src/HOL/Tools/Groebner_Basis/normalizer.ML
author wenzelm
Tue Jun 05 16:26:04 2007 +0200 (2007-06-05)
changeset 23252 67268bb40b21
child 23259 ccee01b8d1c5
permissions -rw-r--r--
Semiring normalization and Groebner Bases.
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(*  Title:      HOL/Tools/Groebner_Basis/normalizer.ML
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    ID:         $Id$
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    Author:     Amine Chaieb, TU Muenchen
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*)
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signature NORMALIZER = 
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sig
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 val mk_cnumber : ctyp -> int -> cterm
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 val mk_cnumeral : int -> cterm
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 val semiring_normalize_conv : Proof.context -> Conv.conv
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 val semiring_normalize_tac : Proof.context -> int -> tactic
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 val semiring_normalize_wrapper :  NormalizerData.entry -> Conv.conv
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 val semiring_normalizers_conv :
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     cterm list -> cterm list * thm list -> cterm list * thm list ->
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     (cterm -> bool) * Conv.conv * Conv.conv * Conv.conv -> (cterm -> Thm.cterm -> bool) ->
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       {add: Conv.conv, mul: Conv.conv, neg: Conv.conv, main: Conv.conv, 
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        pow: Conv.conv, sub: Conv.conv}
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end
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structure Normalizer: NORMALIZER = 
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struct
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open Misc;
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local
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 val pls_const = @{cterm "Numeral.Pls"}
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   and min_const = @{cterm "Numeral.Min"}
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   and bit_const = @{cterm "Numeral.Bit"}
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   and zero = @{cpat "0"}
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   and one = @{cpat "1"}
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 fun mk_cbit 0 = @{cterm "Numeral.bit.B0"}
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  | mk_cbit 1 = @{cterm "Numeral.bit.B1"}
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  | mk_cbit _ = raise CTERM ("mk_cbit", []);
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in
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fun mk_cnumeral 0 = pls_const
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  | mk_cnumeral ~1 = min_const
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  | mk_cnumeral i =
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      let val (q, r) = IntInf.divMod (i, 2)
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      in Thm.capply (Thm.capply bit_const (mk_cnumeral q)) (mk_cbit (IntInf.toInt r)) 
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      end;
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fun mk_cnumber cT = 
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 let 
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  val [nb_of, z, on] = 
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    map (Drule.cterm_rule (instantiate' [SOME cT] [])) [@{cpat "number_of"}, zero, one]
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  fun h 0 = z
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    | h 1 = on
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    | h x = Thm.capply nb_of (mk_cnumeral x)
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 in h end;
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end;
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(* Very basic stuff for terms *)
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val dest_numeral = term_of #> HOLogic.dest_number #> snd;
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val is_numeral = can dest_numeral;
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val numeral01_conv = Simplifier.rewrite
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                         (HOL_basic_ss addsimps [numeral_1_eq_1, numeral_0_eq_0]);
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val zero1_numeral_conv = 
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 Simplifier.rewrite (HOL_basic_ss addsimps [numeral_1_eq_1 RS sym, numeral_0_eq_0 RS sym]);
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val zerone_conv = fn cv => zero1_numeral_conv then_conv cv then_conv numeral01_conv;
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val natarith = [@{thm "add_nat_number_of"}, @{thm "diff_nat_number_of"},
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                @{thm "mult_nat_number_of"}, @{thm "eq_nat_number_of"}, 
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                @{thm "less_nat_number_of"}];
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val nat_add_conv = 
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 zerone_conv 
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  (Simplifier.rewrite 
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    (HOL_basic_ss 
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       addsimps arith_simps @ natarith @ rel_simps
