src/HOL/Groebner_Basis.thy
author haftmann
Thu May 06 17:59:19 2010 +0200 (2010-05-06)
changeset 36714 ae84ddf03c58
parent 36712 2f4c318861b3
child 36716 b09f3ad3208f
permissions -rw-r--r--
dropped duplicate comp_arith
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(*  Title:      HOL/Groebner_Basis.thy
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    Author:     Amine Chaieb, TU Muenchen
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*)
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header {* Semiring normalization and Groebner Bases *}
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theory Groebner_Basis
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imports Numeral_Simprocs Nat_Transfer
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uses
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  "Tools/Groebner_Basis/normalizer.ML"
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  ("Tools/Groebner_Basis/groebner.ML")
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begin
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subsection {* Semiring normalization *}
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setup Normalizer.setup
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locale normalizing_semiring =
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  fixes add mul pwr r0 r1
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  assumes add_a:"(add x (add y z) = add (add x y) z)"
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    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
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    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
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    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
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    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
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    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
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begin
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lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
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proof (induct p)
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  case 0
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  then show ?case by (auto simp add: pwr_0 mul_1)
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next
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  case Suc
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  from this [symmetric] show ?case
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    by (auto simp add: pwr_Suc mul_1 mul_a)
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qed
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lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
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proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
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  fix q x y
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  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
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  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
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    by (simp add: mul_a)
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  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
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  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
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  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
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    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
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qed
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lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
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proof (induct p arbitrary: q)
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  case 0
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  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
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next
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  case Suc
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  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
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qed
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lemma semiring_ops:
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  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
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    and "TERM r0" and "TERM r1" .
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lemma semiring_rules:
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  "add (mul a m) (mul b m) = mul (add a b) m"
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  "add (mul a m) m = mul (add a r1) m"
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  "add m (mul a m) = mul (add a r1) m"
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  "add m m = mul (add r1 r1) m"
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  "add r0 a = a"
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  "add a r0 = a"
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  "mul a b = mul b a"
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  "mul (add a b) c = add (mul a c) (mul b c)"
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  "mul r0 a = r0"
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  "mul a r0 = r0"
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  "mul r1 a = a"
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  "mul a r1 = a"
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  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
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  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
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  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
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  "mul (mul lx ly) rx = mul (mul lx rx) ly"
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  "mul (mul lx ly) rx = mul lx (mul ly rx)"
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  "mul lx (mul rx ry) = mul (mul lx rx) ry"
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  "mul lx (mul rx ry) = mul rx (mul lx ry)"
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  "add (add a b) (add c d) = add (add a c) (add b d)"
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  "add (add a b) c = add a (add b c)"
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  "add a (add c d) = add c (add a d)"
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  "add (add a b) c = add (add a c) b"
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  "add a c = add c a"
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  "add a (add c d) = add (add a c) d"
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  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
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  "mul x (pwr x q) = pwr x (Suc q)"
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  "mul (pwr x q) x = pwr x (Suc q)"
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  "mul x x = pwr x 2"
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  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
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  "pwr (pwr x p) q = pwr x (p * q)"
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  "pwr x 0 = r1"
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  "pwr x 1 = x"
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  "mul x (add y z) = add (mul x y) (mul x z)"
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  "pwr x (Suc q) = mul x (pwr x q)"
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  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
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  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
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proof -
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  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
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next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
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next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
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next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
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next show "add r0 a = a" using add_0 by simp
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next show "add a r0 = a" using add_0 add_c by simp
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next show "mul a b = mul b a" using mul_c by simp
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next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
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next show "mul r0 a = r0" using mul_0 by simp
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next show "mul a r0 = r0" using mul_0 mul_c by simp
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next show "mul r1 a = a" using mul_1 by simp
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next show "mul a r1 = a" using mul_1 mul_c by simp
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next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
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    using mul_c mul_a by simp
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next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
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    using mul_a by simp
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next
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  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
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  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
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  finally
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  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
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    using mul_c by simp
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next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
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next
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  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
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next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
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next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
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next show "add (add a b) (add c d) = add (add a c) (add b d)"
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    using add_c add_a by simp
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next show "add (add a b) c = add a (add b c)" using add_a by simp
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next show "add a (add c d) = add c (add a d)"
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    apply (simp add: add_a) by (simp only: add_c)
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next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
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next show "add a c = add c a" by (rule add_c)
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next show "add a (add c d) = add (add a c) d" using add_a by simp
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next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
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next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
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next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
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next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
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next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
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next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
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next show "pwr x 0 = r1" using pwr_0 .
