src/HOL/Tools/semiring_normalizer.ML

+(*  Title:      HOL/Tools/Groebner_Basis/normalizer.ML
+    Author:     Amine Chaieb, TU Muenchen
+
+Normalization of expressions in semirings.
+*)
+
+signature SEMIRING_NORMALIZER = 
+sig
+  type entry
+  val get: Proof.context -> (thm * entry) list
+  val match: Proof.context -> cterm -> entry option
+  val del: attribute
+  val add: {semiring: cterm list * thm list, ring: cterm list * thm list,
+    field: cterm list * thm list, idom: thm list, ideal: thm list} -> attribute
+  val funs: thm -> {is_const: morphism -> cterm -> bool,
+    dest_const: morphism -> cterm -> Rat.rat,
+    mk_const: morphism -> ctyp -> Rat.rat -> cterm,
+    conv: morphism -> Proof.context -> cterm -> thm} -> declaration
+  val semiring_funs: thm -> declaration
+  val field_funs: thm -> declaration
+
+  val semiring_normalize_conv: Proof.context -> conv
+  val semiring_normalize_ord_conv: Proof.context -> (cterm -> cterm -> bool) -> conv
+  val semiring_normalize_wrapper: Proof.context -> entry -> conv
+  val semiring_normalize_ord_wrapper: Proof.context -> entry
+    -> (cterm -> cterm -> bool) -> conv
+  val semiring_normalizers_conv: cterm list -> cterm list * thm list
+    -> cterm list * thm list -> cterm list * thm list ->
+      (cterm -> bool) * conv * conv * conv -> (cterm -> cterm -> bool) ->
+        {add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
+  val semiring_normalizers_ord_wrapper:  Proof.context -> entry ->
+    (cterm -> cterm -> bool) ->
+      {add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
+
+  val setup: theory -> theory
+end
+
+structure Semiring_Normalizer: SEMIRING_NORMALIZER = 
+struct
+
+(** data **)
+
+type entry =
+ {vars: cterm list,
+  semiring: cterm list * thm list,
+  ring: cterm list * thm list,
+  field: cterm list * thm list,
+  idom: thm list,
+  ideal: thm list} *
+ {is_const: cterm -> bool,
+  dest_const: cterm -> Rat.rat,
+  mk_const: ctyp -> Rat.rat -> cterm,
+  conv: Proof.context -> cterm -> thm};
+
+structure Data = Generic_Data
+(
+  type T = (thm * entry) list;
+  val empty = [];
+  val extend = I;
+  val merge = AList.merge Thm.eq_thm (K true);
+);
+
+val get = Data.get o Context.Proof;
+
+fun match ctxt tm =
+  let
+    fun match_inst
+        ({vars, semiring = (sr_ops, sr_rules), 
+          ring = (r_ops, r_rules), field = (f_ops, f_rules), idom, ideal},
+         fns as {is_const, dest_const, mk_const, conv}) pat =
+       let
+        fun h instT =
+          let
+            val substT = Thm.instantiate (instT, []);
+            val substT_cterm = Drule.cterm_rule substT;
+
+            val vars' = map substT_cterm vars;
+            val semiring' = (map substT_cterm sr_ops, map substT sr_rules);
+            val ring' = (map substT_cterm r_ops, map substT r_rules);
+            val field' = (map substT_cterm f_ops, map substT f_rules);
+            val idom' = map substT idom;
+            val ideal' = map substT ideal;
+
+            val result = ({vars = vars', semiring = semiring', 
+                           ring = ring', field = field', idom = idom', ideal = ideal'}, fns);
+          in SOME result end
+      in (case try Thm.match (pat, tm) of
+           NONE => NONE
+         | SOME (instT, _) => h instT)
+      end;
+
+    fun match_struct (_,
+        entry as ({semiring = (sr_ops, _), ring = (r_ops, _), field = (f_ops, _), ...}, _): entry) =
+      get_first (match_inst entry) (sr_ops @ r_ops @ f_ops);
+  in get_first match_struct (get ctxt) end;
+
+
+(* logical content *)
+
+val semiringN = "semiring";
+val ringN = "ring";
+val idomN = "idom";
+val idealN = "ideal";
+val fieldN = "field";
+
+val del = Thm.declaration_attribute (Data.map o AList.delete Thm.eq_thm);
+
+fun add {semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules), 
+         field = (f_ops, f_rules), idom, ideal} =
+  Thm.declaration_attribute (fn key => fn context => context |> Data.map
+    let
+      val ctxt = Context.proof_of context;
+
+      fun check kind name xs n =
+        null xs orelse length xs = n orelse
+        error ("Expected " ^ string_of_int n ^ " " ^ kind ^ " for " ^ name);
+      val check_ops = check "operations";
+      val check_rules = check "rules";
+
+      val _ =
+        check_ops semiringN sr_ops 5 andalso
+        check_rules semiringN sr_rules 37 andalso
+        check_ops ringN r_ops 2 andalso
+        check_rules ringN r_rules 2 andalso
+        check_ops fieldN f_ops 2 andalso
+        check_rules fieldN f_rules 2 andalso
