src/HOL/Groebner_Basis.thy
author chaieb
Mon, 21 Jul 2008 13:36:39 +0200
changeset 27666 1436d81d1294
parent 26462 dac4e2bce00d
child 28402 09e4aa3ddc25
permissions -rw-r--r--
Relevant rules added to algebra's context
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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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27666
1436d81d1294 Relevant rules added to algebra's context
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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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   286
  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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   292
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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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   295
proof(clarify)
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   296
  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
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   297
    and eq: "add b (mul r c) = add b (mul r d)"
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   298
  hence "mul r c = mul r d" using cnd add_cancel by simp
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   299
  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
67268bb40b21 Semiring normalization and Groebner Bases.
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   301
  thus "False" using add_mul_solve nz cnd by simp
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qed
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parents:
diff changeset
   303
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   304
lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   305
proof-
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   306
  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   307
  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   308
qed
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   309
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   310
declare gb_semiring_axioms' [normalizer del]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   311
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   312
lemmas semiringb_axioms' = semiringb_axioms [normalizer
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   313
  semiring ops: semiring_ops
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   314
  semiring rules: semiring_rules
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   315
  idom rules: noteq_reduce add_scale_eq_noteq]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   316
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   317
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   318
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   319
locale ringb = semiringb + gb_ring + 
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   320
  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   321
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   322
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   323
declare gb_ring_axioms' [normalizer del]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   324
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   325
lemmas ringb_axioms' = ringb_axioms [normalizer
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   326
  semiring ops: semiring_ops
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   327
  semiring rules: semiring_rules
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   328
  ring ops: ring_ops
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   329
  ring rules: ring_rules
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   330
  idom rules: noteq_reduce add_scale_eq_noteq
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   331
  ideal rules: subr0_iff add_r0_iff]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   332
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   333
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   334
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   335
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   336
lemma no_zero_divirors_neq0:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   337
  assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   338
    and ab: "a*b = 0" shows "b = 0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   339
proof -
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   340
  { assume bz: "b \<noteq> 0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   341
    from no_zero_divisors [OF az bz] ab have False by blast }
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   342
  thus "b = 0" by blast
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   343
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   344
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   345
interpretation class_ringb: ringb
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   346
  ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   347
proof(unfold_locales, simp add: ring_simps power_Suc, auto)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   348
  fix w x y z ::"'a::{idom,recpower,number_ring}"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   349
  assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   350
  hence ynz': "y - z \<noteq> 0" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   351
  from p have "w * y + x* z - w*z - x*y = 0" by simp
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   352
  hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   353
  hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   354
  with  no_zero_divirors_neq0 [OF ynz']
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   355
  have "w - x = 0" by blast
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   356
  thus "w = x"  by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   357
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   358
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   359
declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   360
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   361
interpretation natgb: semiringb
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   362
  ["op +" "op *" "op ^" "0::nat" "1"]
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   363
proof (unfold_locales, simp add: ring_simps power_Suc)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   364
  fix w x y z ::"nat"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   365
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   366
    hence "y < z \<or> y > z" by arith
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   367
    moreover {
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   368
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   369
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   370
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   371
      hence "x*k = w*k" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   372
      hence "w = x" using kp by (simp add: mult_cancel2) }
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   373
    moreover {
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   374
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   375
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   376
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   377
      hence "w*k = x*k" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   378
      hence "w = x" using kp by (simp add: mult_cancel2)}
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   379
    ultimately have "w=x" by blast }
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   380
  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   381
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   382
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   383
declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   384
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   385
locale fieldgb = ringb + gb_field
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   386
begin
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   387
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   388
declare gb_field_axioms' [normalizer del]
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   389
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   390
lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   391
  semiring ops: semiring_ops
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   392
  semiring rules: semiring_rules
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   393
  ring ops: ring_ops
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   394
  ring rules: ring_rules
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   395
  idom rules: noteq_reduce add_scale_eq_noteq
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   396
  ideal rules: subr0_iff add_r0_iff]
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   397
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   398
end
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   399
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   400
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
   401
lemmas bool_simps = simp_thms(1-34)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   402
lemma dnf:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   403
    "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   404
    "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   405
  by blast+
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   406
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   407
lemmas weak_dnf_simps = dnf bool_simps
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   408
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   409
lemma nnf_simps:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   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)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   411
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   412
  by blast+
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   413
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   414
lemma PFalse:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   415
    "P \<equiv> False \<Longrightarrow> \<not> P"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   416
    "\<not> P \<Longrightarrow> (P \<equiv> False)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   417
  by auto
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   418
use "Tools/Groebner_Basis/groebner.ML"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   419
23332
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   420
method_setup algebra =
23458
b2267a9e9e28 tuned comments;
wenzelm
parents: 23389
diff changeset
   421
{*
23332
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   422
let
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   423
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   424
 val addN = "add"
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   425
 val delN = "del"
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   426
 val any_keyword = keyword addN || keyword delN
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   427
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   428
in
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   429
fn src => Method.syntax 
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   430
    ((Scan.optional (keyword addN |-- thms) []) -- 
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   431
    (Scan.optional (keyword delN |-- thms) [])) src 
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   432
 #> (fn ((add_ths, del_ths), ctxt) => 
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   433
       Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
23332
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   434
end
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   435
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
27666
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   436
declare dvd_def[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   437
declare dvd_eq_mod_eq_0[symmetric, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   438
declare nat_mod_div_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   439
declare nat_mod_mod_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   440
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   441
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   442
declare zmod_zdiv_equality[symmetric,algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   443
declare zdiv_zmod_equality[symmetric, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   444
declare zdiv_zminus_zminus[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   445
declare zmod_zminus_zminus[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   446
declare zdiv_zminus2[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   447
declare zmod_zminus2[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   448
declare zdiv_zero[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   449
declare zmod_zero[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   450
declare zmod_1[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   451
declare zdiv_1[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   452
declare zmod_minus1_right[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   453
declare zdiv_minus1_right[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   454
declare mod_div_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   455
declare mod_mod_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   456
declare zmod_zmult_self1[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   457
declare zmod_zmult_self2[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   458
declare zmod_eq_0_iff[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   459
declare zdvd_0_left[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   460
declare zdvd1_eq[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   461
declare zmod_eq_dvd_iff[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   462
declare nat_mod_eq_iff[algebra]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   463
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   464
end