src/HOL/Tools/Groebner_Basis/normalizer.ML

normalizer conversions depend on the proof context; added functions for conversion and wrapper that sill depend on the variable ordering (_ord)

(*  Title:      HOL/Tools/Groebner_Basis/normalizer.ML
    ID:         $Id$
    Author:     Amine Chaieb, TU Muenchen
*)

signature NORMALIZER = 
sig
 val mk_cnumber : ctyp -> integer -> cterm
 val mk_cnumeral : integer -> cterm
 val semiring_normalize_conv : Proof.context -> Conv.conv
 val semiring_normalize_ord_conv : Proof.context -> (cterm -> cterm -> bool) -> Conv.conv
 val semiring_normalize_tac : Proof.context -> int -> tactic
 val semiring_normalize_wrapper :  Proof.context -> NormalizerData.entry -> Conv.conv
 val semiring_normalize_ord_wrapper :  Proof.context -> NormalizerData.entry -> (cterm -> cterm -> bool) -> Conv.conv
 val semiring_normalizers_conv :
     cterm list -> cterm list * thm list -> cterm list * thm list ->
     (cterm -> bool) * Conv.conv * Conv.conv * Conv.conv -> (cterm -> cterm -> bool) ->
       {add: Conv.conv, mul: Conv.conv, neg: Conv.conv, main: Conv.conv, 
        pow: Conv.conv, sub: Conv.conv}
end

structure Normalizer: NORMALIZER = 
struct
open Misc;

local
 val pls_const = @{cterm "Numeral.Pls"}
   and min_const = @{cterm "Numeral.Min"}
   and bit_const = @{cterm "Numeral.Bit"}
   and zero = @{cpat "0"}
   and one = @{cpat "1"}
 fun mk_cbit 0 = @{cterm "Numeral.bit.B0"}
  | mk_cbit 1 = @{cterm "Numeral.bit.B1"}
  | mk_cbit _ = raise CTERM ("mk_cbit", []);

in

fun mk_cnumeral 0 = pls_const
  | mk_cnumeral ~1 = min_const
  | mk_cnumeral i =
      let val (q, r) = Integer.divmod i 2
      in Thm.capply (Thm.capply bit_const (mk_cnumeral q)) (mk_cbit (Integer.machine_int r)) end;

fun mk_cnumber cT = 
 let 
  val [nb_of, z, on] = 
    map (Drule.cterm_rule (instantiate' [SOME cT] [])) [@{cpat "number_of"}, zero, one]
  fun h 0 = z
    | h 1 = on
    | h x = Thm.capply nb_of (mk_cnumeral x)
 in h end;
end;


(* Very basic stuff for terms *)
val dest_numeral = term_of #> HOLogic.dest_number #> snd;
val is_numeral = can dest_numeral;

val numeral01_conv = Simplifier.rewrite
                         (HOL_basic_ss addsimps [numeral_1_eq_1, numeral_0_eq_0]);
val zero1_numeral_conv = 
 Simplifier.rewrite (HOL_basic_ss addsimps [numeral_1_eq_1 RS sym, numeral_0_eq_0 RS sym]);
val zerone_conv = fn cv => zero1_numeral_conv then_conv cv then_conv numeral01_conv;
val natarith = [@{thm "add_nat_number_of"}, @{thm "diff_nat_number_of"},
                @{thm "mult_nat_number_of"}, @{thm "eq_nat_number_of"}, 
                @{thm "less_nat_number_of"}];
val nat_add_conv = 
 zerone_conv 
  (Simplifier.rewrite 
    (HOL_basic_ss 
       addsimps arith_simps @ natarith @ rel_simps
             @ [if_False, if_True, add_0, add_Suc, add_number_of_left, Suc_eq_add_numeral_1]
             @ map (fn th => th RS sym) numerals));

val nat_mul_conv = nat_add_conv;
val zeron_tm = @{cterm "0::nat"};
val onen_tm  = @{cterm "1::nat"};
val true_tm = @{cterm "True"};