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next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
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next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
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next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
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next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
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next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
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    by (simp add: nat_number' pwr_Suc mul_pwr)
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qed
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lemmas normalizing_semiring_axioms' =
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  normalizing_semiring_axioms [normalizer
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    semiring ops: semiring_ops
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    semiring rules: semiring_rules]
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end
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sublocale comm_semiring_1
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  < normalizing!: normalizing_semiring plus times power zero one
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proof
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qed (simp_all add: algebra_simps)
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lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
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  by simp
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lemmas semiring_norm =
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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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ML {*
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local
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fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);
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fun int_of_rat x =
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  (case Rat.quotient_of_rat x of (i, 1) => i
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  | _ => error "int_of_rat: bad int");
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val numeral_conv =
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  Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
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  Simplifier.rewrite (HOL_basic_ss addsimps
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    (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
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in
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fun normalizer_funs' key =
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  Normalizer.funs key
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   {is_const = fn phi => numeral_is_const,
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    dest_const = fn phi => fn ct =>
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      Rat.rat_of_int (snd
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        (HOLogic.dest_number (Thm.term_of ct)
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          handle TERM _ => error "ring_dest_const")),
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    mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
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    conv = fn phi => K numeral_conv}
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end
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*}
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declaration {* normalizer_funs' @{thm normalizing.normalizing_semiring_axioms'} *}
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locale normalizing_ring = normalizing_semiring +
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  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and neg :: "'a \<Rightarrow> 'a"
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  assumes neg_mul: "neg x = mul (neg r1) x"
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    and sub_add: "sub x y = add x (neg y)"
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begin
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lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
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lemmas ring_rules = neg_mul sub_add
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lemmas normalizing_ring_axioms' =
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  normalizing_ring_axioms [normalizer
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    semiring ops: semiring_ops
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    semiring rules: semiring_rules
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    ring ops: ring_ops
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    ring rules: ring_rules]
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end
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(*FIXME add class*)