+        check_rules idomN idom 2;
+
+      val mk_meta = Local_Defs.meta_rewrite_rule ctxt;
+      val sr_rules' = map mk_meta sr_rules;
+      val r_rules' = map mk_meta r_rules;
+      val f_rules' = map mk_meta f_rules;
+
+      fun rule i = nth sr_rules' (i - 1);
+
+      val (cx, cy) = Thm.dest_binop (hd sr_ops);
+      val cz = rule 34 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
+      val cn = rule 36 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
+      val ((clx, crx), (cly, cry)) =
+        rule 13 |> Thm.rhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
+      val ((ca, cb), (cc, cd)) =
+        rule 20 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
+      val cm = rule 1 |> Thm.rhs_of |> Thm.dest_arg;
+      val (cp, cq) = rule 26 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_arg;
+
+      val vars = [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry];
+      val semiring = (sr_ops, sr_rules');
+      val ring = (r_ops, r_rules');
+      val field = (f_ops, f_rules');
+      val ideal' = map (symmetric o mk_meta) ideal
+    in
+      AList.delete Thm.eq_thm key #>
+      cons (key, ({vars = vars, semiring = semiring, 
+                          ring = ring, field = field, idom = idom, ideal = ideal'},
+             {is_const = undefined, dest_const = undefined, mk_const = undefined,
+             conv = undefined}))
+    end);
+
+
+(* extra-logical functions *)
+
+fun funs raw_key {is_const, dest_const, mk_const, conv} phi = 
+ Data.map (fn data =>
+  let
+    val key = Morphism.thm phi raw_key;
+    val _ = AList.defined Thm.eq_thm data key orelse
+      raise THM ("No data entry for structure key", 0, [key]);
+    val fns = {is_const = is_const phi, dest_const = dest_const phi,
+      mk_const = mk_const phi, conv = conv phi};
+  in AList.map_entry Thm.eq_thm key (apsnd (K fns)) data end);
+
+fun semiring_funs key = funs key
+   {is_const = fn phi => can HOLogic.dest_number o Thm.term_of,
+    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
+      (case Rat.quotient_of_rat x of (i, 1) => i | _ => error "int_of_rat: bad int"),
+    conv = fn phi => fn _ => 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)}))};
+
+fun field_funs key =
+  let
+    fun numeral_is_const ct =
+      case term_of ct of
+       Const (@{const_name Rings.divide},_) $ a $ b =>
+         can HOLogic.dest_number a andalso can HOLogic.dest_number b
+     | Const (@{const_name Rings.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 Rings.divide},_) $ a $ b=>
+        Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+     | Const (@{const_name Rings.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 funs key
+     {is_const = K numeral_is_const,
+      dest_const = K dest_const,
+      mk_const = mk_const,
+      conv = K (K Numeral_Simprocs.field_comp_conv)}
+  end;
+
+
+
+(** auxiliary **)
+
+fun is_comb ct =
+  (case Thm.term_of ct of
+    _ $ _ => true
+  | _ => false);
+
+val concl = Thm.cprop_of #> Thm.dest_arg;
+
+fun is_binop ct ct' =
+  (case Thm.term_of ct' of
+    c $ _ $ _ => term_of ct aconv c
+  | _ => false);
+
+fun dest_binop ct ct' =
+  if is_binop ct ct' then Thm.dest_binop ct'
+  else raise CTERM ("dest_binop: bad binop", [ct, ct'])
+
+fun inst_thm inst = Thm.instantiate ([], inst);
+
+val dest_numeral = term_of #> HOLogic.dest_number #> snd;
+val is_numeral = can dest_numeral;
+
+val numeral01_conv = Simplifier.rewrite
+                         (HOL_basic_ss addsimps [@{thm numeral_1_eq_1}, @{thm numeral_0_eq_0}]);
+val zero1_numeral_conv = 
+ Simplifier.rewrite (HOL_basic_ss addsimps [@{thm numeral_1_eq_1} RS sym, @{thm numeral_0_eq_0} RS sym]);
+fun zerone_conv 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 @{thms arith_simps} @ natarith @ @{thms rel_simps}
+             @ [@{thm if_False}, @{thm if_True}, @{thm Nat.add_0}, @{thm add_Suc},
+                 @{thm add_number_of_left}, @{thm Suc_eq_plus1}]
+             @ map (fn th => th RS sym) @{thms numerals}));
+
+val zeron_tm = @{cterm "0::nat"};
+val onen_tm  = @{cterm "1::nat"};
+val true_tm = @{cterm "True"};
+
+
+(** normalizing conversions **)
+
+(* core conversion *)
+
+fun semiring_normalizers_conv vars (sr_ops, sr_rules) (r_ops, r_rules) (f_ops, f_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
+    ([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
+    | _ => (TrueI, TrueI, true_tm, true_tm, (fn t => (t,t)), K false, true_tm, true_tm));
+