(* The main function! *)
fun semiring_normalizers_conv vars (sr_ops, sr_rules) (r_ops, r_rules)
  (is_semiring_constant, semiring_add_conv, semiring_mul_conv, semiring_pow_conv) =
let

val [pthm_02, pthm_03, pthm_04, pthm_05, pthm_07, pthm_08,
     pthm_09, pthm_10, pthm_11, pthm_12, pthm_13, pthm_14, pthm_15, pthm_16,
     pthm_17, pthm_18, pthm_19, pthm_21, pthm_22, pthm_23, pthm_24,
     pthm_25, pthm_26, pthm_27, pthm_28, pthm_29, pthm_30, pthm_31, pthm_32,
     pthm_33, pthm_34, pthm_35, pthm_36, pthm_37, pthm_38,pthm_39,pthm_40] = sr_rules;

val [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry] = vars;
val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
val [add_tm, mul_tm, pow_tm] = map (Thm.dest_fun o Thm.dest_fun) [add_pat, mul_pat, pow_pat];

val dest_add = dest_binop add_tm
val dest_mul = dest_binop mul_tm
fun dest_pow tm =
 let val (l,r) = dest_binop pow_tm tm
 in if is_numeral r then (l,r) else raise CTERM ("dest_pow",[tm])
 end;
val is_add = is_binop add_tm
val is_mul = is_binop mul_tm
fun is_pow tm = is_binop pow_tm tm andalso is_numeral(Thm.dest_arg tm);

val (neg_mul,sub_add,sub_tm,neg_tm,dest_sub,is_sub,cx',cy') =
  (case (r_ops, r_rules) of
    ([], []) => (TrueI, TrueI, true_tm, true_tm, (fn t => (t,t)), K false, true_tm, true_tm)
  | ([sub_pat, neg_pat], [neg_mul, sub_add]) =>
      let
        val sub_tm = Thm.dest_fun (Thm.dest_fun sub_pat)
        val neg_tm = Thm.dest_fun neg_pat
        val dest_sub = dest_binop sub_tm
        val is_sub = is_binop sub_tm
      in (neg_mul,sub_add,sub_tm,neg_tm,dest_sub,is_sub, neg_mul |> concl |> Thm.dest_arg,
          sub_add |> concl |> Thm.dest_arg |> Thm.dest_arg)
      end);
in fn variable_order =>
 let

(* Conversion for "x^n * x^m", with either x^n = x and/or x^m = x possible.  *)
(* Also deals with "const * const", but both terms must involve powers of    *)
(* the same variable, or both be constants, or behaviour may be incorrect.   *)

 fun powvar_mul_conv tm =
  let
  val (l,r) = dest_mul tm
  in if is_semiring_constant l andalso is_semiring_constant r
     then semiring_mul_conv tm
     else
      ((let
         val (lx,ln) = dest_pow l
        in
         ((let val (rx,rn) = dest_pow r
               val th1 = inst_thm [(cx,lx),(cp,ln),(cq,rn)] pthm_29
                val (tm1,tm2) = Thm.dest_comb(concl th1) in
               transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)
           handle CTERM _ =>
            (let val th1 = inst_thm [(cx,lx),(cq,ln)] pthm_31
                 val (tm1,tm2) = Thm.dest_comb(concl th1) in
               transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)) end)
       handle CTERM _ =>
           ((let val (rx,rn) = dest_pow r
                val th1 = inst_thm [(cx,rx),(cq,rn)] pthm_30
                val (tm1,tm2) = Thm.dest_comb(concl th1) in
               transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)
           handle CTERM _ => inst_thm [(cx,l)] pthm_32

))
 end;

(* Remove "1 * m" from a monomial, and just leave m.                         *)

 fun monomial_deone th =
       (let val (l,r) = dest_mul(concl th) in
           if l aconvc one_tm
          then transitive th (inst_thm [(ca,r)] pthm_13)  else th end)
       handle CTERM _ => th;

