src/HOL/Groebner_Basis.thy
author chaieb
Mon Jul 21 13:36:39 2008 +0200 (2008-07-21)
changeset 27666 1436d81d1294
parent 26462 dac4e2bce00d
child 28402 09e4aa3ddc25
permissions -rw-r--r--
Relevant rules added to algebra's context
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(*  Title:      HOL/Groebner_Basis.thy
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    ID:         $Id$
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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 NatBin
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uses
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  "Tools/Groebner_Basis/misc.ML"
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  "Tools/Groebner_Basis/normalizer_data.ML"
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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 NormalizerData.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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  includes meta_term_syntax
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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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  by rule+
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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" 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, recpower}" "1"]
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  by unfold_locales (auto simp add: ring_simps power_Suc)
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lemmas nat_arith =
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  add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
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lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
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  by (simp add: numeral_1_eq_1)
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lemmas comp_arith = Let_def arith_simps nat_arith rel_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_add_numeral_1
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  numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
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  iszero_number_of_Bit1 iszero_number_of_Bit0 nonzero_number_of_Min
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  iszero_number_of_Pls iszero_0 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 =
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  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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  NormalizerData.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:
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  includes meta_term_syntax
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  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,recpower,number_ring}" 1 "op -" "uminus"]
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  by unfold_locales simp_all
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declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
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use "Tools/Groebner_Basis/normalizer.ML"
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method_setup sring_norm = {*
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  Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
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*} "semiring normalizer"
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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: "divide x y = mul x (inverse y)"
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     and inverse: "inverse x = divide r1 x"
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begin
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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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end
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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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   308
qed
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declare gb_semiring_axioms' [normalizer del]
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lemmas semiringb_axioms' = semiringb_axioms [normalizer
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  semiring ops: semiring_ops
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  semiring rules: semiring_rules
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  idom rules: noteq_reduce add_scale_eq_noteq]
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end
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locale 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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declare gb_ring_axioms' [normalizer del]
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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]
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wenzelm@23252
   333
end
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chaieb@25250
   335
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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
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   343
qed
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   344
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interpretation class_ringb: ringb
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  ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
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proof(unfold_locales, simp add: ring_simps power_Suc, auto)
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  fix w x y z ::"'a::{idom,recpower,number_ring}"
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  assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
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   350
  hence ynz': "y - z \<noteq> 0" by simp
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   351
  from p have "w * y + x* z - w*z - x*y = 0" by simp
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   352
  hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
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   353
  hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
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   354
  with  no_zero_divirors_neq0 [OF ynz']
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   355
  have "w - x = 0" by blast
