src/HOL/Tools/groebner.ML
author haftmann
Sat Aug 28 16:14:32 2010 +0200 (2010-08-28)
changeset 38864 4abe644fcea5
parent 38795 848be46708dc
child 40718 4d7211968607
permissions -rw-r--r--
formerly unnamed infix equality now named HOL.eq
     1 (*  Title:      HOL/Tools/groebner.ML
     2     Author:     Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 signature GROEBNER =
     6 sig
     7   val ring_and_ideal_conv :
     8     {idom: thm list, ring: cterm list * thm list, field: cterm list * thm list,
     9      vars: cterm list, semiring: cterm list * thm list, ideal : thm list} ->
    10     (cterm -> Rat.rat) -> (Rat.rat -> cterm) ->
    11     conv ->  conv ->
    12      {ring_conv : conv, 
    13      simple_ideal: (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list),
    14      multi_ideal: cterm list -> cterm list -> cterm list -> (cterm * cterm) list,
    15      poly_eq_ss: simpset, unwind_conv : conv}
    16   val ring_tac: thm list -> thm list -> Proof.context -> int -> tactic
    17   val ideal_tac: thm list -> thm list -> Proof.context -> int -> tactic
    18   val algebra_tac: thm list -> thm list -> Proof.context -> int -> tactic
    19   val algebra_method: (Proof.context -> Method.method) context_parser
    20 end
    21 
    22 structure Groebner : GROEBNER =
    23 struct
    24 
    25 open Conv Drule Thm;
    26 
    27 fun is_comb ct =
    28   (case Thm.term_of ct of
    29     _ $ _ => true
    30   | _ => false);
    31 
    32 val concl = Thm.cprop_of #> Thm.dest_arg;
    33 
    34 fun is_binop ct ct' =
    35   (case Thm.term_of ct' of
    36     c $ _ $ _ => term_of ct aconv c
    37   | _ => false);
    38 
    39 fun dest_binary ct ct' =
    40   if is_binop ct ct' then Thm.dest_binop ct'
    41   else raise CTERM ("dest_binary: bad binop", [ct, ct'])
    42 
    43 fun inst_thm inst = Thm.instantiate ([], inst);
    44 
    45 val rat_0 = Rat.zero;
    46 val rat_1 = Rat.one;
    47 val minus_rat = Rat.neg;
    48 val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
    49 fun int_of_rat a =
    50     case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
    51 val lcm_rat = fn x => fn y => Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
    52 
    53 val (eqF_intr, eqF_elim) =
    54   let val [th1,th2] = @{thms PFalse}
    55   in (fn th => th COMP th2, fn th => th COMP th1) end;
    56 
    57 val (PFalse, PFalse') =
    58  let val PFalse_eq = nth @{thms simp_thms} 13
    59  in (PFalse_eq RS iffD1, PFalse_eq RS iffD2) end;
    60 
    61 
    62 (* Type for recording history, i.e. how a polynomial was obtained. *)
    63 
    64 datatype history =
    65    Start of int
    66  | Mmul of (Rat.rat * int list) * history
    67  | Add of history * history;
    68 
    69 
    70 (* Monomial ordering. *)
    71 
    72 fun morder_lt m1 m2=
    73     let fun lexorder l1 l2 =
    74             case (l1,l2) of
    75                 ([],[]) => false
    76               | (x1::o1,x2::o2) => x1 > x2 orelse x1 = x2 andalso lexorder o1 o2
    77               | _ => error "morder: inconsistent monomial lengths"
    78         val n1 = Integer.sum m1
    79         val n2 = Integer.sum m2 in
    80     n1 < n2 orelse n1 = n2 andalso lexorder m1 m2
    81     end;
    82 
    83 fun morder_le m1 m2 = morder_lt m1 m2 orelse (m1 = m2);
    84 
    85 fun morder_gt m1 m2 = morder_lt m2 m1;
    86 
    87 (* Arithmetic on canonical polynomials. *)
    88 
    89 fun grob_neg l = map (fn (c,m) => (minus_rat c,m)) l;
    90 
    91 fun grob_add l1 l2 =
    92   case (l1,l2) of
    93     ([],l2) => l2
    94   | (l1,[]) => l1
    95   | ((c1,m1)::o1,(c2,m2)::o2) =>
    96         if m1 = m2 then
    97           let val c = c1+/c2 val rest = grob_add o1 o2 in
    98           if c =/ rat_0 then rest else (c,m1)::rest end
    99         else if morder_lt m2 m1 then (c1,m1)::(grob_add o1 l2)
   100         else (c2,m2)::(grob_add l1 o2);
   101 
   102 fun grob_sub l1 l2 = grob_add l1 (grob_neg l2);
   103 
   104 fun grob_mmul (c1,m1) (c2,m2) = (c1*/c2, ListPair.map (op +) (m1, m2));
   105 
   106 fun grob_cmul cm pol = map (grob_mmul cm) pol;
   107 
   108 fun grob_mul l1 l2 =
   109   case l1 of
   110     [] => []
   111   | (h1::t1) => grob_add (grob_cmul h1 l2) (grob_mul t1 l2);
   112 
   113 fun grob_inv l =
   114   case l of
   115     [(c,vs)] => if (forall (fn x => x = 0) vs) then
   116                   if (c =/ rat_0) then error "grob_inv: division by zero"
   117                   else [(rat_1 // c,vs)]
   118               else error "grob_inv: non-constant divisor polynomial"
   119   | _ => error "grob_inv: non-constant divisor polynomial";
   120 
   121 fun grob_div l1 l2 =
   122   case l2 of
   123     [(c,l)] => if (forall (fn x => x = 0) l) then
   124                  if c =/ rat_0 then error "grob_div: division by zero"
   125                  else grob_cmul (rat_1 // c,l) l1
   126              else error "grob_div: non-constant divisor polynomial"
   127   | _ => error "grob_div: non-constant divisor polynomial";
   128 
   129 fun grob_pow vars l n =
   130   if n < 0 then error "grob_pow: negative power"
   131   else if n = 0 then [(rat_1,map (fn v => 0) vars)]
   132   else grob_mul l (grob_pow vars l (n - 1));
   133 
   134 fun degree vn p =
   135  case p of
   136   [] => error "Zero polynomial"
   137 | [(c,ns)] => nth ns vn
   138 | (c,ns)::p' => Int.max (nth ns vn, degree vn p');
   139 
   140 fun head_deg vn p = let val d = degree vn p in