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             @ [if_False, if_True, add_0, add_Suc, add_number_of_left, Suc_eq_add_numeral_1]
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             @ map (fn th => th RS sym) numerals));
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val nat_mul_conv = nat_add_conv;
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val zeron_tm = @{cterm "0::nat"};
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val onen_tm  = @{cterm "1::nat"};
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val true_tm = @{cterm "True"};
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(* The main function! *)
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fun semiring_normalizers_conv vars (sr_ops, sr_rules) (r_ops, r_rules)
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  (is_semiring_constant, semiring_add_conv, semiring_mul_conv, semiring_pow_conv) =
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let
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val [pthm_02, pthm_03, pthm_04, pthm_05, pthm_07, pthm_08,
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     pthm_09, pthm_10, pthm_11, pthm_12, pthm_13, pthm_14, pthm_15, pthm_16,
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     pthm_17, pthm_18, pthm_19, pthm_21, pthm_22, pthm_23, pthm_24,
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     pthm_25, pthm_26, pthm_27, pthm_28, pthm_29, pthm_30, pthm_31, pthm_32,
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     pthm_33, pthm_34, pthm_35, pthm_36, pthm_37, pthm_38,pthm_39,pthm_40] = sr_rules;
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val [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry] = vars;
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val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
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val [add_tm, mul_tm, pow_tm] = map (Thm.dest_fun o Thm.dest_fun) [add_pat, mul_pat, pow_pat];
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val dest_add = dest_binop add_tm
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val dest_mul = dest_binop mul_tm
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fun dest_pow tm =
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 let val (l,r) = dest_binop pow_tm tm
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 in if is_numeral r then (l,r) else raise CTERM ("dest_pow",[tm])
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 end;
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val is_add = is_binop add_tm
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val is_mul = is_binop mul_tm
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fun is_pow tm = is_binop pow_tm tm andalso is_numeral(Thm.dest_arg tm);
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val (neg_mul,sub_add,sub_tm,neg_tm,dest_sub,is_sub,cx',cy') =
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  (case (r_ops, r_rules) of
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    ([], []) => (TrueI, TrueI, true_tm, true_tm, (fn t => (t,t)), K false, true_tm, true_tm)
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  | ([sub_pat, neg_pat], [neg_mul, sub_add]) =>
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      let
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        val sub_tm = Thm.dest_fun (Thm.dest_fun sub_pat)
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        val neg_tm = Thm.dest_fun neg_pat
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        val dest_sub = dest_binop sub_tm
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        val is_sub = is_binop sub_tm
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      in (neg_mul,sub_add,sub_tm,neg_tm,dest_sub,is_sub, neg_mul |> concl |> Thm.dest_arg,
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          sub_add |> concl |> Thm.dest_arg |> Thm.dest_arg)
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      end);
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in fn variable_order =>
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 let
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(* Conversion for "x^n * x^m", with either x^n = x and/or x^m = x possible.  *)
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(* Also deals with "const * const", but both terms must involve powers of    *)
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(* the same variable, or both be constants, or behaviour may be incorrect.   *)
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 fun powvar_mul_conv tm =
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  let
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  val (l,r) = dest_mul tm
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  in if is_semiring_constant l andalso is_semiring_constant r
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     then semiring_mul_conv tm
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     else
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      ((let
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         val (lx,ln) = dest_pow l
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        in
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         ((let val (rx,rn) = dest_pow r
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               val th1 = inst_thm [(cx,lx),(cp,ln),(cq,rn)] pthm_29
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                val (tm1,tm2) = Thm.dest_comb(concl th1) in
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               transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)
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           handle CTERM _ =>