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   356
  thus "w = x"  by simp
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   357
qed
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   358
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   359
declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
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   360
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interpretation natgb: semiringb
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  ["op +" "op *" "op ^" "0::nat" "1"]
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proof (unfold_locales, simp add: ring_simps power_Suc)
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   364
  fix w x y z ::"nat"
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   365
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
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   366
    hence "y < z \<or> y > z" by arith
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   367
    moreover {
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   368
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
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   369
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
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   370
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
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   371
      hence "x*k = w*k" by simp
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   372
      hence "w = x" using kp by (simp add: mult_cancel2) }
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   373
    moreover {
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   374
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
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   375
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
nipkow@23477
   376
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
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   377
      hence "w*k = x*k" by simp
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   378
      hence "w = x" using kp by (simp add: mult_cancel2)}
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   379
    ultimately have "w=x" by blast }
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   380
  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
wenzelm@23252
   381
qed
wenzelm@23252
   382
wenzelm@26462
   383
declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
wenzelm@23252
   384
chaieb@23327
   385
locale fieldgb = ringb + gb_field
chaieb@23327
   386
begin
chaieb@23327
   387
wenzelm@26462
   388
declare gb_field_axioms' [normalizer del]
chaieb@23327
   389
wenzelm@26462
   390
lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
chaieb@23327
   391
  semiring ops: semiring_ops
chaieb@23327
   392
  semiring rules: semiring_rules
chaieb@23327
   393
  ring ops: ring_ops
chaieb@23327
   394
  ring rules: ring_rules
chaieb@25250
   395
  idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314
   396
  ideal rules: subr0_iff add_r0_iff]
wenzelm@26314
   397
chaieb@23327
   398
end
chaieb@23327
   399
chaieb@23327
   400
wenzelm@23258
   401
lemmas bool_simps = simp_thms(1-34)
wenzelm@23252
   402
lemma dnf:
wenzelm@23252
   403
    "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
wenzelm@23252
   404
    "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
wenzelm@23252
   405
  by blast+
wenzelm@23252
   406
wenzelm@23252
   407
lemmas weak_dnf_simps = dnf bool_simps
wenzelm@23252
   408
wenzelm@23252
   409
lemma nnf_simps:
wenzelm@23252
   410
    "(\<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)"
wenzelm@23252
   411
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
wenzelm@23252
   412
  by blast+
wenzelm@23252
   413
wenzelm@23252
   414
lemma PFalse:
wenzelm@23252
   415
    "P \<equiv> False \<Longrightarrow> \<not> P"
wenzelm@23252
   416
    "\<not> P \<Longrightarrow> (P \<equiv> False)"
wenzelm@23252
   417
  by auto
wenzelm@23252
   418
use "Tools/Groebner_Basis/groebner.ML"
wenzelm@23252
   419
chaieb@23332
   420
method_setup algebra =
wenzelm@23458
   421
{*
chaieb@23332
   422
let
chaieb@23332
   423
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
chaieb@23332
   424
 val addN = "add"
chaieb@23332
   425
 val delN = "del"
chaieb@23332
   426
 val any_keyword = keyword addN || keyword delN
chaieb@23332
   427
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
chaieb@23332
   428
in
chaieb@23332
   429
fn src => Method.syntax 
chaieb@23332
   430
    ((Scan.optional (keyword addN |-- thms) []) -- 
chaieb@23332
   431
    (Scan.optional (keyword delN |-- thms) [])) src 
chaieb@23332
   432
 #> (fn ((add_ths, del_ths), ctxt) => 
chaieb@25250
   433
       Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
chaieb@23332
   434
end
chaieb@25250
   435
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
chaieb@27666
   436
declare dvd_def[algebra]
chaieb@27666
   437
declare dvd_eq_mod_eq_0[symmetric, algebra]
chaieb@27666
   438
declare nat_mod_div_trivial[algebra]
chaieb@27666
   439
declare nat_mod_mod_trivial[algebra]
chaieb@27666
   440
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
chaieb@27666
   441
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
chaieb@27666
   442
declare zmod_zdiv_equality[symmetric,algebra]
chaieb@27666
   443
declare zdiv_zmod_equality[symmetric, algebra]
chaieb@27666
   444
declare zdiv_zminus_zminus[algebra]
chaieb@27666
   445
declare zmod_zminus_zminus[algebra]
chaieb@27666
   446
declare zdiv_zminus2[algebra]
chaieb@27666
   447
declare zmod_zminus2[algebra]
chaieb@27666
   448
declare zdiv_zero[algebra]
chaieb@27666
   449
declare zmod_zero[algebra]
chaieb@27666
   450
declare zmod_1[algebra]
chaieb@27666
   451
declare zdiv_1[algebra]
chaieb@27666
   452
declare zmod_minus1_right[algebra]
chaieb@27666
   453
declare zdiv_minus1_right[algebra]
chaieb@27666
   454
declare mod_div_trivial[algebra]
chaieb@27666
   455
declare mod_mod_trivial[algebra]
chaieb@27666
   456
declare zmod_zmult_self1[algebra]
chaieb@27666
   457
declare zmod_zmult_self2[algebra]
chaieb@27666
   458
declare zmod_eq_0_iff[algebra]
chaieb@27666
   459
declare zdvd_0_left[algebra]
chaieb@27666
   460
declare zdvd1_eq[algebra]
chaieb@27666
   461
declare zmod_eq_dvd_iff[algebra]
chaieb@27666
   462
declare nat_mod_eq_iff[algebra]
wenzelm@23252
   463
wenzelm@23252
   464
end