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interpretation normalizing!: normalizing_ring plus times power
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  "0::'a::{comm_semiring_1,number_ring}" 1 minus uminus proof
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qed simp_all
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declaration {* normalizer_funs' @{thm normalizing.normalizing_ring_axioms'} *}
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locale normalizing_field = normalizing_ring +
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  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and inverse:: "'a \<Rightarrow> 'a"
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  assumes divide_inverse: "divide x y = mul x (inverse y)"
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     and inverse_divide: "inverse x = divide r1 x"
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begin
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lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
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lemmas field_rules = divide_inverse inverse_divide
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lemmas normalizing_field_axioms' =
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  normalizing_field_axioms [normalizer
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    semiring ops: semiring_ops
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    semiring rules: semiring_rules
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    ring ops: ring_ops
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    ring rules: ring_rules
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    field ops: field_ops
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    field rules: field_rules]
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end
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locale normalizing_semiring_cancel = normalizing_semiring +
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  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
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  and add_mul_solve: "add (mul w y) (mul x z) =
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    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
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begin
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lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
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proof-
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  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
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  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
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    using add_mul_solve by blast
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  finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
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    by simp
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qed
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lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
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  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
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proof(clarify)
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  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
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    and eq: "add b (mul r c) = add b (mul r d)"
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  hence "mul r c = mul r d" using cnd add_cancel by simp
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  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
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    using mul_0 add_cancel by simp
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  thus "False" using add_mul_solve nz cnd by simp
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qed
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lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
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proof-
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  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
chaieb@25250
   284
  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
chaieb@25250
   285
qed
chaieb@25250
   286
haftmann@36712
   287
declare normalizing_semiring_axioms' [normalizer del]
wenzelm@23252
   288
haftmann@36712
   289
lemmas normalizing_semiring_cancel_axioms' =
haftmann@36712
   290
  normalizing_semiring_cancel_axioms [normalizer
haftmann@36712
   291
    semiring ops: semiring_ops
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   292
    semiring rules: semiring_rules
haftmann@36712
   293
    idom rules: noteq_reduce add_scale_eq_noteq]
wenzelm@23252
   294
wenzelm@23252
   295
end
wenzelm@23252
   296
haftmann@36712
   297
locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
chaieb@25250
   298
  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
wenzelm@23252
   299
begin
wenzelm@23252
   300
haftmann@36712