(* Conversion for "(monomial)^n", where n is a numeral.                      *)

 val monomial_pow_conv =
  let
   fun monomial_pow tm bod ntm =
    if not(is_comb bod)
    then reflexive tm
    else
     if is_semiring_constant bod
     then semiring_pow_conv tm
     else
      let
      val (lopr,r) = Thm.dest_comb bod
      in if not(is_comb lopr)
         then reflexive tm
        else
          let
          val (opr,l) = Thm.dest_comb lopr
         in
           if opr aconvc pow_tm andalso is_numeral r
          then
            let val th1 = inst_thm [(cx,l),(cp,r),(cq,ntm)] pthm_34
                val (l,r) = Thm.dest_comb(concl th1)
           in transitive th1 (Drule.arg_cong_rule l (nat_mul_conv r))
           end
           else
            if opr aconvc mul_tm
            then
             let
              val th1 = inst_thm [(cx,l),(cy,r),(cq,ntm)] pthm_33
             val (xy,z) = Thm.dest_comb(concl th1)
              val (x,y) = Thm.dest_comb xy
              val thl = monomial_pow y l ntm
              val thr = monomial_pow z r ntm
             in transitive th1 (combination (Drule.arg_cong_rule x thl) thr)
             end
             else reflexive tm
          end
      end
  in fn tm =>
   let
    val (lopr,r) = Thm.dest_comb tm
    val (opr,l) = Thm.dest_comb lopr
   in if not (opr aconvc pow_tm) orelse not(is_numeral r)
      then raise CTERM ("monomial_pow_conv", [tm])
      else if r aconvc zeron_tm
      then inst_thm [(cx,l)] pthm_35
      else if r aconvc onen_tm
      then inst_thm [(cx,l)] pthm_36
      else monomial_deone(monomial_pow tm l r)
   end
  end;

(* Multiplication of canonical monomials.                                    *)
 val monomial_mul_conv =
  let
   fun powvar tm =
    if is_semiring_constant tm then one_tm
    else
     ((let val (lopr,r) = Thm.dest_comb tm
           val (opr,l) = Thm.dest_comb lopr
       in if opr aconvc pow_tm andalso is_numeral r then l 
          else raise CTERM ("monomial_mul_conv",[tm]) end)
     handle CTERM _ => tm)   (* FIXME !? *)
   fun  vorder x y =
    if x aconvc y then 0
    else
     if x aconvc one_tm then ~1
     else if y aconvc one_tm then 1
      else if variable_order x y then ~1 else 1
   fun monomial_mul tm l r =
    ((let val (lx,ly) = dest_mul l val vl = powvar lx
      in
      ((let
        val (rx,ry) = dest_mul r
         val vr = powvar rx
         val ord = vorder vl vr
        in
         if ord = 0
        then
          let
             val th1 = inst_thm [(clx,lx),(cly,ly),(crx,rx),(cry,ry)] pthm_15
             val (tm1,tm2) = Thm.dest_comb(concl th1)
             val (tm3,tm4) = Thm.dest_comb tm1
             val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv tm4)) tm2
             val th3 = transitive th1 th2
              val  (tm5,tm6) = Thm.dest_comb(concl th3)
              val  (tm7,tm8) = Thm.dest_comb tm6
             val  th4 = monomial_mul tm6 (Thm.dest_arg tm7) tm8
         in  transitive th3 (Drule.arg_cong_rule tm5 th4)
         end
         else
          let val th0 = if ord < 0 then pthm_16 else pthm_17
             val th1 = inst_thm [(clx,lx),(cly,ly),(crx,rx),(cry,ry)] th0
             val (tm1,tm2) = Thm.dest_comb(concl th1)
             val (tm3,tm4) = Thm.dest_comb tm2
         in transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
         end
        end)
       handle CTERM _ =>
        (let val vr = powvar r val ord = vorder vl vr
        in
          if ord = 0 then
           let
           val th1 = inst_thm [(clx,lx),(cly,ly),(crx,r)] pthm_18
                 val (tm1,tm2) = Thm.dest_comb(concl th1)
           val (tm3,tm4) = Thm.dest_comb tm1
           val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv tm4)) tm2
          in transitive th1 th2
          end
          else
          if ord < 0 then
            let val th1 = inst_thm [(clx,lx),(cly,ly),(crx,r)] pthm_19
                val (tm1,tm2) = Thm.dest_comb(concl th1)
                val (tm3,tm4) = Thm.dest_comb tm2
           in transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
           end
           else inst_thm [(ca,l),(cb,r)] pthm_09
        end)) end)
     handle CTERM _ =>
      (let val vl = powvar l in
        ((let
          val (rx,ry) = dest_mul r
          val vr = powvar rx
           val ord = vorder vl vr
         in if ord = 0 then
              let val th1 = inst_thm [(clx,l),(crx,rx),(cry,ry)] pthm_21
                 val (tm1,tm2) = Thm.dest_comb(concl th1)
                 val (tm3,tm4) = Thm.dest_comb tm1
             in transitive th1 (Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv tm4)) tm2)
             end
             else if ord > 0 then
                 let val th1 = inst_thm [(clx,l),(crx,rx),(cry,ry)] pthm_22
                     val (tm1,tm2) = Thm.dest_comb(concl th1)
                    val (tm3,tm4) = Thm.dest_comb tm2
                in transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
                end
             else reflexive tm
         end)
        handle CTERM _ =>
          (let val vr = powvar r
               val  ord = vorder vl vr
          in if ord = 0 then powvar_mul_conv tm
              else if ord > 0 then inst_thm [(ca,l),(cb,r)] pthm_09
              else reflexive tm
          end)) end))
  in fn tm => let val (l,r) = dest_mul tm in monomial_deone(monomial_mul tm l r)
             end
  end;
(* Multiplication by monomial of a polynomial.                               *)