   141  (d,fold (fn (c,r) => fn q => grob_add q [(c, map_index (fn (i,n) => if i = vn then 0 else n) r)]) (filter (fn (c,ns) => c <>/ rat_0 andalso nth ns vn = d) p) []) end;
   142 
   143 val is_zerop = forall (fn (c,ns) => c =/ rat_0 andalso forall (curry (op =) 0) ns);
   144 val grob_pdiv =
   145  let fun pdiv_aux vn (n,a) p k s =
   146   if is_zerop s then (k,s) else
   147   let val (m,b) = head_deg vn s
   148   in if m < n then (k,s) else
   149      let val p' = grob_mul p [(rat_1, map_index (fn (i,v) => if i = vn then m - n else 0)
   150                                                 (snd (hd s)))]
   151      in if a = b then pdiv_aux vn (n,a) p k (grob_sub s p')
   152         else pdiv_aux vn (n,a) p (k + 1) (grob_sub (grob_mul a s) (grob_mul b p'))
   153      end
   154   end
   155  in fn vn => fn s => fn p => pdiv_aux vn (head_deg vn p) p 0 s
   156  end;
   157 
   158 (* Monomial division operation. *)
   159 
   160 fun mdiv (c1,m1) (c2,m2) =
   161   (c1//c2,
   162    map2 (fn n1 => fn n2 => if n1 < n2 then error "mdiv" else n1 - n2) m1 m2);
   163 
   164 (* Lowest common multiple of two monomials. *)
   165 
   166 fun mlcm (c1,m1) (c2,m2) = (rat_1, ListPair.map Int.max (m1, m2));
   167 
   168 (* Reduce monomial cm by polynomial pol, returning replacement for cm.  *)
   169 
   170 fun reduce1 cm (pol,hpol) =
   171   case pol of
   172     [] => error "reduce1"
   173   | cm1::cms => ((let val (c,m) = mdiv cm cm1 in
   174                     (grob_cmul (minus_rat c,m) cms,
   175                      Mmul((minus_rat c,m),hpol)) end)
   176                 handle  ERROR _ => error "reduce1");
   177 
   178 (* Try this for all polynomials in a basis.  *)
   179 fun tryfind f l =
   180     case l of
   181         [] => error "tryfind"
   182       | (h::t) => ((f h) handle ERROR _ => tryfind f t);
   183 
   184 fun reduceb cm basis = tryfind (fn p => reduce1 cm p) basis;
   185 
   186 (* Reduction of a polynomial (always picking largest monomial possible).     *)
   187 
   188 fun reduce basis (pol,hist) =
   189   case pol of
   190     [] => (pol,hist)
   191   | cm::ptl => ((let val (q,hnew) = reduceb cm basis in
   192                    reduce basis (grob_add q ptl,Add(hnew,hist)) end)
   193                handle (ERROR _) =>
   194                    (let val (q,hist') = reduce basis (ptl,hist) in
   195                        (cm::q,hist') end));
   196 
   197 (* Check for orthogonality w.r.t. LCM.                                       *)
   198 
   199 fun orthogonal l p1 p2 =
   200   snd l = snd(grob_mmul (hd p1) (hd p2));
   201 
   202 (* Compute S-polynomial of two polynomials.                                  *)
   203 
   204 fun spoly cm ph1 ph2 =
   205   case (ph1,ph2) of
   206     (([],h),p) => ([],h)
   207   | (p,([],h)) => ([],h)
   208   | ((cm1::ptl1,his1),(cm2::ptl2,his2)) =>
   209         (grob_sub (grob_cmul (mdiv cm cm1) ptl1)
   210                   (grob_cmul (mdiv cm cm2) ptl2),
   211          Add(Mmul(mdiv cm cm1,his1),
   212              Mmul(mdiv (minus_rat(fst cm),snd cm) cm2,his2)));
   213 
   214 (* Make a polynomial monic.                                                  *)
   215 
   216 fun monic (pol,hist) =
   217   if null pol then (pol,hist) else
   218   let val (c',m') = hd pol in
   219   (map (fn (c,m) => (c//c',m)) pol,
   220    Mmul((rat_1 // c',map (K 0) m'),hist)) end;
   221 
   222 (* The most popular heuristic is to order critical pairs by LCM monomial.    *)
   223 
   224 fun forder ((c1,m1),_) ((c2,m2),_) = morder_lt m1 m2;
   225 
   226 fun poly_lt  p q =
   227   case (p,q) of
   228     (p,[]) => false
   229   | ([],q) => true
   230   | ((c1,m1)::o1,(c2,m2)::o2) =>
   231         c1 </ c2 orelse
   232         c1 =/ c2 andalso ((morder_lt m1 m2) orelse m1 = m2 andalso poly_lt o1 o2);
   233 
   234 fun align  ((p,hp),(q,hq)) =
   235   if poly_lt p q then ((p,hp),(q,hq)) else ((q,hq),(p,hp));
   236 fun forall2 p l1 l2 =
   237   case (l1,l2) of
   238     ([],[]) => true
   239   | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
   240   | _ => false;
   241 
   242 fun poly_eq p1 p2 =
   243   forall2 (fn (c1,m1) => fn (c2,m2) => c1 =/ c2 andalso (m1: int list) = m2) p1 p2;
   244 
   245 fun memx ((p1,h1),(p2,h2)) ppairs =
   246   not (exists (fn ((q1,_),(q2,_)) => poly_eq p1 q1 andalso poly_eq p2 q2) ppairs);
   247 
   248 (* Buchberger's second criterion.                                            *)
   249 
   250 fun criterion2 basis (lcm,((p1,h1),(p2,h2))) opairs =
   251   exists (fn g => not(poly_eq (fst g) p1) andalso not(poly_eq (fst g) p2) andalso
   252                    can (mdiv lcm) (hd(fst g)) andalso
   253                    not(memx (align (g,(p1,h1))) (map snd opairs)) andalso
   254                    not(memx (align (g,(p2,h2))) (map snd opairs))) basis;
   255 
   256 (* Test for hitting constant polynomial.                                     *)
   257 
   258 fun constant_poly p =
   259   length p = 1 andalso forall (fn x => x = 0) (snd(hd p));
   260 
   261 (* Grobner basis algorithm.                                                  *)
   262 
   263 (* FIXME: try to get rid of mergesort? *)
   264 fun merge ord l1 l2 =
   265  case l1 of
   266   [] => l2
   267  | h1::t1 =>
   268    case l2 of
   269     [] => l1
   270    | h2::t2 => if ord h1 h2 then h1::(merge ord t1 l2)
   271                else h2::(merge ord l1 t2);
   272 fun mergesort ord l =
   273  let
   274  fun mergepairs l1 l2 =
   275   case (l1,l2) of
   276    ([s],[]) => s
   277  | (l,[]) => mergepairs [] l