+val (divide_inverse, inverse_divide, divide_tm, inverse_tm, is_divide) = 
+  (case (f_ops, f_rules) of 
+   ([divide_pat, inverse_pat], [div_inv, inv_div]) => 
+     let val div_tm = funpow 2 Thm.dest_fun divide_pat
+         val inv_tm = Thm.dest_fun inverse_pat
+     in (div_inv, inv_div, div_tm, inv_tm, is_binop div_tm)
+     end
+   | _ => (TrueI, TrueI, true_tm, true_tm, K false));
+
+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_add_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 (Integer.add o snd) vdegs1 num_0
+      val deg2 = fold (Integer.add 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).       *)
+
+fun num_conv n =
+  nat_add_conv (Thm.capply @{cterm Suc} (Numeral.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.                                                                 *)
+
+fun polynomial_neg_conv 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 (Conv.arg1_conv semiring_mul_conv (concl th1))
+        in transitive th2 (polynomial_monomial_mul_conv (concl th2))
+        end
+   end;
+
+
+(* Subtraction.                                                              *)
+fun polynomial_sub_conv 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 lopr aconvc inverse_tm then
+       let val th1 = Drule.arg_cong_rule lopr (polynomial_conv r)
+       in transitive th1 (semiring_mul_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 divide_tm 
+            then
+              let val th1 = combination (Drule.arg_cong_rule opr (polynomial_conv l)) 
+                                        (polynomial_conv r)
+                  val th2 = (Conv.rewr_conv divide_inverse then_conv polynomial_mul_conv)
+                              (Thm.rhs_of th1)
+              in transitive th1 th2
+              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_exp_ss =
+  HOL_basic_ss addsimps (@{thms nat_number} @ @{thms nat_arith} @ @{thms arith_simps} @ @{thms rel_simps})
+    addsimps [@{thm Let_def}, @{thm if_False}, @{thm if_True}, @{thm Nat.add_0}, @{thm add_Suc}];
+
+fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS;
+
+
+(* various normalizing conversions *)
+
+fun semiring_normalizers_ord_wrapper ctxt ({vars, semiring, ring, field, idom, ideal}, 
+                                     {conv, dest_const, mk_const, is_const}) ord =
+  let
+    val pow_conv =
+      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)
+  in semiring_normalizers_conv vars semiring ring field dat ord end;
+
+fun semiring_normalize_ord_wrapper ctxt ({vars, semiring, ring, field, idom, ideal}, {conv, dest_const, mk_const, is_const}) ord =
+ #main (semiring_normalizers_ord_wrapper ctxt ({vars = vars, semiring = semiring, ring = ring, field = field, idom = idom, ideal = ideal},{conv = conv, dest_const = dest_const, mk_const = mk_const, is_const = is_const}) ord);
+
+fun semiring_normalize_wrapper ctxt data = 
+  semiring_normalize_ord_wrapper ctxt data simple_cterm_ord;
+
+fun semiring_normalize_ord_conv ctxt ord tm =
+  (case 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;
+
+
+(** Isar setup **)
+
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
+fun keyword2 k1 k2 = Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.colon) >> K ();
+fun keyword3 k1 k2 k3 =
+  Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.$$$ k3 -- Args.colon) >> K ();
+
+val opsN = "ops";
+val rulesN = "rules";
+
+val normN = "norm";
+val constN = "const";
+val delN = "del";
+
+val any_keyword =
+  keyword2 semiringN opsN || keyword2 semiringN rulesN ||
+  keyword2 ringN opsN || keyword2 ringN rulesN ||
+  keyword2 fieldN opsN || keyword2 fieldN rulesN ||
+  keyword2 idomN rulesN || keyword2 idealN rulesN;
+
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+val terms = thms >> map Drule.dest_term;
+
+fun optional scan = Scan.optional scan [];
+
+in
+
+val setup =
+  Attrib.setup @{binding normalizer}
+    (Scan.lift (Args.$$$ delN >> K del) ||
+      ((keyword2 semiringN opsN |-- terms) --
+       (keyword2 semiringN rulesN |-- thms)) --
+      (optional (keyword2 ringN opsN |-- terms) --
+       optional (keyword2 ringN rulesN |-- thms)) --
+      (optional (keyword2 fieldN opsN |-- terms) --
+       optional (keyword2 fieldN rulesN |-- thms)) --
+      optional (keyword2 idomN rulesN |-- thms) --
+      optional (keyword2 idealN rulesN |-- thms)
+      >> (fn ((((sr, r), f), id), idl) => 
+             add {semiring = sr, ring = r, field = f, idom = id, ideal = idl}))
+    "semiring normalizer data";
+
+end;
+
+end;