_num_simproc", |
|
504 |
proc = proc2, identifier = []} |
|
505 |
||
506 |
val ord_frac_simproc = |
|
507 |
make_simproc |
|
508 |
{lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, |
|
509 |
@{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"}, |
|
510 |
@{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, |
|
511 |
@{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"}, |
|
512 |
@{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"}, |
|
513 |
@{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}], |
|
514 |
name = "ord_frac_simproc", proc = proc3, identifier = []} |
|
515 |
||
516 |
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, |
|
517 |
@{thm "divide_Numeral1"}, |
|
36305 | 518 |
@{thm "divide_zero"}, @{thm "divide_Numeral0"}, |
28402 | 519 |
@{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"}, |
520 |
@{thm "mult_num_frac"}, @{thm "mult_frac_num"}, |
|
521 |
@{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, |
|
522 |
@{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, |
|
523 |
@{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
|
524 |
@{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym, |
35084 | 525 |
@{thm field_divide_inverse} RS sym, @{thm inverse_divide}, |
36712
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parents:
36702
diff
changeset
|
526 |
Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute})))) |
35084 | 527 |
(@{thm field_divide_inverse} RS sym)] |
28402 | 528 |
|
36712
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parents:
36702
diff
changeset
|
529 |
in |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
530 |
|
2f4c318861b3
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parents:
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changeset
|
531 |
val field_comp_conv = (Simplifier.rewrite |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
532 |
(HOL_basic_ss addsimps @{thms "semiring_norm"} |
35410 | 533 |
addsimps ths addsimps @{thms simp_thms} |
31068
f591144b0f17
modules numeral_simprocs, nat_numeral_simprocs; proper structures for numeral simprocs
haftmann
parents:
31017
diff
changeset
|
534 |
addsimprocs Numeral_Simprocs.field_cancel_numeral_factors |
28402 | 535 |
addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, |
536 |
ord_frac_simproc] |
|
537 |
addcongs [@{thm "if_weak_cong"}])) |
|
538 |
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps |
|
539 |
[@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)})) |
|
36712
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haftmann
parents:
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|
540 |
|
23252 | 541 |
end |
36712
2f4c318861b3
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parents:
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diff
changeset
|
542 |
*} |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
543 |
|
2f4c318861b3
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haftmann
parents:
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diff
changeset
|
544 |
declaration {* |
2f4c318861b3
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parents:
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|
545 |
let |
28402 | 546 |
|
547 |
fun numeral_is_const ct = |
|
548 |
case term_of ct of |
|
35084 | 549 |
Const (@{const_name Rings.divide},_) $ a $ b => |
30866 | 550 |
can HOLogic.dest_number a andalso can HOLogic.dest_number b |
35084 | 551 |
| Const (@{const_name Rings.inverse},_)$t => can HOLogic.dest_number t |
28402 | 552 |
| t => can HOLogic.dest_number t |
553 |
||
554 |
fun dest_const ct = ((case term_of ct of |
|
35084 | 555 |
Const (@{const_name Rings.divide},_) $ a $ b=> |
28402 | 556 |
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) |
35084 | 557 |
| Const (@{const_name Rings.inverse},_)$t => |
30869
71fde5b7b43c
More precise treatement of rational constants by the normalizer for fields
chaieb
parents:
30866
diff
changeset
|
558 |
Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t))) |
28402 | 559 |
| t => Rat.rat_of_int (snd (HOLogic.dest_number t))) |
560 |
handle TERM _ => error "ring_dest_const") |
|
561 |
||
562 |
fun mk_const phi cT x = |
|
563 |
let val (a, b) = Rat.quotient_of_rat x |
|
564 |
in if b = 1 then Numeral.mk_cnumber cT a |
|
565 |
else Thm.capply |
|
566 |
(Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) |
|
567 |
(Numeral.mk_cnumber cT a)) |
|
568 |
(Numeral.mk_cnumber cT b) |
|
569 |
end |
|
570 |
||
571 |
in |
|
36712
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
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diff
changeset
|
572 |
|
2f4c318861b3
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haftmann
parents:
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diff
changeset
|
573 |
Normalizer.funs @{thm normalizing.normalizing_field_cancel_axioms'} |
28402 | 574 |
{is_const = K numeral_is_const, |
575 |
dest_const = K dest_const, |
|
576 |
mk_const = mk_const, |
|
36712
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
577 |
conv = K (K field_comp_conv)} |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
578 |
|
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
579 |
end |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
580 |
*} |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
581 |
|
36716 | 582 |
lemmas comp_arith = semiring_norm (*FIXME*) |
583 |
||
36712
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haftmann
parents:
36702
diff
changeset
|
584 |
|
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
585 |
subsection {* Groebner Bases *} |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
586 |
|
2f4c318861b3
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haftmann
parents:
36702
diff
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|
587 |
lemmas bool_simps = simp_thms(1-34) |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
588 |
|
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