   278  | (l,[s1]) => mergepairs (s1::l) []
   279  | (l,(s1::s2::ss)) => mergepairs ((merge ord s1 s2)::l) ss
   280  in if null l  then []  else mergepairs [] (map (fn x => [x]) l)
   281  end;
   282 
   283 
   284 fun grobner_basis basis pairs =
   285  case pairs of
   286    [] => basis
   287  | (l,(p1,p2))::opairs =>
   288    let val (sph as (sp,hist)) = monic (reduce basis (spoly l p1 p2))
   289    in 
   290     if null sp orelse criterion2 basis (l,(p1,p2)) opairs
   291     then grobner_basis basis opairs
   292     else if constant_poly sp then grobner_basis (sph::basis) []
   293     else 
   294      let 
   295       val rawcps = map (fn p => (mlcm (hd(fst p)) (hd sp),align(p,sph)))
   296                               basis
   297       val newcps = filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q)))
   298                         rawcps
   299      in grobner_basis (sph::basis)
   300                  (merge forder opairs (mergesort forder newcps))
   301      end
   302    end;
   303 
   304 (* Interreduce initial polynomials.                                          *)
   305 
   306 fun grobner_interreduce rpols ipols =
   307   case ipols of
   308     [] => map monic (rev rpols)
   309   | p::ps => let val p' = reduce (rpols @ ps) p in
   310              if null (fst p') then grobner_interreduce rpols ps
   311              else grobner_interreduce (p'::rpols) ps end;
   312 
   313 (* Overall function.                                                         *)
   314 
   315 fun grobner pols =
   316     let val npols = map_index (fn (n, p) => (p, Start n)) pols
   317         val phists = filter (fn (p,_) => not (null p)) npols
   318         val bas = grobner_interreduce [] (map monic phists)
   319         val prs0 = map_product pair bas bas
   320         val prs1 = filter (fn ((x,_),(y,_)) => poly_lt x y) prs0
   321         val prs2 = map (fn (p,q) => (mlcm (hd(fst p)) (hd(fst q)),(p,q))) prs1
   322         val prs3 =
   323             filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q))) prs2 in
   324         grobner_basis bas (mergesort forder prs3) end;
   325 
   326 (* Get proof of contradiction from Grobner basis.                            *)
   327 
   328 fun find p l =
   329   case l of
   330       [] => error "find"
   331     | (h::t) => if p(h) then h else find p t;
   332 
   333 fun grobner_refute pols =
   334   let val gb = grobner pols in
   335   snd(find (fn (p,h) => length p = 1 andalso forall (fn x=> x=0) (snd(hd p))) gb)
   336   end;
   337 
   338 (* Turn proof into a certificate as sum of multipliers.                      *)
   339 (* In principle this is very inefficient: in a heavily shared proof it may   *)
   340 (* make the same calculation many times. Could put in a cache or something.  *)
   341 
   342 fun resolve_proof vars prf =
   343   case prf of
   344     Start(~1) => []
   345   | Start m => [(m,[(rat_1,map (K 0) vars)])]
   346   | Mmul(pol,lin) =>
   347         let val lis = resolve_proof vars lin in
   348             map (fn (n,p) => (n,grob_cmul pol p)) lis end
   349   | Add(lin1,lin2) =>
   350         let val lis1 = resolve_proof vars lin1
   351             val lis2 = resolve_proof vars lin2
   352             val dom = distinct (op =) (union (op =) (map fst lis1) (map fst lis2))
   353         in
   354             map (fn n => let val a = these (AList.lookup (op =) lis1 n)
   355                              val b = these (AList.lookup (op =) lis2 n)
   356                          in (n,grob_add a b) end) dom end;
   357 
   358 (* Run the procedure and produce Weak Nullstellensatz certificate.           *)
   359 
   360 fun grobner_weak vars pols =
   361     let val cert = resolve_proof vars (grobner_refute pols)
   362         val l =
   363             fold_rev (fold_rev (lcm_rat o denominator_rat o fst) o snd) cert (rat_1) in
   364         (l,map (fn (i,p) => (i,map (fn (d,m) => (l*/d,m)) p)) cert) end;
   365 
   366 (* Prove a polynomial is in ideal generated by others, using Grobner basis.  *)
   367 
   368 fun grobner_ideal vars pols pol =
   369   let val (pol',h) = reduce (grobner pols) (grob_neg pol,Start(~1)) in
   370   if not (null pol') then error "grobner_ideal: not in the ideal" else
   371   resolve_proof vars h end;
   372 
   373 (* Produce Strong Nullstellensatz certificate for a power of pol.            *)
   374 
   375 fun grobner_strong vars pols pol =
   376     let val vars' = @{cterm "True"}::vars
   377         val grob_z = [(rat_1,1::(map (fn x => 0) vars))]
   378         val grob_1 = [(rat_1,(map (fn x => 0) vars'))]
   379         fun augment p= map (fn (c,m) => (c,0::m)) p
   380         val pols' = map augment pols
   381         val pol' = augment pol
   382         val allpols = (grob_sub (grob_mul grob_z pol') grob_1)::pols'
   383         val (l,cert) = grobner_weak vars' allpols
   384         val d = fold (fold (Integer.max o hd o snd) o snd) cert 0
   385         fun transform_monomial (c,m) =
   386             grob_cmul (c,tl m) (grob_pow vars pol (d - hd m))
   387         fun transform_polynomial q = fold_rev (grob_add o transform_monomial) q []
   388         val cert' = map (fn (c,q) => (c-1,transform_polynomial q))
   389                         (filter (fn (k,_) => k <> 0) cert) in
   390         (d,l,cert') end;
   391 
   392 
   393 (* Overall parametrized universal procedure for (semi)rings.                 *)
   394 (* We return an ideal_conv and the actual ring prover.                       *)
