src/HOL/Groebner_Basis.thy

merged

(*  Title:      HOL/Groebner_Basis.thy
    Author:     Amine Chaieb, TU Muenchen
*)

header {* Semiring normalization and Groebner Bases *}

theory Groebner_Basis
imports Nat_Numeral
uses
  "Tools/Groebner_Basis/misc.ML"
  "Tools/Groebner_Basis/normalizer_data.ML"
  ("Tools/Groebner_Basis/normalizer.ML")
  ("Tools/Groebner_Basis/groebner.ML")
begin

subsection {* Semiring normalization *}

setup NormalizerData.setup


locale gb_semiring =
  fixes add mul pwr r0 r1
  assumes add_a:"(add x (add y z) = add (add x y) z)"
    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
begin

lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
proof (induct p)
  case 0
  then show ?case by (auto simp add: pwr_0 mul_1)
next
  case Suc
  from this [symmetric] show ?case
    by (auto simp add: pwr_Suc mul_1 mul_a)
qed

lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
  fix q x y
  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
    by (simp add: mul_a)
  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
qed

lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
proof (induct p arbitrary: q)
  case 0
  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
next
  case Suc
  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
qed


subsubsection {* Declaring the abstract theory *}

lemma semiring_ops:
  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
    and "TERM r0" and "TERM r1" .

lemma semiring_rules:
  "add (mul a m) (mul b m) = mul (add a b) m"
  "add (mul a m) m = mul (add a r1) m"
  "add m (mul a m) = mul (add a r1) m"
  "add m m = mul (add r1 r1) m"
  "add r0 a = a"
  "add a r0 = a"
  "mul a b = mul b a"
  "mul (add a b) c = add (mul a c) (mul b c)"
  "mul r0 a = r0"
  "mul a r0 = r0"
  "mul r1 a = a"
  "mul a r1 = a"
  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
  "mul (mul lx ly) rx = mul (mul lx rx) ly"
  "mul (mul lx ly) rx = mul lx (mul ly rx)"
  "mul lx (mul rx ry) = mul (mul lx rx) ry"
  "mul lx (mul rx ry) = mul rx (mul lx ry)"
  "add (add a b) (add c d) = add (add a c) (add b d)"
  "add (add a b) c = add a (add b c)"
  "add a (add c d) = add c (add a d)"
  "add (add a b) c = add (add a c) b"
  "add a c = add c a"
  "add a (add c d) = add (add a c) d"
  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
  "mul x (pwr x q) = pwr x (Suc q)"
  "mul (pwr x q) x = pwr x (Suc q)"
  "mul x x = pwr x 2"
  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
  "pwr (pwr x p) q = pwr x (p * q)"
  "pwr x 0 = r1"
  "pwr x 1 = x"
  "mul x (add y z) = add (mul x y) (mul x z)"
  "pwr x (Suc q) = mul x (pwr x q)"
  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
proof -
  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
next show "add r0 a = a" using add_0 by simp
next show "add a r0 = a" using add_0 add_c by simp
next show "mul a b = mul b a" using mul_c by simp
next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
next show "mul r0 a = r0" using mul_0 by simp
next show "mul a r0 = r0" using mul_0 mul_c by simp
next show "mul r1 a = a" using mul_1 by simp
next show "mul a r1 = a" using mul_1 mul_c by simp
next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
    using mul_c mul_a by simp
next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
    using mul_a by simp
next
  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
  finally
  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
    using mul_c by simp
next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
next
  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
next show "add (add a b) (add c d) = add (add a c) (add b d)"
    using add_c add_a by simp
next show "add (add a b) c = add a (add b c)" using add_a by simp
next show "add a (add c d) = add c (add a d)"
    apply (simp add: add_a) by (simp only: add_c)
next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
next show "add a c = add c a" by (rule add_c)
next show "add a (add c d) = add (add a c) d" using add_a by simp
next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
next show "pwr x 0 = r1" using pwr_0 .
next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
    by (simp add: nat_number pwr_Suc mul_pwr)
qed