 val polynomial_monomial_mul_conv =
  let
   fun pmm_conv tm =
    let val (l,r) = dest_mul tm
    in
    ((let val (y,z) = dest_add r
          val th1 = inst_thm [(cx,l),(cy,y),(cz,z)] pthm_37
          val (tm1,tm2) = Thm.dest_comb(concl th1)
          val (tm3,tm4) = Thm.dest_comb tm1
          val th2 = combination (Drule.arg_cong_rule tm3 (monomial_mul_conv tm4)) (pmm_conv tm2)
      in transitive th1 th2
      end)
     handle CTERM _ => monomial_mul_conv tm)
   end
 in pmm_conv
 end;

(* Addition of two monomials identical except for constant multiples.        *)

fun monomial_add_conv tm =
 let val (l,r) = dest_add tm
 in if is_semiring_constant l andalso is_semiring_constant r
    then semiring_add_conv tm
    else
     let val th1 =
           if is_mul l andalso is_semiring_constant(Thm.dest_arg1 l)
           then if is_mul r andalso is_semiring_constant(Thm.dest_arg1 r) then
                    inst_thm [(ca,Thm.dest_arg1 l),(cm,Thm.dest_arg r), (cb,Thm.dest_arg1 r)] pthm_02
                else inst_thm [(ca,Thm.dest_arg1 l),(cm,r)] pthm_03
           else if is_mul r andalso is_semiring_constant(Thm.dest_arg1 r)
           then inst_thm [(cm,l),(ca,Thm.dest_arg1 r)] pthm_04
           else inst_thm [(cm,r)] pthm_05
         val (tm1,tm2) = Thm.dest_comb(concl th1)
         val (tm3,tm4) = Thm.dest_comb tm1
         val th2 = Drule.arg_cong_rule tm3 (semiring_add_conv tm4)
         val th3 = transitive th1 (Drule.fun_cong_rule th2 tm2)
         val tm5 = concl th3
      in
      if (Thm.dest_arg1 tm5) aconvc zero_tm
      then transitive th3 (inst_thm [(ca,Thm.dest_arg tm5)] pthm_11)
      else monomial_deone th3
     end
 end;

