src/HOL/Groebner_Basis.thy
author haftmann
Wed Oct 28 19:09:47 2009 +0100 (2009-10-28)
changeset 33296 a3924d1069e5
parent 31790 05c92381363c
child 33361 1f18de40b43f
permissions -rw-r--r--
moved theory Divides after theory Nat_Numeral; tuned some proof texts
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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 IntDiv
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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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  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
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    and "TERM r0" and "TERM r1" .
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lemma semiring_rules:
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  "add (mul a m) (mul b m) = mul (add a b) m"
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  "add (mul a m) m = mul (add a r1) m"
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  "add m (mul a m) = mul (add a r1) m"
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  "add m m = mul (add r1 r1) m"
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  "add r0 a = a"
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  "add a r0 = a"
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  "mul a b = mul b a"
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  "mul (add a b) c = add (mul a c) (mul b c)"
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  "mul r0 a = r0"
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  "mul a r0 = r0"
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  "mul r1 a = a"
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  "mul a r1 = a"
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  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
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  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
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  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
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  "mul (mul lx ly) rx = mul (mul lx rx) ly"
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  "mul (mul lx ly) rx = mul lx (mul ly rx)"
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  "mul lx (mul rx ry) = mul (mul lx rx) ry"
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  "mul lx (mul rx ry) = mul rx (mul lx ry)"
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  "add (add a b) (add c d) = add (add a c) (add b d)"
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  "add (add a b) c = add a (add b c)"
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  "add a (add c d) = add c (add a d)"
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  "add (add a b) c = add (add a c) b"
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  "add a c = add c a"
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  "add a (add c d) = add (add a c) d"
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  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
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  "mul x (pwr x q) = pwr x (Suc q)"
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  "mul (pwr x q) x = pwr x (Suc q)"
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  "mul x x = pwr x 2"
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  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
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  "pwr (pwr x p) q = pwr x (p * q)"
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  "pwr x 0 = r1"
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  "pwr x 1 = x"
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  "mul x (add y z) = add (mul x y) (mul x z)"
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  "pwr x (Suc q) = mul x (pwr x q)"
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  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
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  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
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proof -
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  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
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next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
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next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
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next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
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next show "add r0 a = a" using add_0 by simp
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next show "add a r0 = a" using add_0 add_c by simp
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next show "mul a b = mul b a" using mul_c by simp
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next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
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next show "mul r0 a = r0" using mul_0 by simp
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next show "mul a r0 = r0" using mul_0 mul_c by simp
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next show "mul r1 a = a" using mul_1 by simp
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next show "mul a r1 = a" using mul_1 mul_c by simp
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next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
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    using mul_c mul_a by simp
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next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
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    using mul_a by simp
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next
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  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
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  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
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  finally
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  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
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    using mul_c by simp
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next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
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next
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  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
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next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
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next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
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next show "add (add a b) (add c d) = add (add a c) (add b d)"
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    using add_c add_a by simp
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next show "add (add a b) c = add a (add b c)" using add_a by simp
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next show "add a (add c d) = add c (add a d)"
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    apply (simp add: add_a) by (simp only: add_c)
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next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
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next show "add a c = add c a" by (rule add_c)
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next show "add a (add c d) = add (add a c) d" using add_a by simp
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next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
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next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
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next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
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next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
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next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
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next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
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next show "pwr x 0 = r1" using pwr_0 .
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next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
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next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
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next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
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next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
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next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
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    by (simp add: nat_number pwr_Suc mul_pwr)
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qed
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lemmas gb_semiring_axioms' =
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  gb_semiring_axioms [normalizer
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    semiring ops: semiring_ops
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    semiring rules: semiring_rules]
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end
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interpretation class_semiring: gb_semiring
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    "op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1"
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  proof qed (auto simp add: algebra_simps power_Suc)
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lemmas nat_arith =
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  add_nat_number_of
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  diff_nat_number_of
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  mult_nat_number_of
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  eq_nat_number_of
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  less_nat_number_of
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lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
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  by (simp add: numeral_1_eq_1)
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lemmas comp_arith =
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  Let_def arith_simps nat_arith rel_simps neg_simps if_False
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  if_True add_0 add_Suc add_number_of_left mult_number_of_left
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  numeral_1_eq_1[symmetric] Suc_eq_plus1
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  numeral_0_eq_0[symmetric] numerals[symmetric]
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  iszero_simps not_iszero_Numeral1
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lemmas semiring_norm = comp_arith
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ML {*
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local
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open Conv;
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fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);
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fun int_of_rat x =
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  (case Rat.quotient_of_rat x of (i, 1) => i
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  | _ => error "int_of_rat: bad int");
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val numeral_conv =
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  Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
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  Simplifier.rewrite (HOL_basic_ss addsimps
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    (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(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: shows "TERM (sub x y)" and "TERM (neg x)" .
