(*  Title:      HOL/Groebner_Basis.thy

    Author:     Amine Chaieb, TU Muenchen

*)

header {* Semiring normalization and Groebner Bases *}

theory Groebner_Basis

imports NatBin

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, recpower}" "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_add_numeral_1

  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,recpower,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,recpower,number_ring}" "1" "op -" "uminus"

proof(unfold_locales, simp add: algebra_simps power_Suc, auto)

  fix w x y z ::"'a::{idom,recpower,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,recpower,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{* 

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 = []}

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]

local

open Conv

in

val comp_conv = (Simplifier.rewrite

(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}

              addsimps ths addsimps simp_thms

              addsimprocs 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))

 | 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