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            (let val th1 = inst_thm [(cx,lx),(cq,ln)] pthm_31
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                 val (tm1,tm2) = Thm.dest_comb(concl th1) in
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               transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)) end)
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       handle CTERM _ =>
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           ((let val (rx,rn) = dest_pow r
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                val th1 = inst_thm [(cx,rx),(cq,rn)] pthm_30
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                val (tm1,tm2) = Thm.dest_comb(concl th1) in
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               transitive th1 (Drule.arg_cong_rule tm1 (nat_add_conv tm2)) end)
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           handle CTERM _ => inst_thm [(cx,l)] pthm_32
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))
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 end;
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(* Remove "1 * m" from a monomial, and just leave m.                         *)
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 fun monomial_deone th =
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       (let val (l,r) = dest_mul(concl th) in
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           if l aconvc one_tm
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          then transitive th (inst_thm [(ca,r)] pthm_13)  else th end)
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       handle CTERM _ => th;
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(* Conversion for "(monomial)^n", where n is a numeral.                      *)
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 val monomial_pow_conv =
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  let
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   fun monomial_pow tm bod ntm =
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    if not(is_comb bod)
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    then reflexive tm
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    else
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     if is_semiring_constant bod
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     then semiring_pow_conv tm
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     else
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      let
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      val (lopr,r) = Thm.dest_comb bod
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      in if not(is_comb lopr)
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         then reflexive tm
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        else
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          let
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          val (opr,l) = Thm.dest_comb lopr
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         in
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           if opr aconvc pow_tm andalso is_numeral r
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          then
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            let val th1 = inst_thm [(cx,l),(cp,r),(cq,ntm)] pthm_34
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                val (l,r) = Thm.dest_comb(concl th1)
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           in transitive th1 (Drule.arg_cong_rule l (nat_mul_conv r))
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           end
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           else
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            if opr aconvc mul_tm
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            then
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             let
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              val th1 = inst_thm [(cx,l),(cy,r),(cq,ntm)] pthm_33
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             val (xy,z) = Thm.dest_comb(concl th1)
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              val (x,y) = Thm.dest_comb xy
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              val thl = monomial_pow y l ntm
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              val thr = monomial_pow z r ntm
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             in transitive th1 (combination (Drule.arg_cong_rule x thl) thr)
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             end
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             else reflexive tm
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          end
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      end
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  in fn tm =>
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   let
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    val (lopr,r) = Thm.dest_comb tm
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    val (opr,l) = Thm.dest_comb lopr
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   in if not (opr aconvc pow_tm) orelse not(is_numeral r)
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      then raise CTERM ("monomial_pow_conv", [tm])
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      else if r aconvc zeron_tm
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      then inst_thm [(cx,l)] pthm_35
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      else if r aconvc onen_tm
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      then inst_thm [(cx,l)] pthm_36