lemmas gb_semiring_axioms' =
  gb_semiring_axioms [normalizer
    semiring ops: semiring_ops
    semiring rules: semiring_rules]

end

interpretation class_semiring: gb_semiring
    "op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1"
  proof qed (auto simp add: algebra_simps power_Suc)

lemmas nat_arith =
  add_nat_number_of
  diff_nat_number_of
  mult_nat_number_of
  eq_nat_number_of
  less_nat_number_of

lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
  by (simp add: numeral_1_eq_1)

lemmas comp_arith =
  Let_def arith_simps nat_arith rel_simps neg_simps if_False
  if_True add_0 add_Suc add_number_of_left mult_number_of_left
  numeral_1_eq_1[symmetric] Suc_eq_plus1
  numeral_0_eq_0[symmetric] numerals[symmetric]
  iszero_simps not_iszero_Numeral1

lemmas semiring_norm = comp_arith

ML {*
local

open Conv;

fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);

fun int_of_rat x =
  (case Rat.quotient_of_rat x of (i, 1) => i
  | _ => error "int_of_rat: bad int");

val numeral_conv =
  Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
  Simplifier.rewrite (HOL_basic_ss addsimps
    (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));

in

fun normalizer_funs key =
  NormalizerData.funs key
   {is_const = fn phi => numeral_is_const,
    dest_const = fn phi => fn ct =>
      Rat.rat_of_int (snd
        (HOLogic.dest_number (Thm.term_of ct)
          handle TERM _ => error "ring_dest_const")),
    mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
    conv = fn phi => K numeral_conv}

end
*}

declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}


locale gb_ring = gb_semiring +
  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    and neg :: "'a \<Rightarrow> 'a"
  assumes neg_mul: "neg x = mul (neg r1) x"
    and sub_add: "sub x y = add x (neg y)"
begin

lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .

lemmas ring_rules = neg_mul sub_add

lemmas gb_ring_axioms' =
  gb_ring_axioms [normalizer
    semiring ops: semiring_ops
    semiring rules: semiring_rules
    ring ops: ring_ops
    ring rules: ring_rules]

end


interpretation class_ring: gb_ring "op +" "op *" "op ^"
    "0::'a::{comm_semiring_1,number_ring}" 1 "op -" "uminus"
  proof qed simp_all


declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}

use "Tools/Groebner_Basis/normalizer.ML"


method_setup sring_norm = {*
  Scan.succeed (SIMPLE_METHOD' o Normalizer.semiring_normalize_tac)
*} "semiring normalizer"


locale gb_field = gb_ring +
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    and inverse:: "'a \<Rightarrow> 'a"
  assumes divide_inverse: "divide x y = mul x (inverse y)"
     and inverse_divide: "inverse x = divide r1 x"
begin

lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .

lemmas field_rules = divide_inverse inverse_divide

lemmas gb_field_axioms' =
  gb_field_axioms [normalizer
    semiring ops: semiring_ops
    semiring rules: semiring_rules
    ring ops: ring_ops
    ring rules: ring_rules
    field ops: field_ops
    field rules: field_rules]

end


subsection {* Groebner Bases *}

locale semiringb = gb_semiring +
  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
  and add_mul_solve: "add (mul w y) (mul x z) =
    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
begin

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)"
proof-
  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
    using add_mul_solve by blast
  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)"
    by simp
qed

lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
proof(clarify)
  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
    and eq: "add b (mul r c) = add b (mul r d)"
  hence "mul r c = mul r d" using cnd add_cancel by simp
  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
    using mul_0 add_cancel by simp
  thus "False" using add_mul_solve nz cnd by simp
qed

lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
proof-
  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
qed

declare gb_semiring_axioms' [normalizer del]

lemmas semiringb_axioms' = semiringb_axioms [normalizer
  semiring ops: semiring_ops
  semiring rules: semiring_rules
  idom rules: noteq_reduce add_scale_eq_noteq]

end

locale ringb = semiringb + gb_ring + 
  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
begin

declare gb_ring_axioms' [normalizer del]