   395 
   396 fun refute_disj rfn tm =
   397  case term_of tm of
   398   Const(@{const_name HOL.disj},_)$l$r =>
   399    compose_single(refute_disj rfn (dest_arg tm),2,compose_single(refute_disj rfn (dest_arg1 tm),2,disjE))
   400   | _ => rfn tm ;
   401 
   402 val notnotD = @{thm notnotD};
   403 fun mk_binop ct x y = capply (capply ct x) y
   404 
   405 val mk_comb = capply;
   406 fun is_neg t =
   407     case term_of t of
   408       (Const(@{const_name Not},_)$p) => true
   409     | _  => false;
   410 fun is_eq t =
   411  case term_of t of
   412  (Const(@{const_name HOL.eq},_)$_$_) => true
   413 | _  => false;
   414 
   415 fun end_itlist f l =
   416   case l of
   417         []     => error "end_itlist"
   418       | [x]    => x
   419       | (h::t) => f h (end_itlist f t);
   420 
   421 val list_mk_binop = fn b => end_itlist (mk_binop b);
   422 
   423 val list_dest_binop = fn b =>
   424  let fun h acc t =
   425   ((let val (l,r) = dest_binary b t in h (h acc r) l end)
   426    handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *)
   427  in h []
   428  end;
   429 
   430 val strip_exists =
   431  let fun h (acc, t) =
   432       case (term_of t) of
   433        Const(@{const_name Ex},_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
   434      | _ => (acc,t)
   435  in fn t => h ([],t)
   436  end;
   437 
   438 fun is_forall t =
   439  case term_of t of
   440   (Const(@{const_name All},_)$Abs(_,_,_)) => true
   441 | _ => false;
   442 
   443 val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
   444 val bool_simps = @{thms bool_simps};
   445 val nnf_simps = @{thms nnf_simps};
   446 val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
   447 val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms weak_dnf_simps});
   448 val initial_conv =
   449     Simplifier.rewrite
   450      (HOL_basic_ss addsimps nnf_simps
   451        addsimps [not_all, not_ex]
   452        addsimps map (fn th => th RS sym) (@{thms ex_simps} @ @{thms all_simps}));
   453 
   454 val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
   455 
   456 val cTrp = @{cterm "Trueprop"};
   457 val cConj = @{cterm HOL.conj};
   458 val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
   459 val assume_Trueprop = mk_comb cTrp #> assume;
   460 val list_mk_conj = list_mk_binop cConj;
   461 val conjs = list_dest_binop cConj;
   462 val mk_neg = mk_comb cNot;
   463 
   464 fun striplist dest = 
   465  let
   466   fun h acc x = case try dest x of
   467     SOME (a,b) => h (h acc b) a
   468   | NONE => x::acc
   469  in h [] end;
   470 fun list_mk_binop b = foldr1 (fn (s,t) => Thm.capply (Thm.capply b s) t);
   471 
   472 val eq_commute = mk_meta_eq @{thm eq_commute};
   473 
   474 fun sym_conv eq = 
   475  let val (l,r) = Thm.dest_binop eq
   476  in instantiate' [SOME (ctyp_of_term l)] [SOME l, SOME r] eq_commute
   477  end;
   478 
   479   (* FIXME : copied from cqe.ML -- complex QE*)
   480 fun conjuncts ct =
   481  case term_of ct of
   482   @{term HOL.conj}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
   483 | _ => [ct];
   484 
   485 fun fold1 f = foldr1 (uncurry f);
   486 
   487 val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm HOL.conj} c) c') ;
   488 
   489 fun mk_conj_tab th = 
   490  let fun h acc th = 
   491    case prop_of th of
   492    @{term "Trueprop"}$(@{term HOL.conj}$p$q) => 
   493      h (h acc (th RS conjunct2)) (th RS conjunct1)
   494   | @{term "Trueprop"}$p => (p,th)::acc
   495 in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
   496 
   497 fun is_conj (@{term HOL.conj}$_$_) = true
   498   | is_conj _ = false;
   499 
   500 fun prove_conj tab cjs = 
   501  case cjs of 
   502    [c] => if is_conj (term_of c) then prove_conj tab (conjuncts c) else tab c
   503  | c::cs => conjI OF [prove_conj tab [c], prove_conj tab cs];
   504 
   505 fun conj_ac_rule eq = 
   506  let 
   507   val (l,r) = Thm.dest_equals eq
   508   val ctabl = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} l))
   509   val ctabr = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} r))
   510   fun tabl c = the (Termtab.lookup ctabl (term_of c))
   511   fun tabr c = the (Termtab.lookup ctabr (term_of c))
   512   val thl  = prove_conj tabl (conjuncts r) |> implies_intr_hyps
   513   val thr  = prove_conj tabr (conjuncts l) |> implies_intr_hyps
   514   val eqI = instantiate' [] [SOME l, SOME r] @{thm iffI}
   515  in implies_elim (implies_elim eqI thl) thr |> mk_meta_eq end;
   516 
   517  (* END FIXME.*)
   518 
   519    (* Conversion for the equivalence of existential statements where 
   520       EX quantifiers are rearranged differently *)
   521  fun ext T = cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
   522  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
   523 
   524 fun choose v th th' = case concl_of th of 
   525   @{term Trueprop} $ (Const(@{const_name Ex},_)$_) => 
   526    let
   527     val p = (funpow 2 Thm.dest_arg o cprop_of) th
   528     val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
   529     val th0 = fconv_rule (Thm.beta_conversion true)
   530         (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
   531     val pv = (Thm.rhs_of o Thm.beta_conversion true) 
   532           (Thm.capply @{cterm Trueprop} (Thm.capply p v))
   533     val th1 = forall_intr v (implies_intr pv th')
   534    in implies_elim (implies_elim th0 th) th1  end
   535 | _ => error ""
   536 
   537 fun simple_choose v th = 