   301
declare normalizing_ring_axioms' [normalizer del]
wenzelm@23252
   302
haftmann@36712
   303
lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
wenzelm@23252
   304
  semiring ops: semiring_ops
wenzelm@23252
   305
  semiring rules: semiring_rules
wenzelm@23252
   306
  ring ops: ring_ops
wenzelm@23252
   307
  ring rules: ring_rules
chaieb@25250
   308
  idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314
   309
  ideal rules: subr0_iff add_r0_iff]
wenzelm@23252
   310
wenzelm@23252
   311
end
wenzelm@23252
   312
haftmann@36712
   313
lemma (in no_zero_divisors) prod_eq_zero_eq_zero:
haftmann@36712
   314
  assumes "a * b = 0" and "a \<noteq> 0"
haftmann@36712
   315
  shows "b = 0"
haftmann@36712
   316
proof (rule classical)
haftmann@36712
   317
  assume "b \<noteq> 0" with `a \<noteq> 0` no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@36712
   318
  with `a * b = 0` show ?thesis by simp
wenzelm@23252
   319
qed
wenzelm@23252
   320
haftmann@36712
   321
(*FIXME introduce class*)
haftmann@36712
   322
interpretation normalizing!: normalizing_ring_cancel
haftmann@31017
   323
  "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
huffman@35216
   324
proof(unfold_locales, simp add: algebra_simps, auto)
haftmann@31017
   325
  fix w x y z ::"'a::{idom,number_ring}"
wenzelm@23252
   326
  assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
wenzelm@23252
   327
  hence ynz': "y - z \<noteq> 0" by simp
wenzelm@23252
   328
  from p have "w * y + x* z - w*z - x*y = 0" by simp
nipkow@29667
   329
  hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
nipkow@29667
   330
  hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
haftmann@36712
   331
  with  prod_eq_zero_eq_zero [OF _ ynz']
wenzelm@23252
   332
  have "w - x = 0" by blast
wenzelm@23252
   333
  thus "w = x"  by simp
wenzelm@23252
   334
qed
wenzelm@23252
   335
haftmann@36712
   336
declaration {* normalizer_funs' @{thm normalizing.normalizing_ring_cancel_axioms'} *}
wenzelm@23252
   337
haftmann@36712
   338
interpretation normalizing_nat!: normalizing_semiring_cancel
ballarin@29223
   339
  "op +" "op *" "op ^" "0::nat" "1"
huffman@35216
   340
proof (unfold_locales, simp add: algebra_simps)
wenzelm@23252
   341
  fix w x y z ::"nat"
wenzelm@23252
   342
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
wenzelm@23252
   343
    hence "y < z \<or> y > z" by arith
wenzelm@23252
   344
    moreover {
wenzelm@23252
   345
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
wenzelm@23252
   346
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
nipkow@29667
   347
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
wenzelm@23252
   348
      hence "x*k = w*k" by simp
huffman@35216
   349
      hence "w = x" using kp by simp }
wenzelm@23252
   350
    moreover {
wenzelm@23252
   351
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
wenzelm@23252
   352
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
nipkow@29667
   353
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
wenzelm@23252
   354
      hence "w*k = x*k" by simp
huffman@35216
   355
      hence "w = x" using kp by simp }
wenzelm@23252
   356
    ultimately have "w=x" by blast }
wenzelm@23252
   357
  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
wenzelm@23252
   358
qed
wenzelm@23252
   359
haftmann@36712
   360
declaration {* normalizer_funs' @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
wenzelm@23252
   361
haftmann@36712
   362
locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
chaieb@23327
   363
begin
chaieb@23327
   364
haftmann@36712
   365
declare normalizing_field_axioms' [normalizer del]
chaieb@23327
   366
haftmann@36712
   367
lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
chaieb@23327
   368
  semiring ops: semiring_ops
chaieb@23327
   369
  semiring rules: semiring_rules
chaieb@23327
   370
  ring ops: ring_ops
chaieb@23327
   371
  ring rules: ring_rules
chaieb@30866
   372
  field ops: field_ops
chaieb@30866
   373
  field rules: field_rules
chaieb@25250
   374
  idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314
   375
  ideal rules: subr0_iff add_r0_iff]
wenzelm@26314
   376
chaieb@23327
   377
end
chaieb@23327
   378
haftmann@36712
   379
(*FIXME introduce class*)
haftmann@36712
   380
interpretation normalizing!: normalizing_field_cancel "op +" "op *" "op ^"
haftmann@36712
   381
  "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse"
haftmann@36712
   382
apply (unfold_locales) by (simp_all add: divide_inverse)
haftmann@28402
   383
haftmann@36409
   384
lemma divide_Numeral1: "(x::'a::{field, number_ring}) / Numeral1 = x" by simp
haftmann@36409
   385
lemma divide_Numeral0: "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
haftmann@28402
   386
  by simp
haftmann@36409
   387
lemma mult_frac_frac: "((x::'a::field_inverse_zero) / y) * (z / w) = (x*z) / (y*w)"
haftmann@28402
   388
  by simp
haftmann@36409
   389
lemma mult_frac_num: "((x::'a::field_inverse_zero) / y) * z  = (x*z) / y"
haftmann@36712
   390
  by (fact times_divide_eq_left)
haftmann@36409
   391
lemma mult_num_frac: "((x::'a::field_inverse_zero) / y) * z  = (x*z) / y"