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lemmas ring_rules = neg_mul sub_add
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lemmas gb_ring_axioms' =
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  gb_ring_axioms [normalizer
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    semiring ops: semiring_ops
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    semiring rules: semiring_rules
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    ring ops: ring_ops
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    ring rules: ring_rules]
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end
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interpretation class_ring: gb_ring "op +" "op *" "op ^"
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    "0::'a::{comm_semiring_1,number_ring}" 1 "op -" "uminus"
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  proof qed simp_all
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declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
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use "Tools/Groebner_Basis/normalizer.ML"
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method_setup sring_norm = {*
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  Scan.succeed (SIMPLE_METHOD' o Normalizer.semiring_normalize_tac)
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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_inverse: "divide x y = mul x (inverse y)"
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     and inverse_divide: "inverse x = divide r1 x"
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begin
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lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
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lemmas field_rules = divide_inverse inverse_divide
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lemmas gb_field_axioms' =
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  gb_field_axioms [normalizer
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    semiring ops: semiring_ops
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    semiring rules: semiring_rules
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    ring ops: ring_ops
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    ring rules: ring_rules
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    field ops: field_ops
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    field rules: field_rules]
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end
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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)"
wenzelm@23252
   294
    using add_mul_solve by blast
wenzelm@23252
   295
  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)"
wenzelm@23252
   296
    by simp
wenzelm@23252
   297
qed
wenzelm@23252
   298
wenzelm@23252
   299
lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
wenzelm@23252
   300
  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
wenzelm@23252
   301
proof(clarify)
wenzelm@23252
   302
  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
wenzelm@23252
   303
    and eq: "add b (mul r c) = add b (mul r d)"
wenzelm@23252
   304
  hence "mul r c = mul r d" using cnd add_cancel by simp
wenzelm@23252
   305
  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
wenzelm@23252
   306
    using mul_0 add_cancel by simp
wenzelm@23252
   307
  thus "False" using add_mul_solve nz cnd by simp
wenzelm@23252
   308
qed
wenzelm@23252
   309
chaieb@25250
   310
lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
chaieb@25250
   311
proof-
chaieb@25250
   312
  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
chaieb@25250
   313
  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
chaieb@25250
   314
qed
chaieb@25250
   315
wenzelm@26462
   316
declare gb_semiring_axioms' [normalizer del]
wenzelm@23252
   317
wenzelm@26462
   318
lemmas semiringb_axioms' = semiringb_axioms [normalizer
wenzelm@23252
   319
  semiring ops: semiring_ops
wenzelm@23252
   320
  semiring rules: semiring_rules
wenzelm@26314
   321
  idom rules: noteq_reduce add_scale_eq_noteq]
wenzelm@23252
   322
wenzelm@23252
   323
end
wenzelm@23252
   324
chaieb@25250
   325
locale ringb = semiringb + gb_ring + 
chaieb@25250
   326
  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
wenzelm@23252
   327
begin
wenzelm@23252
   328
wenzelm@26462
   329
declare gb_ring_axioms' [normalizer del]
wenzelm@23252
   330
wenzelm@26462
   331
lemmas ringb_axioms' = ringb_axioms [normalizer
wenzelm@23252
   332
  semiring ops: semiring_ops
wenzelm@23252
   333
  semiring rules: semiring_rules
wenzelm@23252
   334
  ring ops: ring_ops
wenzelm@23252
   335
  ring rules: ring_rules
chaieb@25250
   336
  idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314
   337
  ideal rules: subr0_iff add_r0_iff]
wenzelm@23252
   338
wenzelm@23252
   339
end
wenzelm@23252
   340
chaieb@25250
   341
wenzelm@23252
   342
lemma no_zero_divirors_neq0:
wenzelm@23252
   343
  assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
wenzelm@23252
   344
    and ab: "a*b = 0" shows "b = 0"
wenzelm@23252
   345
proof -
wenzelm@23252
   346
  { assume bz: "b \<noteq> 0"
wenzelm@23252
   347
    from no_zero_divisors [OF az bz] ab have False by blast }
wenzelm@23252
   348
  thus "b = 0" by blast
wenzelm@23252
   349
qed
wenzelm@23252
   350
wenzelm@30729
   351
interpretation class_ringb: ringb
haftmann@31017
   352
  "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
nipkow@29667
   353
proof(unfold_locales, simp add: algebra_simps power_Suc, auto)
haftmann@31017
   354
  fix w x y z ::"'a::{idom,number_ring}"
wenzelm@23252
   355
  assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
wenzelm@23252
   356
  hence ynz': "y - z \<noteq> 0" by simp
wenzelm@23252
   357
  from p have "w * y + x* z - w*z - x*y = 0" by simp
nipkow@29667
   358
  hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
nipkow@29667
   359
  hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
wenzelm@23252
   360
  with  no_zero_divirors_neq0 [OF ynz']
wenzelm@23252
   361
  have "w - x = 0" by blast
wenzelm@23252
   362
  thus "w = x"  by simp
wenzelm@23252
   363
qed
wenzelm@23252
   364
wenzelm@26462
   365
declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
wenzelm@23252
   366
wenzelm@30729
   367
interpretation natgb: semiringb
ballarin@29223
   368
  "op +" "op *" "op ^" "0::nat" "1"
nipkow@29667
   369
proof (unfold_locales, simp add: algebra_simps power_Suc)
wenzelm@23252
   370
  fix w x y z ::"nat"
wenzelm@23252
   371
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
wenzelm@23252
   372
    hence "y < z \<or> y > z" by arith
wenzelm@23252
   373
    moreover {
wenzelm@23252
   374
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
wenzelm@23252
   375
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
nipkow@29667
   376
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
wenzelm@23252
   377
      hence "x*k = w*k" by simp
wenzelm@23252
   378
      hence "w = x" using kp by (simp add: mult_cancel2) }
wenzelm@23252
   379
    moreover {
wenzelm@23252
   380
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
wenzelm@23252
   381
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
nipkow@29667
   382
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
wenzelm@23252
   383
      hence "w*k = x*k" by simp
wenzelm@23252
   384
      hence "w = x" using kp by (simp add: mult_cancel2)}
wenzelm@23252
   385
    ultimately have "w=x" by blast }
wenzelm@23252
   386
  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
wenzelm@23252
   387
qed
wenzelm@23252
   388
wenzelm@26462
   389
declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
wenzelm@23252
   390
chaieb@23327
   391
locale fieldgb = ringb + gb_field
chaieb@23327
   392
begin
chaieb@23327
   393
wenzelm@26462
   394
declare gb_field_axioms' [normalizer del]
chaieb@23327
   395
wenzelm@26462
   396
lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
chaieb@23327
   397
  semiring ops: semiring_ops
chaieb@23327
   398
  semiring rules: semiring_rules
chaieb@23327
   399
  ring ops: ring_ops
chaieb@23327
   400
  ring rules: ring_rules
chaieb@30866
   401
  field ops: field_ops
chaieb@30866
   402
  field rules: field_rules
chaieb@25250
   403
  idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314
   404
  ideal rules: subr0_iff add_r0_iff]
wenzelm@26314
   405
chaieb@23327
   406
end
chaieb@23327
   407
chaieb@23327
   408
wenzelm@23258
   409
lemmas bool_simps = simp_thms(1-34)
wenzelm@23252
   410
lemma dnf:
wenzelm@23252
   411
    "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
wenzelm@23252
   412
    "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
wenzelm@23252
   413
  by blast+
wenzelm@23252
   414
wenzelm@23252
   415
lemmas weak_dnf_simps = dnf bool_simps
wenzelm@23252
   416
wenzelm@23252
   417
lemma nnf_simps:
wenzelm@23252
   418
    "(\<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
   419
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
wenzelm@23252
   420
  by blast+
wenzelm@23252
   421
wenzelm@23252
   422
lemma PFalse:
wenzelm@23252
   423
    "P \<equiv> False \<Longrightarrow> \<not> P"
wenzelm@23252
   424
    "\<not> P \<Longrightarrow> (P \<equiv> False)"
wenzelm@23252
   425
  by auto
wenzelm@23252
   426
use "Tools/Groebner_Basis/groebner.ML"
wenzelm@23252
   427
chaieb@23332
   428
method_setup algebra =
wenzelm@23458
   429
{*
chaieb@23332
   430
let
chaieb@23332
   431
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
chaieb@23332