   538    choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
   539 
   540 
   541  fun mkexi v th = 
   542   let 
   543    val p = Thm.cabs v (Thm.dest_arg (Thm.cprop_of th))
   544   in implies_elim 
   545     (fconv_rule (Thm.beta_conversion true) (instantiate' [SOME (ctyp_of_term v)] [SOME p, SOME v] @{thm exI}))
   546       th
   547   end
   548  fun ex_eq_conv t = 
   549   let 
   550   val (p0,q0) = Thm.dest_binop t
   551   val (vs',P) = strip_exists p0 
   552   val (vs,_) = strip_exists q0 
   553    val th = assume (Thm.capply @{cterm Trueprop} P)
   554    val th1 =  implies_intr_hyps (fold simple_choose vs' (fold mkexi vs th))
   555    val th2 =  implies_intr_hyps (fold simple_choose vs (fold mkexi vs' th))
   556    val p = (Thm.dest_arg o Thm.dest_arg1 o cprop_of) th1
   557    val q = (Thm.dest_arg o Thm.dest_arg o cprop_of) th1
   558   in implies_elim (implies_elim (instantiate' [] [SOME p, SOME q] iffI) th1) th2
   559      |> mk_meta_eq
   560   end;
   561 
   562 
   563  fun getname v = case term_of v of 
   564   Free(s,_) => s
   565  | Var ((s,_),_) => s
   566  | _ => "x"
   567  fun mk_eq s t = Thm.capply (Thm.capply @{cterm "op == :: bool => _"} s) t
   568  fun mkeq s t = Thm.capply @{cterm Trueprop} (Thm.capply (Thm.capply @{cterm "op = :: bool => _"} s) t)
   569  fun mk_exists v th = arg_cong_rule (ext (ctyp_of_term v))
   570    (Thm.abstract_rule (getname v) v th)
   571  val simp_ex_conv = 
   572      Simplifier.rewrite (HOL_basic_ss addsimps @{thms simp_thms(39)})
   573 
   574 fun frees t = Thm.add_cterm_frees t [];
   575 fun free_in v t = member op aconvc (frees t) v;
   576 
   577 val vsubst = let
   578  fun vsubst (t,v) tm =  
   579    (Thm.rhs_of o Thm.beta_conversion false) (Thm.capply (Thm.cabs v tm) t)
   580 in fold vsubst end;
   581 
   582 
   583 (** main **)
   584 
   585 fun ring_and_ideal_conv
   586   {vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules), 
   587    field = (f_ops, f_rules), idom, ideal}
   588   dest_const mk_const ring_eq_conv ring_normalize_conv =
   589 let
   590   val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
   591   val [ring_add_tm, ring_mul_tm, ring_pow_tm] =
   592     map dest_fun2 [add_pat, mul_pat, pow_pat];
   593 
   594   val (ring_sub_tm, ring_neg_tm) =
   595     (case r_ops of
   596      [sub_pat, neg_pat] => (dest_fun2 sub_pat, dest_fun neg_pat)
   597     |_  => (@{cterm "True"}, @{cterm "True"}));
   598 
   599   val (field_div_tm, field_inv_tm) =
   600     (case f_ops of
   601        [div_pat, inv_pat] => (dest_fun2 div_pat, dest_fun inv_pat)
   602      | _ => (@{cterm "True"}, @{cterm "True"}));
   603 
   604   val [idom_thm, neq_thm] = idom;
   605   val [idl_sub, idl_add0] = 
   606      if length ideal = 2 then ideal else [eq_commute, eq_commute]
   607   fun ring_dest_neg t =
   608     let val (l,r) = dest_comb t 
   609     in if Term.could_unify(term_of l,term_of ring_neg_tm) then r 
   610        else raise CTERM ("ring_dest_neg", [t])
   611     end
   612 
   613  val ring_mk_neg = fn tm => mk_comb (ring_neg_tm) (tm);
   614  fun field_dest_inv t =
   615     let val (l,r) = dest_comb t in
   616         if Term.could_unify(term_of l, term_of field_inv_tm) then r 
   617         else raise CTERM ("field_dest_inv", [t])
   618     end
   619  val ring_dest_add = dest_binary ring_add_tm;
   620  val ring_mk_add = mk_binop ring_add_tm;
   621  val ring_dest_sub = dest_binary ring_sub_tm;
   622  val ring_mk_sub = mk_binop ring_sub_tm;
   623  val ring_dest_mul = dest_binary ring_mul_tm;
   624  val ring_mk_mul = mk_binop ring_mul_tm;
   625  val field_dest_div = dest_binary field_div_tm;
   626  val field_mk_div = mk_binop field_div_tm;
   627  val ring_dest_pow = dest_binary ring_pow_tm;
   628  val ring_mk_pow = mk_binop ring_pow_tm ;
   629  fun grobvars tm acc =
   630     if can dest_const tm then acc
   631     else if can ring_dest_neg tm then grobvars (dest_arg tm) acc
   632     else if can ring_dest_pow tm then grobvars (dest_arg1 tm) acc
   633     else if can ring_dest_add tm orelse can ring_dest_sub tm
   634             orelse can ring_dest_mul tm
   635     then grobvars (dest_arg1 tm) (grobvars (dest_arg tm) acc)
   636     else if can field_dest_inv tm
   637          then
   638           let val gvs = grobvars (dest_arg tm) [] 
   639           in if null gvs then acc else tm::acc
   640           end
   641     else if can field_dest_div tm then
   642          let val lvs = grobvars (dest_arg1 tm) acc
   643              val gvs = grobvars (dest_arg tm) []
   644           in if null gvs then lvs else tm::acc
   645           end 
   646     else tm::acc ;
   647 
   648 fun grobify_term vars tm =
   649 ((if not (member (op aconvc) vars tm) then raise CTERM ("Not a variable", [tm]) else
   650      [(rat_1,map (fn i => if i aconvc tm then 1 else 0) vars)])
   651 handle  CTERM _ =>
   652  ((let val x = dest_const tm
   653  in if x =/ rat_0 then [] else [(x,map (fn v => 0) vars)]
   654  end)
   655  handle ERROR _ =>
   656   ((grob_neg(grobify_term vars (ring_dest_neg tm)))
   657   handle CTERM _ =>
   658    (
   659    (grob_inv(grobify_term vars (field_dest_inv tm)))
   660    handle CTERM _ => 
   661     ((let val (l,r) = ring_dest_add tm
   662     in grob_add (grobify_term vars l) (grobify_term vars r)
   663     end)
   664     handle CTERM _ =>
   665      ((let val (l,r) = ring_dest_sub tm
   666      in grob_sub (grobify_term vars l) (grobify_term vars r)
   667      end)
   668      handle  CTERM _ =>