haftmann@36712
   392
  by (fact times_divide_eq_left)
haftmann@28402
   393
haftmann@28402
   394
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
haftmann@28402
   395
haftmann@36409
   396
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::field_inverse_zero) / y + z = (x + z*y) / y"
haftmann@28402
   397
  by (simp add: add_divide_distrib)
haftmann@36409
   398
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::field_inverse_zero) / y = (x + z*y) / y"
haftmann@28402
   399
  by (simp add: add_divide_distrib)
haftmann@35084
   400
haftmann@36712
   401
ML {* 
haftmann@28402
   402
local
haftmann@28402
   403
 val zr = @{cpat "0"}
haftmann@28402
   404
 val zT = ctyp_of_term zr
haftmann@28402
   405
 val geq = @{cpat "op ="}
haftmann@28402
   406
 val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
haftmann@28402
   407
 val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
haftmann@28402
   408
 val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
haftmann@28402
   409
 val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
haftmann@28402
   410
haftmann@28402
   411
 fun prove_nz ss T t =
haftmann@28402
   412
    let
haftmann@28402
   413
      val z = instantiate_cterm ([(zT,T)],[]) zr
haftmann@28402
   414
      val eq = instantiate_cterm ([(eqT,T)],[]) geq
wenzelm@35410
   415
      val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
haftmann@28402
   416
           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
haftmann@28402
   417
                  (Thm.capply (Thm.capply eq t) z)))
haftmann@28402
   418
    in equal_elim (symmetric th) TrueI
haftmann@28402
   419
    end
haftmann@28402
   420
haftmann@28402
   421
 fun proc phi ss ct =
haftmann@28402
   422
  let
haftmann@28402
   423
    val ((x,y),(w,z)) =
haftmann@28402
   424
         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
haftmann@28402
   425
    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
haftmann@28402
   426
    val T = ctyp_of_term x
haftmann@28402
   427
    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
haftmann@28402
   428
    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
haftmann@28402
   429
  in SOME (implies_elim (implies_elim th y_nz) z_nz)
haftmann@28402
   430
  end
haftmann@28402
   431
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
haftmann@28402
   432
haftmann@28402
   433
 fun proc2 phi ss ct =
haftmann@28402
   434
  let
haftmann@28402
   435
    val (l,r) = Thm.dest_binop ct
haftmann@28402
   436
    val T = ctyp_of_term l
haftmann@28402
   437
  in (case (term_of l, term_of r) of
haftmann@35084
   438
      (Const(@{const_name Rings.divide},_)$_$_, _) =>
haftmann@28402
   439
        let val (x,y) = Thm.dest_binop l val z = r
haftmann@28402
   440
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
haftmann@28402
   441
            val ynz = prove_nz ss T y
haftmann@28402
   442
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
haftmann@28402
   443
        end
haftmann@35084
   444
     | (_, Const (@{const_name Rings.divide},_)$_$_) =>
haftmann@28402
   445
        let val (x,y) = Thm.dest_binop r val z = l
haftmann@28402
   446
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
haftmann@28402
   447
            val ynz = prove_nz ss T y
haftmann@28402
   448
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
haftmann@28402
   449
        end
haftmann@28402
   450
     | _ => NONE)
haftmann@28402
   451
  end
haftmann@28402
   452
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
haftmann@28402
   453
haftmann@35084
   454
 fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
haftmann@28402
   455
   | is_number t = can HOLogic.dest_number t
haftmann@28402
   456
haftmann@28402
   457
 val is_number = is_number o term_of
haftmann@28402
   458
haftmann@28402
   459
 fun proc3 phi ss ct =
haftmann@28402
   460
  (case term_of ct of
haftmann@35092
   461
    Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
haftmann@28402
   462
      let
haftmann@28402
   463
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402
   464
        val _ = map is_number [a,b,c]
haftmann@28402
   465
        val T = ctyp_of_term c
haftmann@28402
   466
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
haftmann@28402
   467
      in SOME (mk_meta_eq th) end
haftmann@35092
   468
  | Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
haftmann@28402
   469
      let
haftmann@28402
   470
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402
   471
        val _ = map is_number [a,b,c]
haftmann@28402
   472
        val T = ctyp_of_term c
haftmann@28402
   473
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
haftmann@28402
   474
      in SOME (mk_meta_eq th) end
haftmann@35084
   475
  | Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
haftmann@28402
   476
      let
haftmann@28402
   477
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402
   478
        val _ = map is_number [a,b,c]
haftmann@28402
   479
        val T = ctyp_of_term c
haftmann@28402
   480
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
haftmann@28402
   481
      in SOME (mk_meta_eq th) end
haftmann@35092
   482
  | Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
haftmann@28402
   483
    let
haftmann@28402
   484
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402
   485
        val _ = map is_number [a,b,c]
haftmann@28402
   486
        val T = ctyp_of_term c
haftmann@28402
   487
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
haftmann@28402
   488
      in SOME (mk_meta_eq th) end
haftmann@35092
   489
  | Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
haftmann@28402
   490
    let
haftmann@28402
   491
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402
   492
        val _ = map is_number [a,b,c]
haftmann@28402
   493
        val T = ctyp_of_term c
haftmann@28402
   494
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
haftmann@28402
   495
      in SOME (mk_meta_eq th) end
haftmann@35084
   496
  | Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
haftmann@28402
   497
    let
haftmann@28402
   498
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402
   499
        val _ = map is_number [a,b,c]
haftmann@28402
   500
        val T = ctyp_of_term c
haftmann@28402
   501
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
haftmann@28402
   502
      in SOME (mk_meta_eq th) end
haftmann@28402
   503
  | _ => NONE)
haftmann@28402
   504
  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
haftmann@28402
   505
haftmann@28402
   506
val add_frac_frac_simproc =
haftmann@28402
   507
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
haftmann@28402
   508
                     name = "add_frac_frac_simproc",
haftmann@28402
   509
                     proc = proc, identifier = []}
haftmann@28402
   510
haftmann@28402
   511
val add_frac_num_simproc =
haftmann@28402
   512
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
haftmann@28402
   513
                     name = "add_frac_num_simproc",
haftmann@28402
   514
                     proc = proc2, identifier = []}
haftmann@28402
   515
haftmann@28402
   516
val ord_frac_simproc =
haftmann@28402
   517
  make_simproc
haftmann@28402
   518
    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
haftmann@28402
   519
             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
haftmann@28402
   520
             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
haftmann@28402
   521
             @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
haftmann@28402
   522
             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
haftmann@28402
   523
             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
haftmann@28402
   524
             name = "ord_frac_simproc", proc = proc3, identifier = []}
haftmann@28402
   525
haftmann@28402
   526
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
haftmann@28402
   527
           @{thm "divide_Numeral1"},
haftmann@36305
   528
           @{thm "divide_zero"}, @{thm "divide_Numeral0"},
haftmann@28402
   529
           @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
haftmann@28402
   530
           @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
haftmann@28402
   531
           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
haftmann@28402
   532
           @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
haftmann@28402
   533
           @{thm "diff_def"}, @{thm "minus_divide_left"},
chaieb@30869
   534
           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
haftmann@35084
   535
           @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
haftmann@36712
   536
           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
haftmann@35084
   537
           (@{thm field_divide_inverse} RS sym)]
haftmann@28402
   538
haftmann@36712
   539
in
haftmann@36712
   540
haftmann@36712
   541
val field_comp_conv = (Simplifier.rewrite
haftmann@36712
   542
(HOL_basic_ss addsimps @{thms "semiring_norm"}
wenzelm@35410
   543
              addsimps ths addsimps @{thms simp_thms}
haftmann@31068
   544
              addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@28402
   545
               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
haftmann@28402
   546
                            ord_frac_simproc]
haftmann@28402
   547
                addcongs [@{thm "if_weak_cong"}]))
haftmann@28402
   548
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
haftmann@28402
   549
  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
haftmann@36712
   550
wenzelm@23252
   551
end
haftmann@36712
   552
*}
haftmann@36712
   553
haftmann@36712
   554
declaration {*
haftmann@36712
   555
let
haftmann@28402
   556
haftmann@28402
   557
fun numeral_is_const ct =
haftmann@28402
   558
  case term_of ct of
haftmann@35084
   559
   Const (@{const_name Rings.divide},_) $ a $ b =>
chaieb@30866
   560
     can HOLogic.dest_number a andalso can HOLogic.dest_number b
haftmann@35084
   561
 | Const (@{const_name Rings.inverse},_)$t => can HOLogic.dest_number t
haftmann@28402
   562
 | t => can HOLogic.dest_number t
haftmann@28402
   563
haftmann@28402
   564
fun dest_const ct = ((case term_of ct of
haftmann@35084
   565
   Const (@{const_name Rings.divide},_) $ a $ b=>
haftmann@28402
   566