(* Ordering on monomials.                                                    *)

fun striplist dest =
 let fun strip x acc =
   ((let val (l,r) = dest x in
        strip l (strip r acc) end)
    handle CTERM _ => x::acc)    (* FIXME !? *)
 in fn x => strip x []
 end;


fun powervars tm =
 let val ptms = striplist dest_mul tm
 in if is_semiring_constant (hd ptms) then tl ptms else ptms
 end;
val num_0 = 0;
val num_1 = 1;
fun dest_varpow tm =
 ((let val (x,n) = dest_pow tm in (x,dest_numeral n) end)
   handle CTERM _ =>
   (tm,(if is_semiring_constant tm then num_0 else num_1)));

val morder =
 let fun lexorder l1 l2 =
  case (l1,l2) of
    ([],[]) => 0
  | (vps,[]) => ~1
  | ([],vps) => 1
  | (((x1,n1)::vs1),((x2,n2)::vs2)) =>
     if variable_order x1 x2 then 1
     else if variable_order x2 x1 then ~1
     else if n1 < n2 then ~1
     else if n2 < n1 then 1
     else lexorder vs1 vs2
 in fn tm1 => fn tm2 =>
  let val vdegs1 = map dest_varpow (powervars tm1)
      val vdegs2 = map dest_varpow (powervars tm2)
      val deg1 = fold_rev ((curry (op +)) o snd) vdegs1 num_0
      val deg2 = fold_rev ((curry (op +)) o snd) vdegs2 num_0
  in if deg1 < deg2 then ~1 else if deg1 > deg2 then 1
                            else lexorder vdegs1 vdegs2
  end
 end;

(* Addition of two polynomials.                                              *)

val polynomial_add_conv =
 let
 fun dezero_rule th =
  let
   val tm = concl th
  in
   if not(is_add tm) then th else
   let val (lopr,r) = Thm.dest_comb tm
       val l = Thm.dest_arg lopr
   in
    if l aconvc zero_tm
    then transitive th (inst_thm [(ca,r)] pthm_07)   else
        if r aconvc zero_tm
        then transitive th (inst_thm [(ca,l)] pthm_08)  else th
   end
  end
 fun padd tm =
  let
   val (l,r) = dest_add tm
  in
   if l aconvc zero_tm then inst_thm [(ca,r)] pthm_07
   else if r aconvc zero_tm then inst_thm [(ca,l)] pthm_08
   else
    if is_add l
    then
     let val (a,b) = dest_add l
     in
     if is_add r then
      let val (c,d) = dest_add r
          val ord = morder a c
      in
       if ord = 0 then
        let val th1 = inst_thm [(ca,a),(cb,b),(cc,c),(cd,d)] pthm_23
            val (tm1,tm2) = Thm.dest_comb(concl th1)
            val (tm3,tm4) = Thm.dest_comb tm1
            val th2 = Drule.arg_cong_rule tm3 (monomial_add_conv tm4)
        in dezero_rule (transitive th1 (combination th2 (padd tm2)))
        end
       else (* ord <> 0*)
        let val th1 =
                if ord > 0 then inst_thm [(ca,a),(cb,b),(cc,r)] pthm_24
                else inst_thm [(ca,l),(cc,c),(cd,d)] pthm_25
            val (tm1,tm2) = Thm.dest_comb(concl th1)
        in dezero_rule (transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
        end
      end
     else (* not (is_add r)*)
      let val ord = morder a r
      in
       if ord = 0 then
        let val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_26
            val (tm1,tm2) = Thm.dest_comb(concl th1)
            val (tm3,tm4) = Thm.dest_comb tm1
            val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (monomial_add_conv tm4)) tm2
        in dezero_rule (transitive th1 th2)
        end
       else (* ord <> 0*)
        if ord > 0 then
          let val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_24
              val (tm1,tm2) = Thm.dest_comb(concl th1)
          in dezero_rule (transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
          end
        else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
      end
    end
   else (* not (is_add l)*)
    if is_add r then
      let val (c,d) = dest_add r
          val  ord = morder l c
      in
       if ord = 0 then
         let val th1 = inst_thm [(ca,l),(cc,c),(cd,d)] pthm_28
             val (tm1,tm2) = Thm.dest_comb(concl th1)
             val (tm3,tm4) = Thm.dest_comb tm1
             val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (monomial_add_conv tm4)) tm2
         in dezero_rule (transitive th1 th2)
         end
       else
        if ord > 0 then reflexive tm
        else
         let val th1 = inst_thm [(ca,l),(cc,c),(cd,d)] pthm_25
             val (tm1,tm2) = Thm.dest_comb(concl th1)
         in dezero_rule (transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
         end
      end
    else
     let val ord = morder l r
     in
      if ord = 0 then monomial_add_conv tm
      else if ord > 0 then dezero_rule(reflexive tm)
      else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
     end
  end
 in padd
 end;