   669       ((let val (l,r) = ring_dest_mul tm
   670       in grob_mul (grobify_term vars l) (grobify_term vars r)
   671       end)
   672        handle CTERM _ =>
   673         (  (let val (l,r) = field_dest_div tm
   674           in grob_div (grobify_term vars l) (grobify_term vars r)
   675           end)
   676          handle CTERM _ =>
   677           ((let val (l,r) = ring_dest_pow tm
   678           in grob_pow vars (grobify_term vars l) ((term_of #> HOLogic.dest_number #> snd) r)
   679           end)
   680            handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
   681 val eq_tm = idom_thm |> concl |> dest_arg |> dest_arg |> dest_fun2;
   682 val dest_eq = dest_binary eq_tm;
   683 
   684 fun grobify_equation vars tm =
   685     let val (l,r) = dest_binary eq_tm tm
   686     in grob_sub (grobify_term vars l) (grobify_term vars r)
   687     end;
   688 
   689 fun grobify_equations tm =
   690  let
   691   val cjs = conjs tm
   692   val  rawvars = fold_rev (fn eq => fn a =>
   693                                        grobvars (dest_arg1 eq) (grobvars (dest_arg eq) a)) cjs []
   694   val vars = sort (fn (x, y) => Term_Ord.term_ord(term_of x,term_of y))
   695                   (distinct (op aconvc) rawvars)
   696  in (vars,map (grobify_equation vars) cjs)
   697  end;
   698 
   699 val holify_polynomial =
   700  let fun holify_varpow (v,n) =
   701   if n = 1 then v else ring_mk_pow v (Numeral.mk_cnumber @{ctyp nat} n)  (* FIXME *)
   702  fun holify_monomial vars (c,m) =
   703   let val xps = map holify_varpow (filter (fn (_,n) => n <> 0) (vars ~~ m))
   704    in end_itlist ring_mk_mul (mk_const c :: xps)
   705   end
   706  fun holify_polynomial vars p =
   707      if null p then mk_const (rat_0)
   708      else end_itlist ring_mk_add (map (holify_monomial vars) p)
   709  in holify_polynomial
   710  end ;
   711 val idom_rule = simplify (HOL_basic_ss addsimps [idom_thm]);
   712 fun prove_nz n = eqF_elim
   713                  (ring_eq_conv(mk_binop eq_tm (mk_const n) (mk_const(rat_0))));
   714 val neq_01 = prove_nz (rat_1);
   715 fun neq_rule n th = [prove_nz n, th] MRS neq_thm;
   716 fun mk_add th1 = combination(arg_cong_rule ring_add_tm th1);
   717 
   718 fun refute tm =
   719  if tm aconvc false_tm then assume_Trueprop tm else
   720  ((let
   721    val (nths0,eths0) = List.partition (is_neg o concl) (HOLogic.conj_elims (assume_Trueprop tm))
   722    val  nths = filter (is_eq o dest_arg o concl) nths0
   723    val eths = filter (is_eq o concl) eths0
   724   in
   725    if null eths then
   726     let
   727       val th1 = end_itlist (fn th1 => fn th2 => idom_rule(HOLogic.conj_intr th1 th2)) nths
   728       val th2 = Conv.fconv_rule
   729                 ((arg_conv #> arg_conv)
   730                      (binop_conv ring_normalize_conv)) th1
   731       val conc = th2 |> concl |> dest_arg
   732       val (l,r) = conc |> dest_eq
   733     in implies_intr (mk_comb cTrp tm)
   734                     (equal_elim (arg_cong_rule cTrp (eqF_intr th2))
   735                            (reflexive l |> mk_object_eq))
   736     end
   737    else
   738    let
   739     val (vars,l,cert,noteqth) =(
   740      if null nths then
   741       let val (vars,pols) = grobify_equations(list_mk_conj(map concl eths))
   742           val (l,cert) = grobner_weak vars pols
   743       in (vars,l,cert,neq_01)
   744       end
   745      else
   746       let
   747        val nth = end_itlist (fn th1 => fn th2 => idom_rule(HOLogic.conj_intr th1 th2)) nths
   748        val (vars,pol::pols) =
   749           grobify_equations(list_mk_conj(dest_arg(concl nth)::map concl eths))
   750        val (deg,l,cert) = grobner_strong vars pols pol
   751        val th1 = Conv.fconv_rule((arg_conv o arg_conv)(binop_conv ring_normalize_conv)) nth
   752        val th2 = funpow deg (idom_rule o HOLogic.conj_intr th1) neq_01
   753       in (vars,l,cert,th2)
   754       end)
   755     val cert_pos = map (fn (i,p) => (i,filter (fn (c,m) => c >/ rat_0) p)) cert
   756     val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (minus_rat c,m))
   757                                             (filter (fn (c,m) => c </ rat_0) p))) cert
   758     val  herts_pos = map (fn (i,p) => (i,holify_polynomial vars p)) cert_pos
   759     val  herts_neg = map (fn (i,p) => (i,holify_polynomial vars p)) cert_neg
   760     fun thm_fn pols =
   761         if null pols then reflexive(mk_const rat_0) else
   762         end_itlist mk_add
   763             (map (fn (i,p) => arg_cong_rule (mk_comb ring_mul_tm p)
   764               (nth eths i |> mk_meta_eq)) pols)
   765     val th1 = thm_fn herts_pos
   766     val th2 = thm_fn herts_neg
   767     val th3 = HOLogic.conj_intr(mk_add (symmetric th1) th2 |> mk_object_eq) noteqth
   768     val th4 = Conv.fconv_rule ((arg_conv o arg_conv o binop_conv) ring_normalize_conv)
   769                                (neq_rule l th3)
   770     val (l,r) = dest_eq(dest_arg(concl th4))
   771    in implies_intr (mk_comb cTrp tm)
   772                         (equal_elim (arg_cong_rule cTrp (eqF_intr th4))
   773                    (reflexive l |> mk_object_eq))
   774    end
   775   end) handle ERROR _ => raise CTERM ("Gorbner-refute: unable to refute",[tm]))
   776 
   777 fun ring tm =
   778  let
   779   fun mk_forall x p =
   780       mk_comb (cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (cabs x p)
   781   val avs = add_cterm_frees tm []
   782   val P' = fold mk_forall avs tm