   432
 val addN = "add"
chaieb@23332
   433
 val delN = "del"
chaieb@23332
   434
 val any_keyword = keyword addN || keyword delN
chaieb@23332
   435
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
chaieb@23332
   436
in
wenzelm@30549
   437
  ((Scan.optional (keyword addN |-- thms) []) -- 
wenzelm@30549
   438
   (Scan.optional (keyword delN |-- thms) [])) >>
wenzelm@30549
   439
  (fn (add_ths, del_ths) => fn ctxt =>
wenzelm@30510
   440
       SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
chaieb@23332
   441
end
chaieb@25250
   442
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
chaieb@27666
   443
declare dvd_def[algebra]
chaieb@27666
   444
declare dvd_eq_mod_eq_0[symmetric, algebra]
nipkow@30027
   445
declare mod_div_trivial[algebra]
nipkow@30027
   446
declare mod_mod_trivial[algebra]
chaieb@27666
   447
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
chaieb@27666
   448
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
chaieb@27666
   449
declare zmod_zdiv_equality[symmetric,algebra]
chaieb@27666
   450
declare zdiv_zmod_equality[symmetric, algebra]
chaieb@27666
   451
declare zdiv_zminus_zminus[algebra]
chaieb@27666
   452
declare zmod_zminus_zminus[algebra]
chaieb@27666
   453
declare zdiv_zminus2[algebra]
chaieb@27666
   454
declare zmod_zminus2[algebra]
chaieb@27666
   455
declare zdiv_zero[algebra]
chaieb@27666
   456
declare zmod_zero[algebra]
nipkow@30031
   457
declare mod_by_1[algebra]
nipkow@30031
   458
declare div_by_1[algebra]
chaieb@27666
   459
declare zmod_minus1_right[algebra]
chaieb@27666
   460
declare zdiv_minus1_right[algebra]
chaieb@27666
   461
declare mod_div_trivial[algebra]
chaieb@27666
   462
declare mod_mod_trivial[algebra]
nipkow@30034
   463
declare mod_mult_self2_is_0[algebra]
nipkow@30034
   464
declare mod_mult_self1_is_0[algebra]
chaieb@27666
   465
declare zmod_eq_0_iff[algebra]
nipkow@30042
   466
declare dvd_0_left_iff[algebra]
chaieb@27666
   467
declare zdvd1_eq[algebra]
chaieb@27666
   468
declare zmod_eq_dvd_iff[algebra]
chaieb@27666
   469
declare nat_mod_eq_iff[algebra]
wenzelm@23252
   470
haftmann@28402
   471
subsection{* Groebner Bases for fields *}
haftmann@28402
   472
wenzelm@30729
   473
interpretation class_fieldgb:
haftmann@31017
   474
  fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
haftmann@28402
   475
haftmann@28402
   476
lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
haftmann@28402
   477
lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
haftmann@28402
   478
  by simp
haftmann@28402
   479
lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
haftmann@28402
   480
  by simp
haftmann@28402
   481
lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
haftmann@28402
   482
  by simp
haftmann@28402
   483
lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
haftmann@28402
   484
  by simp
haftmann@28402
   485
haftmann@28402
   486
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
haftmann@28402
   487
haftmann@28402
   488
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
haftmann@28402
   489
  by (simp add: add_divide_distrib)
haftmann@28402
   490
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
haftmann@28402
   491
  by (simp add: add_divide_distrib)
chaieb@30869
   492
ML{* let open Conv in fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))   (@{thm divide_inverse} RS sym)end*}
haftmann@28402
   493
ML{* 
haftmann@28402
   494
local
haftmann@28402
   495
 val zr = @{cpat "0"}
haftmann@28402
   496
 val zT = ctyp_of_term zr
haftmann@28402
   497
 val geq = @{cpat "op ="}
haftmann@28402
   498
 val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
haftmann@28402
   499
 val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
haftmann@28402
   500
 val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
haftmann@28402
   501
 val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
haftmann@28402
   502
haftmann@28402
   503
 fun prove_nz ss T t =
haftmann@28402
   504
    let
haftmann@28402
   505
      val z = instantiate_cterm ([(zT,T)],[]) zr
haftmann@28402
   506
      val eq = instantiate_cterm ([(eqT,T)],[]) geq
haftmann@28402
   507
      val th = Simplifier.rewrite (ss addsimps simp_thms)
haftmann@28402
   508
           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
haftmann@28402
   509
                  (Thm.capply (Thm.capply eq t) z)))
haftmann@28402
   510
    in equal_elim (symmetric th) TrueI
haftmann@28402
   511
    end
haftmann@28402
   512
haftmann@28402
   513
 fun proc phi ss ct =
haftmann@28402
   514
  let
haftmann@28402
   515
    val ((x,y),(w,z)) =
haftmann@28402
   516
         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
haftmann@28402
   517
    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
haftmann@28402
   518
    val T = ctyp_of_term x