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      else monomial_deone(monomial_pow tm l r)
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   end
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  end;
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(* Multiplication of canonical monomials.                                    *)
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 val monomial_mul_conv =
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  let
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   fun powvar tm =
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    if is_semiring_constant tm then one_tm
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    else
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     ((let val (lopr,r) = Thm.dest_comb tm
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           val (opr,l) = Thm.dest_comb lopr
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       in if opr aconvc pow_tm andalso is_numeral r then l 
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          else raise CTERM ("monomial_mul_conv",[tm]) end)
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     handle CTERM _ => tm)   (* FIXME !? *)
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   fun  vorder x y =
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    if x aconvc y then 0
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    else
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     if x aconvc one_tm then ~1
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     else if y aconvc one_tm then 1
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      else if variable_order x y then ~1 else 1
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   fun monomial_mul tm l r =
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    ((let val (lx,ly) = dest_mul l val vl = powvar lx
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      in
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      ((let
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        val (rx,ry) = dest_mul r
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         val vr = powvar rx
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         val ord = vorder vl vr
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        in
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         if ord = 0
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        then
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          let
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             val th1 = inst_thm [(clx,lx),(cly,ly),(crx,rx),(cry,ry)] pthm_15
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             val (tm1,tm2) = Thm.dest_comb(concl th1)
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             val (tm3,tm4) = Thm.dest_comb tm1
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             val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv tm4)) tm2
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             val th3 = transitive th1 th2
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              val  (tm5,tm6) = Thm.dest_comb(concl th3)
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              val  (tm7,tm8) = Thm.dest_comb tm6
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             val  th4 = monomial_mul tm6 (Thm.dest_arg tm7) tm8
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         in  transitive th3 (Drule.arg_cong_rule tm5 th4)
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         end
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         else
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          let val th0 = if ord < 0 then pthm_16 else pthm_17
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             val th1 = inst_thm [(clx,lx),(cly,ly),(crx,rx),(cry,ry)] th0
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             val (tm1,tm2) = Thm.dest_comb(concl th1)
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             val (tm3,tm4) = Thm.dest_comb tm2
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         in transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
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         end
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        end)
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       handle CTERM _ =>
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        (let val vr = powvar r val ord = vorder vl vr
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        in
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          if ord = 0 then
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           let
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           val th1 = inst_thm [(clx,lx),(cly,ly),(crx,r)] pthm_18
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                 val (tm1,tm2) = Thm.dest_comb(concl th1)
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           val (tm3,tm4) = Thm.dest_comb tm1
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           val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv tm4)) tm2
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          in transitive th1 th2
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          end
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          else
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          if ord < 0 then
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            let val th1 = inst_thm [(clx,lx),(cly,ly),(crx,r)] pthm_19
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                val (tm1,tm2) = Thm.dest_comb(concl th1)