   783   val th1 = initial_conv(mk_neg P')
   784   val (evs,bod) = strip_exists(concl th1) in
   785    if is_forall bod then raise CTERM("ring: non-universal formula",[tm])
   786    else
   787    let
   788     val th1a = weak_dnf_conv bod
   789     val boda = concl th1a
   790     val th2a = refute_disj refute boda
   791     val th2b = [mk_object_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans
   792     val th2 = fold (fn v => fn th => (forall_intr v th) COMP allI) evs (th2b RS PFalse)
   793     val th3 = equal_elim
   794                 (Simplifier.rewrite (HOL_basic_ss addsimps [not_ex RS sym])
   795                           (th2 |> cprop_of)) th2
   796     in specl avs
   797              ([[[mk_object_eq th1, th3 RS PFalse'] MRS trans] MRS PFalse] MRS notnotD)
   798    end
   799  end
   800 fun ideal tms tm ord =
   801  let
   802   val rawvars = fold_rev grobvars (tm::tms) []
   803   val vars = sort ord (distinct (fn (x,y) => (term_of x) aconv (term_of y)) rawvars)
   804   val pols = map (grobify_term vars) tms
   805   val pol = grobify_term vars tm
   806   val cert = grobner_ideal vars pols pol
   807  in map_range (fn n => these (AList.lookup (op =) cert n) |> holify_polynomial vars)
   808    (length pols)
   809  end
   810 
   811 fun poly_eq_conv t = 
   812  let val (a,b) = Thm.dest_binop t
   813  in fconv_rule (arg_conv (arg1_conv ring_normalize_conv)) 
   814      (instantiate' [] [SOME a, SOME b] idl_sub)
   815  end
   816  val poly_eq_simproc = 
   817   let 
   818    fun proc phi  ss t = 
   819     let val th = poly_eq_conv t
   820     in if Thm.is_reflexive th then NONE else SOME th
   821     end
   822    in make_simproc {lhss = [Thm.lhs_of idl_sub], 
   823                 name = "poly_eq_simproc", proc = proc, identifier = []}
   824    end;
   825   val poly_eq_ss = HOL_basic_ss addsimps @{thms simp_thms}
   826                         addsimprocs [poly_eq_simproc]
   827 
   828  local
   829   fun is_defined v t =
   830   let 
   831    val mons = striplist(dest_binary ring_add_tm) t 
   832   in member (op aconvc) mons v andalso 
   833     forall (fn m => v aconvc m 
   834           orelse not(member (op aconvc) (Thm.add_cterm_frees m []) v)) mons
   835   end
   836 
   837   fun isolate_variable vars tm =
   838   let 
   839    val th = poly_eq_conv tm
   840    val th' = (sym_conv then_conv poly_eq_conv) tm
   841    val (v,th1) = 
   842    case find_first(fn v=> is_defined v (Thm.dest_arg1 (Thm.rhs_of th))) vars of
   843     SOME v => (v,th')
   844    | NONE => (the (find_first 
   845           (fn v => is_defined v (Thm.dest_arg1 (Thm.rhs_of th'))) vars) ,th)
   846    val th2 = transitive th1 
   847         (instantiate' []  [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v] 
   848           idl_add0)
   849    in fconv_rule(funpow 2 arg_conv ring_normalize_conv) th2
   850    end
   851  in
   852  fun unwind_polys_conv tm =
   853  let 
   854   val (vars,bod) = strip_exists tm
   855   val cjs = striplist (dest_binary @{cterm HOL.conj}) bod
   856   val th1 = (the (get_first (try (isolate_variable vars)) cjs) 
   857              handle Option => raise CTERM ("unwind_polys_conv",[tm]))
   858   val eq = Thm.lhs_of th1
   859   val bod' = list_mk_binop @{cterm HOL.conj} (eq::(remove op aconvc eq cjs))
   860   val th2 = conj_ac_rule (mk_eq bod bod')
   861   val th3 = transitive th2 
   862          (Drule.binop_cong_rule @{cterm HOL.conj} th1 
   863                 (reflexive (Thm.dest_arg (Thm.rhs_of th2))))
   864   val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3))
   865   val vars' = (remove op aconvc v vars) @ [v]
   866   val th4 = fconv_rule (arg_conv simp_ex_conv) (mk_exists v th3)
   867   val th5 = ex_eq_conv (mk_eq tm (fold mk_ex (remove op aconvc v vars) (Thm.lhs_of th4)))
   868  in transitive th5 (fold mk_exists (remove op aconvc v vars) th4)
   869  end;
   870 end
   871 
   872 local
   873  fun scrub_var v m =
   874   let 
   875    val ps = striplist ring_dest_mul m 
   876    val ps' = remove op aconvc v ps
   877   in if null ps' then one_tm else fold1 ring_mk_mul ps'
   878   end
   879  fun find_multipliers v mons =
   880   let 
   881    val mons1 = filter (fn m => free_in v m) mons 
   882    val mons2 = map (scrub_var v) mons1 
   883    in  if null mons2 then zero_tm else fold1 ring_mk_add mons2
   884   end
   885 
   886  fun isolate_monomials vars tm =
   887  let 
   888   val (cmons,vmons) =
   889     List.partition (fn m => null (inter (op aconvc) vars (frees m)))
   890                    (striplist ring_dest_add tm)
   891   val cofactors = map (fn v => find_multipliers v vmons) vars
   892   val cnc = if null cmons then zero_tm
   893              else Thm.capply ring_neg_tm
   894                     (list_mk_binop ring_add_tm cmons) 
   895   in (cofactors,cnc)
   896   end;
   897 
   898 fun isolate_variables evs ps eq =
   899  let 
   900   val vars = filter (fn v => free_in v eq) evs
   901   val (qs,p) = isolate_monomials vars eq
   902   val rs = ideal (qs @ ps) p 
   903               (fn (s,t) => Term_Ord.term_ord (term_of s, term_of t))
   904  in (eq, take (length qs) rs ~~ vars)
   905  end;
   906  fun subst_in_poly i p = Thm.rhs_of (ring_normalize_conv (vsubst i p));
   907 in
   908  fun solve_idealism evs ps eqs =
   909   if null evs then [] else
   910   let 
   911    val (eq,cfs) = get_first (try (isolate_variables evs ps)) eqs |> the
   912    val evs' = subtract op aconvc evs (map snd cfs)