haftmann@28402
   519
    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
haftmann@28402
   520
    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
haftmann@28402
   521
  in SOME (implies_elim (implies_elim th y_nz) z_nz)
haftmann@28402
   522
  end
haftmann@28402
   523
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
haftmann@28402
   524
haftmann@28402
   525
 fun proc2 phi ss ct =
haftmann@28402
   526
  let
haftmann@28402
   527
    val (l,r) = Thm.dest_binop ct
haftmann@28402
   528
    val T = ctyp_of_term l
haftmann@28402
   529
  in (case (term_of l, term_of r) of
haftmann@28402
   530
      (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
haftmann@28402
   531
        let val (x,y) = Thm.dest_binop l val z = r
haftmann@28402
   532
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
haftmann@28402
   533
            val ynz = prove_nz ss T y
haftmann@28402
   534
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
haftmann@28402
   535
        end
haftmann@28402
   536
     | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402
   537
        let val (x,y) = Thm.dest_binop r val z = l
haftmann@28402
   538
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
haftmann@28402
   539
            val ynz = prove_nz ss T y
haftmann@28402
   540
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
haftmann@28402
   541
        end
haftmann@28402
   542
     | _ => NONE)
haftmann@28402
   543
  end
haftmann@28402
   544
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
haftmann@28402
   545
haftmann@28402
   546
 fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
haftmann@28402
   547
   | is_number t = can HOLogic.dest_number t
haftmann@28402
   548
haftmann@28402
   549
 val is_number = is_number o term_of
haftmann@28402
   550
haftmann@28402
   551
 fun proc3 phi ss ct =
haftmann@28402
   552
  (case term_of ct of
haftmann@28402
   553
    Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
haftmann@28402
   554
      let
haftmann@28402
   555
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402
   556
        val _ = map is_number [a,b,c]
haftmann@28402
   557
        val T = ctyp_of_term c
haftmann@28402
   558
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
haftmann@28402
   559
      in SOME (mk_meta_eq th) end
haftmann@28402
   560
  | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
haftmann@28402
   561
      let
haftmann@28402
   562
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402
   563
        val _ = map is_number [a,b,c]
haftmann@28402
   564
        val T = ctyp_of_term c
haftmann@28402
   565
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
haftmann@28402
   566
      in SOME (mk_meta_eq th) end
haftmann@28402
   567
  | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
haftmann@28402
   568
      let
haftmann@28402
   569
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402
   570
        val _ = map is_number [a,b,c]
haftmann@28402
   571
        val T = ctyp_of_term c
haftmann@28402
   572
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
haftmann@28402
   573
      in SOME (mk_meta_eq th) end
haftmann@28402
   574
  | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402
   575
    let
haftmann@28402
   576
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402
   577
        val _ = map is_number [a,b,c]
haftmann@28402
   578
        val T = ctyp_of_term c
haftmann@28402
   579
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
haftmann@28402
   580
      in SOME (mk_meta_eq th) end
haftmann@28402
   581
  | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402
   582
    let
haftmann@28402
   583
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402
   584
        val _ = map is_number [a,b,c]
haftmann@28402
   585
        val T = ctyp_of_term c
haftmann@28402
   586
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
haftmann@28402
   587
      in SOME (mk_meta_eq th) end
haftmann@28402
   588
  | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402
   589
    let
haftmann@28402
   590
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402
   591
        val _ = map is_number [a,b,c]
haftmann@28402
   592
        val T = ctyp_of_term c
haftmann@28402
   593
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
haftmann@28402
   594
      in SOME (mk_meta_eq th) end
haftmann@28402
   595
  | _ => NONE)
haftmann@28402
   596
  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
haftmann@28402
   597
haftmann@28402
   598
val add_frac_frac_simproc =
haftmann@28402
   599
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
haftmann@28402
   600
                     name = "add_frac_frac_simproc",
haftmann@28402
   601
                     proc = proc, identifier = []}
haftmann@28402
   602
haftmann@28402
   603