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                val (tm3,tm4) = Thm.dest_comb tm2
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           in transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
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           end
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           else inst_thm [(ca,l),(cb,r)] pthm_09
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        end)) end)
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     handle CTERM _ =>
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      (let val vl = powvar l in
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        ((let
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          val (rx,ry) = dest_mul r
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          val vr = powvar rx
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           val ord = vorder vl vr
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         in if ord = 0 then
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              let val th1 = inst_thm [(clx,l),(crx,rx),(cry,ry)] pthm_21
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                 val (tm1,tm2) = Thm.dest_comb(concl th1)
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                 val (tm3,tm4) = Thm.dest_comb tm1
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             in transitive th1 (Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (powvar_mul_conv tm4)) tm2)
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             end
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             else if ord > 0 then
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                 let val th1 = inst_thm [(clx,l),(crx,rx),(cry,ry)] pthm_22
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                     val (tm1,tm2) = Thm.dest_comb(concl th1)
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                    val (tm3,tm4) = Thm.dest_comb tm2
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                in transitive th1 (Drule.arg_cong_rule tm1 (monomial_mul tm2 (Thm.dest_arg tm3) tm4))
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                end
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             else reflexive tm
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         end)
wenzelm@23252
   298
        handle CTERM _ =>
wenzelm@23252
   299
          (let val vr = powvar r
wenzelm@23252
   300
               val  ord = vorder vl vr
wenzelm@23252
   301
          in if ord = 0 then powvar_mul_conv tm
wenzelm@23252
   302
              else if ord > 0 then inst_thm [(ca,l),(cb,r)] pthm_09
wenzelm@23252
   303
              else reflexive tm
wenzelm@23252
   304
          end)) end))
wenzelm@23252
   305
  in fn tm => let val (l,r) = dest_mul tm in monomial_deone(monomial_mul tm l r)
wenzelm@23252
   306
             end
wenzelm@23252
   307
  end;
wenzelm@23252
   308
(* Multiplication by monomial of a polynomial.                               *)
wenzelm@23252
   309
wenzelm@23252
   310
 val polynomial_monomial_mul_conv =
wenzelm@23252
   311
  let
wenzelm@23252
   312
   fun pmm_conv tm =
wenzelm@23252
   313
    let val (l,r) = dest_mul tm
wenzelm@23252
   314
    in
wenzelm@23252
   315
    ((let val (y,z) = dest_add r
wenzelm@23252
   316
          val th1 = inst_thm [(cx,l),(cy,y),(cz,z)] pthm_37
wenzelm@23252
   317
          val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   318
          val (tm3,tm4) = Thm.dest_comb tm1
wenzelm@23252
   319
          val th2 = combination (Drule.arg_cong_rule tm3 (monomial_mul_conv tm4)) (pmm_conv tm2)
wenzelm@23252
   320
      in transitive th1 th2
wenzelm@23252
   321
      end)
wenzelm@23252
   322
     handle CTERM _ => monomial_mul_conv tm)
wenzelm@23252
   323
   end
wenzelm@23252
   324
 in pmm_conv
wenzelm@23252
   325
 end;
wenzelm@23252
   326
wenzelm@23252
   327
(* Addition of two monomials identical except for constant multiples.        *)
wenzelm@23252
   328
wenzelm@23252
   329
fun monomial_add_conv tm =
wenzelm@23252
   330
 let val (l,r) = dest_add tm
wenzelm@23252
   331
 in if is_semiring_constant l andalso is_semiring_constant r
wenzelm@23252
   332
    then semiring_add_conv tm
wenzelm@23252
   333
    else
wenzelm@23252
   334
     let val th1 =
wenzelm@23252
   335
           if is_mul l andalso is_semiring_constant(Thm.dest_arg1 l)
wenzelm@23252
   336
           then if is_mul r andalso is_semiring_constant(Thm.dest_arg1 r) then
wenzelm@23252
   337
                    inst_thm [(ca,Thm.dest_arg1 l),(cm,Thm.dest_arg r), (cb,Thm.dest_arg1 r)] pthm_02
wenzelm@23252
   338
                else inst_thm [(ca,Thm.dest_arg1 l),(cm,r)] pthm_03
wenzelm@23252
   339
           else if is_mul r andalso is_semiring_constant(Thm.dest_arg1 r)
wenzelm@23252
   340
           then inst_thm [(cm,l),(ca,Thm.dest_arg1 r)] pthm_04
wenzelm@23252
   341
           else inst_thm [(cm,r)] pthm_05
wenzelm@23252
   342
         val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   343
         val (tm3,tm4) = Thm.dest_comb tm1
wenzelm@23252
   344
         val th2 = Drule.arg_cong_rule tm3 (semiring_add_conv tm4)
wenzelm@23252
   345
         val th3 = transitive th1 (Drule.fun_cong_rule th2 tm2)
wenzelm@23252
   346
         val tm5 = concl th3
wenzelm@23252
   347
      in
wenzelm@23252
   348
      if (Thm.dest_arg1 tm5) aconvc zero_tm
wenzelm@23252
   349
      then transitive th3 (inst_thm [(ca,Thm.dest_arg tm5)] pthm_11)
wenzelm@23252
   350
      else monomial_deone th3
wenzelm@23252
   351
     end
wenzelm@23252
   352
 end;
wenzelm@23252
   353