(* Multiplication of two polynomials.                                        *)

val polynomial_mul_conv =
 let
  fun pmul tm =
   let val (l,r) = dest_mul tm
   in
    if not(is_add l) then polynomial_monomial_mul_conv tm
    else
     if not(is_add r) then
      let val th1 = inst_thm [(ca,l),(cb,r)] pthm_09
      in transitive th1 (polynomial_monomial_mul_conv(concl th1))
      end
     else
       let val (a,b) = dest_add l
           val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_10
           val (tm1,tm2) = Thm.dest_comb(concl th1)
           val (tm3,tm4) = Thm.dest_comb tm1
           val th2 = Drule.arg_cong_rule tm3 (polynomial_monomial_mul_conv tm4)
           val th3 = transitive th1 (combination th2 (pmul tm2))
       in transitive th3 (polynomial_add_conv (concl th3))
       end
   end
 in fn tm =>
   let val (l,r) = dest_mul tm
   in
    if l aconvc zero_tm then inst_thm [(ca,r)] pthm_11
    else if r aconvc zero_tm then inst_thm [(ca,l)] pthm_12
    else if l aconvc one_tm then inst_thm [(ca,r)] pthm_13
    else if r aconvc one_tm then inst_thm [(ca,l)] pthm_14
    else pmul tm
   end
 end;

(* Power of polynomial (optimized for the monomial and trivial cases).       *)

val Succ = @{cterm "Suc"};
val num_conv = fn n =>
        nat_add_conv (Thm.capply (Succ) (mk_cnumber @{ctyp "nat"} ((dest_numeral n) - 1)))
                     |> Thm.symmetric;


val polynomial_pow_conv =
 let
  fun ppow tm =
    let val (l,n) = dest_pow tm
    in
     if n aconvc zeron_tm then inst_thm [(cx,l)] pthm_35
     else if n aconvc onen_tm then inst_thm [(cx,l)] pthm_36
     else
         let val th1 = num_conv n
             val th2 = inst_thm [(cx,l),(cq,Thm.dest_arg (concl th1))] pthm_38
             val (tm1,tm2) = Thm.dest_comb(concl th2)
             val th3 = transitive th2 (Drule.arg_cong_rule tm1 (ppow tm2))
             val th4 = transitive (Drule.arg_cong_rule (Thm.dest_fun tm) th1) th3
         in transitive th4 (polynomial_mul_conv (concl th4))
         end
    end
 in fn tm =>
       if is_add(Thm.dest_arg1 tm) then ppow tm else monomial_pow_conv tm
 end;

(* Negation.                                                                 *)

val polynomial_neg_conv =
 fn tm =>
   let val (l,r) = Thm.dest_comb tm in
        if not (l aconvc neg_tm) then raise CTERM ("polynomial_neg_conv",[tm]) else
        let val th1 = inst_thm [(cx',r)] neg_mul
            val th2 = transitive th1 (arg1_conv semiring_mul_conv (concl th1))
        in transitive th2 (polynomial_monomial_mul_conv (concl th2))
        end
   end;