   913    val eqs' = map (subst_in_poly cfs) (remove op aconvc eq eqs)
   914   in cfs @ solve_idealism evs' ps eqs'
   915   end;
   916 end;
   917 
   918 
   919 in {ring_conv = ring, simple_ideal = ideal, multi_ideal = solve_idealism, 
   920     poly_eq_ss = poly_eq_ss, unwind_conv = unwind_polys_conv}
   921 end;
   922 
   923 
   924 fun find_term bounds tm =
   925   (case term_of tm of
   926     Const (@{const_name HOL.eq}, T) $ _ $ _ =>
   927       if domain_type T = HOLogic.boolT then find_args bounds tm
   928       else dest_arg tm
   929   | Const (@{const_name Not}, _) $ _ => find_term bounds (dest_arg tm)
   930   | Const (@{const_name All}, _) $ _ => find_body bounds (dest_arg tm)
   931   | Const (@{const_name Ex}, _) $ _ => find_body bounds (dest_arg tm)
   932   | Const (@{const_name HOL.conj}, _) $ _ $ _ => find_args bounds tm
   933   | Const (@{const_name HOL.disj}, _) $ _ $ _ => find_args bounds tm
   934   | Const (@{const_name HOL.implies}, _) $ _ $ _ => find_args bounds tm
   935   | @{term "op ==>"} $_$_ => find_args bounds tm
   936   | Const("op ==",_)$_$_ => find_args bounds tm
   937   | @{term Trueprop}$_ => find_term bounds (dest_arg tm)
   938   | _ => raise TERM ("find_term", []))
   939 and find_args bounds tm =
   940   let val (t, u) = Thm.dest_binop tm
   941   in (find_term bounds t handle TERM _ => find_term bounds u) end
   942 and find_body bounds b =
   943   let val (_, b') = dest_abs (SOME (Name.bound bounds)) b
   944   in find_term (bounds + 1) b' end;
   945 
   946 
   947 fun get_ring_ideal_convs ctxt form = 
   948  case try (find_term 0) form of
   949   NONE => NONE
   950 | SOME tm =>
   951   (case Semiring_Normalizer.match ctxt tm of
   952     NONE => NONE
   953   | SOME (res as (theory, {is_const, dest_const, 
   954           mk_const, conv = ring_eq_conv})) =>
   955      SOME (ring_and_ideal_conv theory
   956           dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
   957           (Semiring_Normalizer.semiring_normalize_wrapper ctxt res)))
   958 
   959 fun ring_solve ctxt form =
   960   (case try (find_term 0 (* FIXME !? *)) form of
   961     NONE => reflexive form
   962   | SOME tm =>
   963       (case Semiring_Normalizer.match ctxt tm of
   964         NONE => reflexive form
   965       | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
   966         #ring_conv (ring_and_ideal_conv theory
   967           dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
   968           (Semiring_Normalizer.semiring_normalize_wrapper ctxt res)) form));
   969 
   970 fun presimplify ctxt add_thms del_thms = asm_full_simp_tac (Simplifier.context ctxt
   971   (HOL_basic_ss addsimps (Algebra_Simplification.get ctxt) delsimps del_thms addsimps add_thms));
   972 
   973 fun ring_tac add_ths del_ths ctxt =
   974   Object_Logic.full_atomize_tac
   975   THEN' presimplify ctxt add_ths del_ths
   976   THEN' CSUBGOAL (fn (p, i) =>
   977     rtac (let val form = Object_Logic.dest_judgment p
   978           in case get_ring_ideal_convs ctxt form of
   979            NONE => reflexive form
   980           | SOME thy => #ring_conv thy form
   981           end) i
   982       handle TERM _ => no_tac
   983         | CTERM _ => no_tac
   984         | THM _ => no_tac);
   985 
   986 local
   987  fun lhs t = case term_of t of
   988   Const(@{const_name HOL.eq},_)$_$_ => Thm.dest_arg1 t
   989  | _=> raise CTERM ("ideal_tac - lhs",[t])
   990  fun exitac NONE = no_tac
   991    | exitac (SOME y) = rtac (instantiate' [SOME (ctyp_of_term y)] [NONE,SOME y] exI) 1
   992 in 
   993 fun ideal_tac add_ths del_ths ctxt = 
   994   presimplify ctxt add_ths del_ths
   995  THEN'
   996  CSUBGOAL (fn (p, i) =>
   997   case get_ring_ideal_convs ctxt p of
   998    NONE => no_tac
   999  | SOME thy => 
  1000   let
  1001    fun poly_exists_tac {asms = asms, concl = concl, prems = prems,
  1002             params = params, context = ctxt, schematics = scs} = 
  1003     let
  1004      val (evs,bod) = strip_exists (Thm.dest_arg concl)
  1005      val ps = map_filter (try (lhs o Thm.dest_arg)) asms 
  1006      val cfs = (map swap o #multi_ideal thy evs ps) 
  1007                    (map Thm.dest_arg1 (conjuncts bod))
  1008      val ws = map (exitac o AList.lookup op aconvc cfs) evs
  1009     in EVERY (rev ws) THEN Method.insert_tac prems 1 
  1010         THEN ring_tac add_ths del_ths ctxt 1
  1011    end
  1012   in  
  1013      clarify_tac @{claset} i 
  1014      THEN Object_Logic.full_atomize_tac i 
  1015      THEN asm_full_simp_tac (Simplifier.context ctxt (#poly_eq_ss thy)) i 
  1016      THEN clarify_tac @{claset} i 
  1017      THEN (REPEAT (CONVERSION (#unwind_conv thy) i))
  1018      THEN SUBPROOF poly_exists_tac ctxt i
  1019   end
  1020  handle TERM _ => no_tac
  1021      | CTERM _ => no_tac
  1022      | THM _ => no_tac); 
  1023 end;
  1024 
  1025 fun algebra_tac add_ths del_ths ctxt i = 
  1026  ring_tac add_ths del_ths ctxt i ORELSE ideal_tac add_ths del_ths ctxt i
  1027  
  1028 local
  1029 
  1030 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  1031 val addN = "add"
  1032 val delN = "del"
  1033 val any_keyword = keyword addN || keyword delN
  1034 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
  1035 
  1036 in
  1037 
  1038 val algebra_method = ((Scan.optional (keyword addN |-- thms) []) -- 
  1039    (Scan.optional (keyword delN |-- thms) [])) >>
  1040   (fn (add_ths, del_ths) => fn ctxt =>
  1041        SIMPLE_METHOD' (algebra_tac add_ths del_ths ctxt))
  1042 
  1043 end;
  1044 
  1045 end;