wenzelm@23252
   354
(* Ordering on monomials.                                                    *)
wenzelm@23252
   355
wenzelm@23252
   356
fun striplist dest =
wenzelm@23252
   357
 let fun strip x acc =
wenzelm@23252
   358
   ((let val (l,r) = dest x in
wenzelm@23252
   359
        strip l (strip r acc) end)
wenzelm@23252
   360
    handle CTERM _ => x::acc)    (* FIXME !? *)
wenzelm@23252
   361
 in fn x => strip x []
wenzelm@23252
   362
 end;
wenzelm@23252
   363
wenzelm@23252
   364
wenzelm@23252
   365
fun powervars tm =
wenzelm@23252
   366
 let val ptms = striplist dest_mul tm
wenzelm@23252
   367
 in if is_semiring_constant (hd ptms) then tl ptms else ptms
wenzelm@23252
   368
 end;
wenzelm@23252
   369
val num_0 = 0;
wenzelm@23252
   370
val num_1 = 1;
wenzelm@23252
   371
fun dest_varpow tm =
wenzelm@23252
   372
 ((let val (x,n) = dest_pow tm in (x,dest_numeral n) end)
wenzelm@23252
   373
   handle CTERM _ =>
wenzelm@23252
   374
   (tm,(if is_semiring_constant tm then num_0 else num_1)));
wenzelm@23252
   375
wenzelm@23252
   376
val morder =
wenzelm@23252
   377
 let fun lexorder l1 l2 =
wenzelm@23252
   378
  case (l1,l2) of
wenzelm@23252
   379
    ([],[]) => 0
wenzelm@23252
   380
  | (vps,[]) => ~1
wenzelm@23252
   381
  | ([],vps) => 1
wenzelm@23252
   382
  | (((x1,n1)::vs1),((x2,n2)::vs2)) =>
wenzelm@23252
   383
     if variable_order x1 x2 then 1
wenzelm@23252
   384
     else if variable_order x2 x1 then ~1
wenzelm@23252
   385
     else if n1 < n2 then ~1
wenzelm@23252
   386
     else if n2 < n1 then 1
wenzelm@23252
   387
     else lexorder vs1 vs2
wenzelm@23252
   388
 in fn tm1 => fn tm2 =>
wenzelm@23252
   389
  let val vdegs1 = map dest_varpow (powervars tm1)
wenzelm@23252
   390
      val vdegs2 = map dest_varpow (powervars tm2)
wenzelm@23252
   391
      val deg1 = fold_rev ((curry (op +)) o snd) vdegs1 num_0
wenzelm@23252
   392
      val deg2 = fold_rev ((curry (op +)) o snd) vdegs2 num_0
wenzelm@23252
   393
  in if deg1 < deg2 then ~1 else if deg1 > deg2 then 1
wenzelm@23252
   394
                            else lexorder vdegs1 vdegs2
wenzelm@23252
   395
  end
wenzelm@23252
   396
 end;
wenzelm@23252
   397
wenzelm@23252
   398
(* Addition of two polynomials.                                              *)
wenzelm@23252
   399
wenzelm@23252
   400
val polynomial_add_conv =
wenzelm@23252
   401
 let
wenzelm@23252
   402
 fun dezero_rule th =
wenzelm@23252
   403
  let
wenzelm@23252
   404
   val tm = concl th
wenzelm@23252
   405
  in
wenzelm@23252
   406
   if not(is_add tm) then th else
wenzelm@23252
   407
   let val (lopr,r) = Thm.dest_comb tm
wenzelm@23252
   408
       val l = Thm.dest_arg lopr
wenzelm@23252
   409
   in
wenzelm@23252
   410
    if l aconvc zero_tm
wenzelm@23252
   411
    then transitive th (inst_thm [(ca,r)] pthm_07)   else
wenzelm@23252
   412
        if r aconvc zero_tm
wenzelm@23252
   413
        then transitive th (inst_thm [(ca,l)] pthm_08)  else th
wenzelm@23252
   414
   end
wenzelm@23252
   415
  end
wenzelm@23252
   416
 fun padd tm =
wenzelm@23252
   417
  let
wenzelm@23252
   418
   val (l,r) = dest_add tm
wenzelm@23252
   419
  in
wenzelm@23252
   420
   if l aconvc zero_tm then inst_thm [(ca,r)] pthm_07
wenzelm@23252
   421
   else if r aconvc zero_tm then inst_thm [(ca,l)] pthm_08
wenzelm@23252
   422
   else
wenzelm@23252
   423
    if is_add l
wenzelm@23252
   424
    then
wenzelm@23252
   425
     let val (a,b) = dest_add l
wenzelm@23252
   426
     in
wenzelm@23252
   427
     if is_add r then
wenzelm@23252
   428
      let val (c,d) = dest_add r
wenzelm@23252
   429
          val ord = morder a c
wenzelm@23252
   430
      in
wenzelm@23252
   431
       if ord = 0 then
wenzelm@23252
   432
        let val th1 = inst_thm [(ca,a),(cb,b),(cc,c),(cd,d)] pthm_23
wenzelm@23252
   433
            val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   434
            val (tm3,tm4) = Thm.dest_comb tm1
wenzelm@23252
   435
            val th2 = Drule.arg_cong_rule tm3 (monomial_add_conv tm4)
wenzelm@23252
   436
        in dezero_rule (transitive th1 (combination th2 (padd tm2)))
wenzelm@23252
   437
        end
wenzelm@23252
   438
       else (* ord <> 0*)
wenzelm@23252
   439
        let val th1 =
wenzelm@23252
   440
                if ord > 0 then inst_thm [(ca,a),(cb,b),(cc,r)] pthm_24
wenzelm@23252
   441
                else inst_thm [(ca,l),(cc,c),(cd,d)] pthm_25
wenzelm@23252
   442
            val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   443
        in dezero_rule (transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
wenzelm@23252
   444
        end
wenzelm@23252
   445
      end
wenzelm@23252
   446
     else (* not (is_add r)*)
wenzelm@23252
   447
      let val ord = morder a r
wenzelm@23252
   448
      in
wenzelm@23252
   449
       if ord = 0 then
wenzelm@23252
   450
        let val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_26
wenzelm@23252
   451
            val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   452
            val (tm3,tm4) = Thm.dest_comb tm1
wenzelm@23252
   453
            val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (monomial_add_conv tm4)) tm2
wenzelm@23252
   454
        in dezero_rule (transitive th1 th2)
wenzelm@23252
   455
        end
wenzelm@23252
   456
       else (* ord <> 0*)
wenzelm@23252
   457
        if ord > 0 then
wenzelm@23252
   458
          let val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_24
wenzelm@23252
   459
              val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   460
          in dezero_rule (transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
wenzelm@23252
   461
          end
wenzelm@23252
   462
        else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