(* Subtraction.                                                              *)
val polynomial_sub_conv = fn tm =>
  let val (l,r) = dest_sub tm
      val th1 = inst_thm [(cx',l),(cy',r)] sub_add
      val (tm1,tm2) = Thm.dest_comb(concl th1)
      val th2 = Drule.arg_cong_rule tm1 (polynomial_neg_conv tm2)
  in transitive th1 (transitive th2 (polynomial_add_conv (concl th2)))
  end;

(* Conversion from HOL term.                                                 *)

fun polynomial_conv tm =
 if is_semiring_constant tm then semiring_add_conv tm
 else if not(is_comb tm) then reflexive tm
 else
  let val (lopr,r) = Thm.dest_comb tm
  in if lopr aconvc neg_tm then
       let val th1 = Drule.arg_cong_rule lopr (polynomial_conv r)
       in transitive th1 (polynomial_neg_conv (concl th1))
       end
     else
       if not(is_comb lopr) then reflexive tm
       else
         let val (opr,l) = Thm.dest_comb lopr
         in if opr aconvc pow_tm andalso is_numeral r
            then
              let val th1 = Drule.fun_cong_rule (Drule.arg_cong_rule opr (polynomial_conv l)) r
              in transitive th1 (polynomial_pow_conv (concl th1))
              end
            else
              if opr aconvc add_tm orelse opr aconvc mul_tm orelse opr aconvc sub_tm
              then
               let val th1 = combination (Drule.arg_cong_rule opr (polynomial_conv l)) (polynomial_conv r)
                   val f = if opr aconvc add_tm then polynomial_add_conv
                      else if opr aconvc mul_tm then polynomial_mul_conv
                      else polynomial_sub_conv
               in transitive th1 (f (concl th1))
               end
              else reflexive tm
         end
  end;
 in
   {main = polynomial_conv,
    add = polynomial_add_conv,
    mul = polynomial_mul_conv,
    pow = polynomial_pow_conv,
    neg = polynomial_neg_conv,
    sub = polynomial_sub_conv}
 end
end;

val nat_arith = @{thms "nat_arith"};
val nat_exp_ss = HOL_basic_ss addsimps (nat_number @ nat_arith @ arith_simps @ rel_simps)
                              addsimps [Let_def, if_False, if_True, add_0, add_Suc];

fun simple_cterm_ord t u = Term.term_ord (term_of t, term_of u) = LESS;
fun semiring_normalize_ord_wrapper ctxt ({vars, semiring, ring, idom}, 
                                     {conv, dest_const, mk_const, is_const}) ord =
  let
    val pow_conv =
      arg_conv (Simplifier.rewrite nat_exp_ss)
      then_conv Simplifier.rewrite
        (HOL_basic_ss addsimps [nth (snd semiring) 31, nth (snd semiring) 34])
      then_conv conv ctxt
    val dat = (is_const, conv ctxt, conv ctxt, pow_conv)
    val {main, ...} = semiring_normalizers_conv vars semiring ring dat ord
  in main end;

fun semiring_normalize_wrapper ctxt data = 
  semiring_normalize_ord_wrapper ctxt data simple_cterm_ord;

fun semiring_normalize_ord_conv ctxt ord tm =
  (case NormalizerData.match ctxt tm of
    NONE => reflexive tm
  | SOME res => semiring_normalize_ord_wrapper ctxt res ord tm);
 

fun semiring_normalize_conv ctxt = semiring_normalize_ord_conv ctxt simple_cterm_ord;

fun semiring_normalize_tac ctxt = SUBGOAL (fn (goal, i) =>
  rtac (semiring_normalize_conv ctxt
    (cterm_of (ProofContext.theory_of ctxt) (fst (Logic.dest_equals goal)))) i);
end;