lemmas ringb_axioms' = ringb_axioms [normalizer
  semiring ops: semiring_ops
  semiring rules: semiring_rules
  ring ops: ring_ops
  ring rules: ring_rules
  idom rules: noteq_reduce add_scale_eq_noteq
  ideal rules: subr0_iff add_r0_iff]

end


lemma no_zero_divirors_neq0:
  assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
    and ab: "a*b = 0" shows "b = 0"
proof -
  { assume bz: "b \<noteq> 0"
    from no_zero_divisors [OF az bz] ab have False by blast }
  thus "b = 0" by blast
qed

interpretation class_ringb: ringb
  "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
proof(unfold_locales, simp add: algebra_simps power_Suc, auto)
  fix w x y z ::"'a::{idom,number_ring}"
  assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
  hence ynz': "y - z \<noteq> 0" by simp
  from p have "w * y + x* z - w*z - x*y = 0" by simp
  hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
  hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
  with  no_zero_divirors_neq0 [OF ynz']
  have "w - x = 0" by blast
  thus "w = x"  by simp
qed

declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}

interpretation natgb: semiringb
  "op +" "op *" "op ^" "0::nat" "1"
proof (unfold_locales, simp add: algebra_simps power_Suc)
  fix w x y z ::"nat"
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
    hence "y < z \<or> y > z" by arith
    moreover {
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
      hence "x*k = w*k" by simp
      hence "w = x" using kp by (simp add: mult_cancel2) }
    moreover {
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
      hence "w*k = x*k" by simp
      hence "w = x" using kp by (simp add: mult_cancel2)}
    ultimately have "w=x" by blast }
  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
qed

declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}

locale fieldgb = ringb + gb_field
begin

declare gb_field_axioms' [normalizer del]

lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
  semiring ops: semiring_ops
  semiring rules: semiring_rules
  ring ops: ring_ops
  ring rules: ring_rules
  field ops: field_ops
  field rules: field_rules
  idom rules: noteq_reduce add_scale_eq_noteq
  ideal rules: subr0_iff add_r0_iff]

end


lemmas bool_simps = simp_thms(1-34)
lemma dnf:
    "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
    "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
  by blast+

lemmas weak_dnf_simps = dnf bool_simps

lemma nnf_simps:
    "(\<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)"
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
  by blast+

lemma PFalse:
    "P \<equiv> False \<Longrightarrow> \<not> P"
    "\<not> P \<Longrightarrow> (P \<equiv> False)"
  by auto
use "Tools/Groebner_Basis/groebner.ML"

method_setup algebra =
{*
let
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
 val addN = "add"
 val delN = "del"
 val any_keyword = keyword addN || keyword delN
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
in
  ((Scan.optional (keyword addN |-- thms) []) -- 
   (Scan.optional (keyword delN |-- thms) [])) >>
  (fn (add_ths, del_ths) => fn ctxt =>
       SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
end
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
declare dvd_def[algebra]
declare dvd_eq_mod_eq_0[symmetric, algebra]
declare mod_div_trivial[algebra]
declare mod_mod_trivial[algebra]
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
declare zmod_zdiv_equality[symmetric,algebra]
declare zdiv_zmod_equality[symmetric, algebra]
declare zdiv_zminus_zminus[algebra]
declare zmod_zminus_zminus[algebra]
declare zdiv_zminus2[algebra]
declare zmod_zminus2[algebra]
declare zdiv_zero[algebra]
declare zmod_zero[algebra]
declare mod_by_1[algebra]
declare div_by_1[algebra]
declare zmod_minus1_right[algebra]
declare zdiv_minus1_right[algebra]
declare mod_div_trivial[algebra]
declare mod_mod_trivial[algebra]
declare mod_mult_self2_is_0[algebra]
declare mod_mult_self1_is_0[algebra]
declare zmod_eq_0_iff[algebra]
declare dvd_0_left_iff[algebra]
declare zdvd1_eq[algebra]
declare zmod_eq_dvd_iff[algebra]
declare nat_mod_eq_iff[algebra]