wenzelm@23252
   463
      end
wenzelm@23252
   464
    end
wenzelm@23252
   465
   else (* not (is_add l)*)
wenzelm@23252
   466
    if is_add r then
wenzelm@23252
   467
      let val (c,d) = dest_add r
wenzelm@23252
   468
          val  ord = morder l c
wenzelm@23252
   469
      in
wenzelm@23252
   470
       if ord = 0 then
wenzelm@23252
   471
         let val th1 = inst_thm [(ca,l),(cc,c),(cd,d)] pthm_28
wenzelm@23252
   472
             val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   473
             val (tm3,tm4) = Thm.dest_comb tm1
wenzelm@23252
   474
             val th2 = Drule.fun_cong_rule (Drule.arg_cong_rule tm3 (monomial_add_conv tm4)) tm2
wenzelm@23252
   475
         in dezero_rule (transitive th1 th2)
wenzelm@23252
   476
         end
wenzelm@23252
   477
       else
wenzelm@23252
   478
        if ord > 0 then reflexive tm
wenzelm@23252
   479
        else
wenzelm@23252
   480
         let val th1 = inst_thm [(ca,l),(cc,c),(cd,d)] pthm_25
wenzelm@23252
   481
             val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   482
         in dezero_rule (transitive th1 (Drule.arg_cong_rule tm1 (padd tm2)))
wenzelm@23252
   483
         end
wenzelm@23252
   484
      end
wenzelm@23252
   485
    else
wenzelm@23252
   486
     let val ord = morder l r
wenzelm@23252
   487
     in
wenzelm@23252
   488
      if ord = 0 then monomial_add_conv tm
wenzelm@23252
   489
      else if ord > 0 then dezero_rule(reflexive tm)
wenzelm@23252
   490
      else dezero_rule (inst_thm [(ca,l),(cc,r)] pthm_27)
wenzelm@23252
   491
     end
wenzelm@23252
   492
  end
wenzelm@23252
   493
 in padd
wenzelm@23252
   494
 end;
wenzelm@23252
   495
wenzelm@23252
   496
(* Multiplication of two polynomials.                                        *)
wenzelm@23252
   497
wenzelm@23252
   498
val polynomial_mul_conv =
wenzelm@23252
   499
 let
wenzelm@23252
   500
  fun pmul tm =
wenzelm@23252
   501
   let val (l,r) = dest_mul tm
wenzelm@23252
   502
   in
wenzelm@23252
   503
    if not(is_add l) then polynomial_monomial_mul_conv tm
wenzelm@23252
   504
    else
wenzelm@23252
   505
     if not(is_add r) then
wenzelm@23252
   506
      let val th1 = inst_thm [(ca,l),(cb,r)] pthm_09
wenzelm@23252
   507
      in transitive th1 (polynomial_monomial_mul_conv(concl th1))
wenzelm@23252
   508
      end
wenzelm@23252
   509
     else
wenzelm@23252
   510
       let val (a,b) = dest_add l
wenzelm@23252
   511
           val th1 = inst_thm [(ca,a),(cb,b),(cc,r)] pthm_10
wenzelm@23252
   512
           val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   513
           val (tm3,tm4) = Thm.dest_comb tm1
wenzelm@23252
   514
           val th2 = Drule.arg_cong_rule tm3 (polynomial_monomial_mul_conv tm4)
wenzelm@23252
   515
           val th3 = transitive th1 (combination th2 (pmul tm2))
wenzelm@23252
   516
       in transitive th3 (polynomial_add_conv (concl th3))
wenzelm@23252
   517
       end
wenzelm@23252
   518
   end
wenzelm@23252
   519
 in fn tm =>
wenzelm@23252
   520
   let val (l,r) = dest_mul tm
wenzelm@23252
   521
   in
wenzelm@23252
   522
    if l aconvc zero_tm then inst_thm [(ca,r)] pthm_11
wenzelm@23252
   523
    else if r aconvc zero_tm then inst_thm [(ca,l)] pthm_12
wenzelm@23252
   524
    else if l aconvc one_tm then inst_thm [(ca,r)] pthm_13
wenzelm@23252
   525
    else if r aconvc one_tm then inst_thm [(ca,l)] pthm_14
wenzelm@23252
   526
    else pmul tm
wenzelm@23252
   527
   end
wenzelm@23252
   528
 end;
wenzelm@23252
   529
wenzelm@23252
   530
(* Power of polynomial (optimized for the monomial and trivial cases).       *)
wenzelm@23252
   531
wenzelm@23252
   532
val Succ = @{cterm "Suc"};
wenzelm@23252
   533
val num_conv = fn n =>
wenzelm@23252
   534
        nat_add_conv (Thm.capply (Succ) (mk_cnumber @{ctyp "nat"} ((dest_numeral n) - 1)))
wenzelm@23252
   535
                     |> Thm.symmetric;
wenzelm@23252
   536
wenzelm@23252
   537
wenzelm@23252
   538
val polynomial_pow_conv =
wenzelm@23252
   539
 let
wenzelm@23252
   540
  fun ppow tm =
wenzelm@23252
   541
    let val (l,n) = dest_pow tm
wenzelm@23252
   542
    in
wenzelm@23252
   543
     if n aconvc zeron_tm then inst_thm [(cx,l)] pthm_35
wenzelm@23252
   544
     else if n aconvc onen_tm then inst_thm [(cx,l)] pthm_36
wenzelm@23252
   545
     else
wenzelm@23252
   546
         let val th1 = num_conv n
wenzelm@23252
   547
             val th2 = inst_thm [(cx,l),(cq,Thm.dest_arg (concl th1))] pthm_38
wenzelm@23252
   548
             val (tm1,tm2) = Thm.dest_comb(concl th2)
wenzelm@23252
   549
             val th3 = transitive th2 (Drule.arg_cong_rule tm1 (ppow tm2))
wenzelm@23252
   550
             val th4 = transitive (Drule.arg_cong_rule (Thm.dest_fun tm) th1) th3
wenzelm@23252
   551
         in transitive th4 (polynomial_mul_conv (concl th4))
wenzelm@23252
   552
         end
wenzelm@23252
   553
    end
wenzelm@23252
   554
 in fn tm =>
wenzelm@23252
   555
       if is_add(Thm.dest_arg1 tm) then ppow tm else monomial_pow_conv tm
wenzelm@23252
   556
 end;
wenzelm@23252
   557
wenzelm@23252
   558
(* Negation.                                                                 *)
wenzelm@23252
   559
wenzelm@23252
   560
val polynomial_neg_conv =
wenzelm@23252
   561
 fn tm =>
wenzelm@23252
   562
   let val (l,r) = Thm.dest_comb tm in
wenzelm@23252
   563
        if not (l aconvc neg_tm) then raise CTERM ("polynomial_neg_conv",[tm]) else
wenzelm@23252
   564
        let val th1 = inst_thm [(cx',r)] neg_mul
wenzelm@23252
   565
            val th2 = transitive th1 (arg1_conv semiring_mul_conv (concl th1))
wenzelm@23252
   566
        in transitive th2 (polynomial_monomial_mul_conv (concl th2))
wenzelm@23252
   567
        end
wenzelm@23252