589 |
lemma dnf: |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
590 |
"(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))" |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
591 |
"(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
592 |
by blast+ |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
593 |
|
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
594 |
lemmas weak_dnf_simps = dnf bool_simps |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
595 |
|
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
596 |
lemma nnf_simps: |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
597 |
"(\<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)" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
598 |
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
599 |
by blast+ |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
600 |
|
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
601 |
lemma PFalse: |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
602 |
"P \<equiv> False \<Longrightarrow> \<not> P" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
603 |
"\<not> P \<Longrightarrow> (P \<equiv> False)" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
604 |
by auto |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
605 |
|
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
606 |
ML {* |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
607 |
structure Algebra_Simplification = Named_Thms( |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
608 |
val name = "algebra" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
609 |
val description = "pre-simplification rules for algebraic methods" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
610 |
) |
28402 | 611 |
*} |
612 |
||
36712
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
613 |
setup Algebra_Simplification.setup |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
614 |
|
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
615 |
declare dvd_def[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
616 |
declare dvd_eq_mod_eq_0[symmetric, algebra] |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
617 |
declare mod_div_trivial[algebra] |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
618 |
declare mod_mod_trivial[algebra] |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
619 |
declare conjunct1[OF DIVISION_BY_ZERO, algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
620 |
declare conjunct2[OF DIVISION_BY_ZERO, algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
621 |
declare zmod_zdiv_equality[symmetric,algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
622 |
declare zdiv_zmod_equality[symmetric, algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
623 |
declare zdiv_zminus_zminus[algebra] |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
624 |
declare zmod_zminus_zminus[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
625 |
declare zdiv_zminus2[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
626 |
declare zmod_zminus2[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
627 |
declare zdiv_zero[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
628 |
declare zmod_zero[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
629 |
declare mod_by_1[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
630 |
declare div_by_1[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
631 |
declare zmod_minus1_right[algebra] |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
632 |
declare zdiv_minus1_right[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
633 |
declare mod_div_trivial[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
634 |
declare mod_mod_trivial[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
635 |
declare mod_mult_self2_is_0[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
636 |
declare mod_mult_self1_is_0[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
637 |
declare zmod_eq_0_iff[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
638 |
declare dvd_0_left_iff[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
639 |
declare zdvd1_eq[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
640 |
declare zmod_eq_dvd_iff[algebra] |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
641 |
declare nat_mod_eq_iff[algebra] |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
642 |
|
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
643 |
use "Tools/Groebner_Basis/groebner.ML" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
644 |
|
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
645 |
method_setup algebra = |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
646 |
{* |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
647 |
let |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
648 |
fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
649 |
val addN = "add" |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
650 |
val delN = "del" |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
651 |
val any_keyword = keyword addN || keyword delN |
2f4c318861b3
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haftmann
parents:
36702
diff
changeset
|
652 |
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
653 |
in |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
654 |
((Scan.optional (keyword addN |-- thms) []) -- |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
655 |
(Scan.optional (keyword delN |-- thms) [])) >> |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
656 |
(fn (add_ths, del_ths) => fn ctxt => |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
657 |
SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt)) |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
658 |
end |
2f4c318861b3
avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents:
36702
diff
changeset
|
659 |
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases" |
28402 | 660 |
|
661 |
end |