val add_frac_num_simproc =
haftmann@28402
   604
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
haftmann@28402
   605
                     name = "add_frac_num_simproc",
haftmann@28402
   606
                     proc = proc2, identifier = []}
haftmann@28402
   607
haftmann@28402
   608
val ord_frac_simproc =
haftmann@28402
   609
  make_simproc
haftmann@28402
   610
    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
haftmann@28402
   611
             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
haftmann@28402
   612
             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
haftmann@28402
   613
             @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
haftmann@28402
   614
             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
haftmann@28402
   615
             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
haftmann@28402
   616
             name = "ord_frac_simproc", proc = proc3, identifier = []}
haftmann@28402
   617
chaieb@30869
   618
local
chaieb@30869
   619
open Conv
chaieb@30869
   620
in
chaieb@30869
   621
haftmann@28402
   622
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
haftmann@28402
   623
           @{thm "divide_Numeral1"},
haftmann@28402
   624
           @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
haftmann@28402
   625
           @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
haftmann@28402
   626
           @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
haftmann@28402
   627
           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
haftmann@28402
   628
           @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
haftmann@28402
   629
           @{thm "diff_def"}, @{thm "minus_divide_left"},
chaieb@30869
   630
           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
chaieb@30869
   631
           @{thm divide_inverse} RS sym, @{thm inverse_divide}, 
chaieb@30869
   632
           fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))   
chaieb@30869
   633
           (@{thm divide_inverse} RS sym)]
haftmann@28402
   634
haftmann@28402
   635
val comp_conv = (Simplifier.rewrite
haftmann@28402
   636
(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
huffman@28987
   637
              addsimps ths addsimps simp_thms
haftmann@31068
   638
              addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@28402
   639
               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
haftmann@28402
   640
                            ord_frac_simproc]
haftmann@28402
   641
                addcongs [@{thm "if_weak_cong"}]))
haftmann@28402
   642
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
haftmann@28402
   643
  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
wenzelm@23252
   644
end
haftmann@28402
   645
haftmann@28402
   646
fun numeral_is_const ct =
haftmann@28402
   647
  case term_of ct of
haftmann@28402
   648
   Const (@{const_name "HOL.divide"},_) $ a $ b =>
chaieb@30866
   649
     can HOLogic.dest_number a andalso can HOLogic.dest_number b
chaieb@30866
   650
 | Const (@{const_name "HOL.inverse"},_)$t => can HOLogic.dest_number t
haftmann@28402
   651
 | t => can HOLogic.dest_number t
haftmann@28402
   652
haftmann@28402
   653
fun dest_const ct = ((case term_of ct of
haftmann@28402
   654
   Const (@{const_name "HOL.divide"},_) $ a $ b=>
haftmann@28402
   655
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
chaieb@30869
   656
 | Const (@{const_name "HOL.inverse"},_)$t => 
chaieb@30869
   657
               Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
haftmann@28402
   658
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
haftmann@28402
   659
   handle TERM _ => error "ring_dest_const")
haftmann@28402
   660
haftmann@28402
   661
fun mk_const phi cT x =
haftmann@28402
   662
 let val (a, b) = Rat.quotient_of_rat x
haftmann@28402
   663
 in if b = 1 then Numeral.mk_cnumber cT a
haftmann@28402
   664
    else Thm.capply
haftmann@28402
   665
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
haftmann@28402
   666
                     (Numeral.mk_cnumber cT a))
haftmann@28402
   667
         (Numeral.mk_cnumber cT b)
haftmann@28402
   668
  end
haftmann@28402
   669
haftmann@28402
   670
in
haftmann@28402
   671
 val field_comp_conv = comp_conv;
haftmann@28402
   672
 val fieldgb_declaration = 
haftmann@28402
   673
  NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
haftmann@28402
   674
   {is_const = K numeral_is_const,
haftmann@28402
   675
    dest_const = K dest_const,
haftmann@28402
   676
    mk_const = mk_const,
haftmann@28402
   677
    conv = K (K comp_conv)}
haftmann@28402
   678
end;
haftmann@28402
   679
*}
haftmann@28402
   680
haftmann@28402
   681
declaration fieldgb_declaration
haftmann@28402
   682
haftmann@28402
   683
end