   568
   end;
wenzelm@23252
   569
wenzelm@23252
   570
wenzelm@23252
   571
(* Subtraction.                                                              *)
wenzelm@23252
   572
val polynomial_sub_conv = fn tm =>
wenzelm@23252
   573
  let val (l,r) = dest_sub tm
wenzelm@23252
   574
      val th1 = inst_thm [(cx',l),(cy',r)] sub_add
wenzelm@23252
   575
      val (tm1,tm2) = Thm.dest_comb(concl th1)
wenzelm@23252
   576
      val th2 = Drule.arg_cong_rule tm1 (polynomial_neg_conv tm2)
wenzelm@23252
   577
  in transitive th1 (transitive th2 (polynomial_add_conv (concl th2)))
wenzelm@23252
   578
  end;
wenzelm@23252
   579
wenzelm@23252
   580
(* Conversion from HOL term.                                                 *)
wenzelm@23252
   581
wenzelm@23252
   582
fun polynomial_conv tm =
wenzelm@23252
   583
 if not(is_comb tm) orelse is_semiring_constant tm
wenzelm@23252
   584
 then reflexive tm
wenzelm@23252
   585
 else
wenzelm@23252
   586
  let val (lopr,r) = Thm.dest_comb tm
wenzelm@23252
   587
  in if lopr aconvc neg_tm then
wenzelm@23252
   588
       let val th1 = Drule.arg_cong_rule lopr (polynomial_conv r)
wenzelm@23252
   589
       in transitive th1 (polynomial_neg_conv (concl th1))
wenzelm@23252
   590
       end
wenzelm@23252
   591
     else
wenzelm@23252
   592
       if not(is_comb lopr) then reflexive tm
wenzelm@23252
   593
       else
wenzelm@23252
   594
         let val (opr,l) = Thm.dest_comb lopr
wenzelm@23252
   595
         in if opr aconvc pow_tm andalso is_numeral r
wenzelm@23252
   596
            then
wenzelm@23252
   597
              let val th1 = Drule.fun_cong_rule (Drule.arg_cong_rule opr (polynomial_conv l)) r
wenzelm@23252
   598
              in transitive th1 (polynomial_pow_conv (concl th1))
wenzelm@23252
   599
              end
wenzelm@23252
   600
            else
wenzelm@23252
   601
              if opr aconvc add_tm orelse opr aconvc mul_tm orelse opr aconvc sub_tm
wenzelm@23252
   602
              then
wenzelm@23252
   603
               let val th1 = combination (Drule.arg_cong_rule opr (polynomial_conv l)) (polynomial_conv r)
wenzelm@23252
   604
                   val f = if opr aconvc add_tm then polynomial_add_conv
wenzelm@23252
   605
                      else if opr aconvc mul_tm then polynomial_mul_conv
wenzelm@23252
   606
                      else polynomial_sub_conv
wenzelm@23252
   607
               in transitive th1 (f (concl th1))
wenzelm@23252
   608
               end
wenzelm@23252
   609
              else reflexive tm
wenzelm@23252
   610
         end
wenzelm@23252
   611
  end;
wenzelm@23252
   612
 in
wenzelm@23252
   613
   {main = polynomial_conv,
wenzelm@23252
   614
    add = polynomial_add_conv,
wenzelm@23252
   615
    mul = polynomial_mul_conv,
wenzelm@23252
   616
    pow = polynomial_pow_conv,
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   617
    neg = polynomial_neg_conv,
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   618
    sub = polynomial_sub_conv}
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   619
 end
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   620
end;
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   621
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   622
val nat_arith = @{thms "nat_arith"};
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   623
val nat_exp_ss = HOL_basic_ss addsimps (nat_number @ nat_arith @ arith_simps @ rel_simps)
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   624
                              addsimps [Let_def, if_False, if_True, add_0, add_Suc];
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   625
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   626
fun semiring_normalize_wrapper ({vars, semiring, ring, idom}, 
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   627
                                     {conv, dest_const, mk_const, is_const}) =
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   628
  let
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   629
    fun ord t u = Term.term_ord (term_of t, term_of u) = LESS
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   630
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   631
    val pow_conv =
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   632
      arg_conv (Simplifier.rewrite nat_exp_ss)
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   633
      then_conv Simplifier.rewrite
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   634
        (HOL_basic_ss addsimps [nth (snd semiring) 31, nth (snd semiring) 34])
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   635
      then_conv conv
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   636
    val dat = (is_const, conv, conv, pow_conv)
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   637
    val {main, ...} = semiring_normalizers_conv vars semiring ring dat ord
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   638
  in main end;
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   639
wenzelm@23252
   640
fun semiring_normalize_conv ctxt tm =
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   641
  (case NormalizerData.match ctxt tm of
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   642
    NONE => reflexive tm
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   643
  | SOME res => semiring_normalize_wrapper res tm);
wenzelm@23252
   644
wenzelm@23252
   645
wenzelm@23252
   646
fun semiring_normalize_tac ctxt = SUBGOAL (fn (goal, i) =>
wenzelm@23252
   647
  rtac (semiring_normalize_conv ctxt
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   648
    (cterm_of (ProofContext.theory_of ctxt) (fst (Logic.dest_equals goal)))) i);
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   649
end;