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
haftmann@35084
   567
 | Const (@{const_name Rings.inverse},_)$t => 
chaieb@30869
   568
               Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
haftmann@28402
   569
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
haftmann@28402
   570
   handle TERM _ => error "ring_dest_const")
haftmann@28402
   571
haftmann@28402
   572
fun mk_const phi cT x =
haftmann@28402
   573
 let val (a, b) = Rat.quotient_of_rat x
haftmann@28402
   574
 in if b = 1 then Numeral.mk_cnumber cT a
haftmann@28402
   575
    else Thm.capply
haftmann@28402
   576
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
haftmann@28402
   577
                     (Numeral.mk_cnumber cT a))
haftmann@28402
   578
         (Numeral.mk_cnumber cT b)
haftmann@28402
   579
  end
haftmann@28402
   580
haftmann@28402
   581
in
haftmann@36712
   582
 
haftmann@36712
   583
  Normalizer.funs @{thm normalizing.normalizing_field_cancel_axioms'}
haftmann@28402
   584
   {is_const = K numeral_is_const,
haftmann@28402
   585
    dest_const = K dest_const,
haftmann@28402
   586
    mk_const = mk_const,
haftmann@36712
   587
    conv = K (K field_comp_conv)}
haftmann@36712
   588
haftmann@36712
   589
end
haftmann@36712
   590
*}
haftmann@36712
   591
haftmann@36712
   592
haftmann@36712
   593
subsection {* Groebner Bases *}
haftmann@36712
   594
haftmann@36712
   595
lemmas bool_simps = simp_thms(1-34)
haftmann@36712
   596
haftmann@36712
   597
lemma dnf:
haftmann@36712
   598
    "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
haftmann@36712
   599
    "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
haftmann@36712
   600
  by blast+
haftmann@36712
   601
haftmann@36712
   602
lemmas weak_dnf_simps = dnf bool_simps
haftmann@36712
   603
haftmann@36712
   604
lemma nnf_simps:
haftmann@36712
   605
    "(\<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)"
haftmann@36712
   606
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
haftmann@36712
   607
  by blast+
haftmann@36712
   608
haftmann@36712
   609
lemma PFalse:
haftmann@36712
   610
    "P \<equiv> False \<Longrightarrow> \<not> P"
haftmann@36712
   611
    "\<not> P \<Longrightarrow> (P \<equiv> False)"
haftmann@36712
   612
  by auto
haftmann@36712
   613
haftmann@36712
   614
ML {*
haftmann@36712
   615
structure Algebra_Simplification = Named_Thms(
haftmann@36712
   616
  val name = "algebra"
haftmann@36712
   617
  val description = "pre-simplification rules for algebraic methods"
haftmann@36712
   618
)
haftmann@28402
   619
*}
haftmann@28402
   620
haftmann@36712
   621
setup Algebra_Simplification.setup
haftmann@36712
   622
haftmann@36712
   623
declare dvd_def[algebra]
haftmann@36712
   624
declare dvd_eq_mod_eq_0[symmetric, algebra]
haftmann@36712
   625
declare mod_div_trivial[algebra]
haftmann@36712
   626
declare mod_mod_trivial[algebra]
haftmann@36712
   627
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
haftmann@36712
   628
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
haftmann@36712
   629
declare zmod_zdiv_equality[symmetric,algebra]
haftmann@36712
   630
declare zdiv_zmod_equality[symmetric, algebra]
haftmann@36712
   631
declare zdiv_zminus_zminus[algebra]
haftmann@36712
   632
declare zmod_zminus_zminus[algebra]
haftmann@36712
   633
declare zdiv_zminus2[algebra]
haftmann@36712
   634
declare zmod_zminus2[algebra]
haftmann@36712
   635
declare zdiv_zero[algebra]
haftmann@36712
   636
declare zmod_zero[algebra]
haftmann@36712
   637
declare mod_by_1[algebra]
haftmann@36712
   638
declare div_by_1[algebra]
haftmann@36712
   639
declare zmod_minus1_right[algebra]
haftmann@36712
   640
declare zdiv_minus1_right[algebra]
haftmann@36712
   641
declare mod_div_trivial[algebra]
haftmann@36712
   642
declare mod_mod_trivial[algebra]
haftmann@36712
   643
declare mod_mult_self2_is_0[algebra]
haftmann@36712
   644
declare mod_mult_self1_is_0[algebra]
haftmann@36712
   645
declare zmod_eq_0_iff[algebra]
haftmann@36712
   646
declare dvd_0_left_iff[algebra]
haftmann@36712
   647
declare zdvd1_eq[algebra]
haftmann@36712
   648
declare zmod_eq_dvd_iff[algebra]
haftmann@36712
   649
declare nat_mod_eq_iff[algebra]
haftmann@36712
   650
haftmann@36712
   651
use "Tools/Groebner_Basis/groebner.ML"
haftmann@36712
   652
haftmann@36712
   653
method_setup algebra =
haftmann@36712
   654
{*
haftmann@36712
   655
let
haftmann@36712
   656
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
haftmann@36712
   657
 val addN = "add"
haftmann@36712
   658
 val delN = "del"
haftmann@36712
   659
 val any_keyword = keyword addN || keyword delN
haftmann@36712
   660
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
haftmann@36712
   661
in
haftmann@36712
   662
  ((Scan.optional (keyword addN |-- thms) []) -- 
haftmann@36712
   663
   (Scan.optional (keyword delN |-- thms) [])) >>
haftmann@36712
   664
  (fn (add_ths, del_ths) => fn ctxt =>
haftmann@36712
   665
       SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
haftmann@36712
   666
end
haftmann@36712
   667
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
haftmann@28402
   668
haftmann@28402
   669
end