subsection{* Groebner Bases for fields *}

interpretation class_fieldgb:
  fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)

lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
  by simp
lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
  by simp
lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
  by simp
lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
  by simp

lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp

lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
  by (simp add: add_divide_distrib)
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
  by (simp add: add_divide_distrib)
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*}
ML{* 
local
 val zr = @{cpat "0"}
 val zT = ctyp_of_term zr
 val geq = @{cpat "op ="}
 val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
 val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
 val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
 val add_num_frac = mk_meta_eq @{thm "add_num_frac"}

 fun prove_nz ss T t =
    let
      val z = instantiate_cterm ([(zT,T)],[]) zr
      val eq = instantiate_cterm ([(eqT,T)],[]) geq
      val th = Simplifier.rewrite (ss addsimps simp_thms)
           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
                  (Thm.capply (Thm.capply eq t) z)))
    in equal_elim (symmetric th) TrueI
    end

 fun proc phi ss ct =
  let
    val ((x,y),(w,z)) =
         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
    val T = ctyp_of_term x
    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
  in SOME (implies_elim (implies_elim th y_nz) z_nz)
  end
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE

 fun proc2 phi ss ct =
  let
    val (l,r) = Thm.dest_binop ct
    val T = ctyp_of_term l
  in (case (term_of l, term_of r) of
      (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
        let val (x,y) = Thm.dest_binop l val z = r
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
            val ynz = prove_nz ss T y
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
        end
     | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
        let val (x,y) = Thm.dest_binop r val z = l
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
            val ynz = prove_nz ss T y
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
        end
     | _ => NONE)
  end
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE

 fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
   | is_number t = can HOLogic.dest_number t

 val is_number = is_number o term_of

 fun proc3 phi ss ct =
  (case term_of ct of
    Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
      in SOME (mk_meta_eq th) end
  | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
      in SOME (mk_meta_eq th) end
  | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
      in SOME (mk_meta_eq th) end
  | _ => NONE)
  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE

val add_frac_frac_simproc =
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
                     name = "add_frac_frac_simproc",
                     proc = proc, identifier = []}

val add_frac_num_simproc =
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
                     name = "add_frac_num_simproc",
                     proc = proc2, identifier = []}

val ord_frac_simproc =
  make_simproc
    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
             @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
             name = "ord_frac_simproc", proc = proc3, identifier = []}

local
open Conv
in

val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
           @{thm "divide_Numeral1"},
           @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
           @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
           @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
           @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
           @{thm "diff_def"}, @{thm "minus_divide_left"},
           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
           @{thm divide_inverse} RS sym, @{thm inverse_divide}, 
           fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))   
           (@{thm divide_inverse} RS sym)]

val comp_conv = (Simplifier.rewrite
(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
              addsimps ths addsimps simp_thms
              addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
                            ord_frac_simproc]
                addcongs [@{thm "if_weak_cong"}]))
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
end

fun numeral_is_const ct =
  case term_of ct of
   Const (@{const_name "HOL.divide"},_) $ a $ b =>
     can HOLogic.dest_number a andalso can HOLogic.dest_number b
 | Const (@{const_name "HOL.inverse"},_)$t => can HOLogic.dest_number t
 | t => can HOLogic.dest_number t

fun dest_const ct = ((case term_of ct of
   Const (@{const_name "HOL.divide"},_) $ a $ b=>
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
 | Const (@{const_name "HOL.inverse"},_)$t => 
               Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
   handle TERM _ => error "ring_dest_const")

fun mk_const phi cT x =
 let val (a, b) = Rat.quotient_of_rat x
 in if b = 1 then Numeral.mk_cnumber cT a
    else Thm.capply
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
                     (Numeral.mk_cnumber cT a))
         (Numeral.mk_cnumber cT b)
  end

in
 val field_comp_conv = comp_conv;
 val fieldgb_declaration = 
  NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
   {is_const = K numeral_is_const,
    dest_const = K dest_const,
    mk_const = mk_const,
    conv = K (K comp_conv)}
end;
*}

declaration fieldgb_declaration

end