# HG changeset patch
# User haftmann
# Date 1273241545 -7200
# Node ID cf558aeb35b0a97287b2b57b3d2f90d86e035633
# Parent  7f1da69cacb3a1bfab8112fd2d7b0d2c8bfb3495
delete Groebner_Basis directory -- only one file left

diff -r 7f1da69cacb3 -r cf558aeb35b0 src/HOL/Groebner_Basis.thy
--- a/src/HOL/Groebner_Basis.thy	Fri May 07 15:05:52 2010 +0200
+++ b/src/HOL/Groebner_Basis.thy	Fri May 07 16:12:25 2010 +0200
@@ -7,7 +7,7 @@
 theory Groebner_Basis
 imports Semiring_Normalization
 uses
-  ("Tools/Groebner_Basis/groebner.ML")
+  ("Tools/groebner.ML")
 begin
 
 subsection {* Groebner Bases *}
@@ -40,7 +40,7 @@
 
 setup Algebra_Simplification.setup
 
-use "Tools/Groebner_Basis/groebner.ML"
+use "Tools/groebner.ML"
 
 method_setup algebra = Groebner.algebra_method
   "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
diff -r 7f1da69cacb3 -r cf558aeb35b0 src/HOL/Tools/Groebner_Basis/groebner.ML
--- a/src/HOL/Tools/Groebner_Basis/groebner.ML	Fri May 07 15:05:52 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1045 +0,0 @@
-(*  Title:      HOL/Tools/Groebner_Basis/groebner.ML
-    Author:     Amine Chaieb, TU Muenchen
-*)
-
-signature GROEBNER =
-sig
-  val ring_and_ideal_conv :
-    {idom: thm list, ring: cterm list * thm list, field: cterm list * thm list,
-     vars: cterm list, semiring: cterm list * thm list, ideal : thm list} ->
-    (cterm -> Rat.rat) -> (Rat.rat -> cterm) ->
-    conv ->  conv ->
-     {ring_conv : conv, 
-     simple_ideal: (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list),
-     multi_ideal: cterm list -> cterm list -> cterm list -> (cterm * cterm) list,
-     poly_eq_ss: simpset, unwind_conv : conv}
-  val ring_tac: thm list -> thm list -> Proof.context -> int -> tactic
-  val ideal_tac: thm list -> thm list -> Proof.context -> int -> tactic
-  val algebra_tac: thm list -> thm list -> Proof.context -> int -> tactic
-  val algebra_method: (Proof.context -> Method.method) context_parser
-end
-
-structure Groebner : GROEBNER =
-struct
-
-open Conv Drule Thm;
-
-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_binary ct ct' =
-  if is_binop ct ct' then Thm.dest_binop ct'
-  else raise CTERM ("dest_binary: bad binop", [ct, ct'])
-
-fun inst_thm inst = Thm.instantiate ([], inst);
-
-val rat_0 = Rat.zero;
-val rat_1 = Rat.one;
-val minus_rat = Rat.neg;
-val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
-fun int_of_rat a =
-    case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
-val lcm_rat = fn x => fn y => Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
-
-val (eqF_intr, eqF_elim) =
-  let val [th1,th2] = @{thms PFalse}
-  in (fn th => th COMP th2, fn th => th COMP th1) end;
-
-val (PFalse, PFalse') =
- let val PFalse_eq = nth @{thms simp_thms} 13
- in (PFalse_eq RS iffD1, PFalse_eq RS iffD2) end;
-
-
-(* Type for recording history, i.e. how a polynomial was obtained. *)
-
-datatype history =
-   Start of int
- | Mmul of (Rat.rat * int list) * history
- | Add of history * history;
-
-
-(* Monomial ordering. *)
-
-fun morder_lt m1 m2=
-    let fun lexorder l1 l2 =
-            case (l1,l2) of
-                ([],[]) => false
-              | (x1::o1,x2::o2) => x1 > x2 orelse x1 = x2 andalso lexorder o1 o2
-              | _ => error "morder: inconsistent monomial lengths"
-        val n1 = Integer.sum m1
-        val n2 = Integer.sum m2 in
-    n1 < n2 orelse n1 = n2 andalso lexorder m1 m2
-    end;
-
-fun morder_le m1 m2 = morder_lt m1 m2 orelse (m1 = m2);
-
-fun morder_gt m1 m2 = morder_lt m2 m1;
-
-(* Arithmetic on canonical polynomials. *)
-
-fun grob_neg l = map (fn (c,m) => (minus_rat c,m)) l;
-
-fun grob_add l1 l2 =
-  case (l1,l2) of
-    ([],l2) => l2
-  | (l1,[]) => l1
-  | ((c1,m1)::o1,(c2,m2)::o2) =>
-        if m1 = m2 then
-          let val c = c1+/c2 val rest = grob_add o1 o2 in
-          if c =/ rat_0 then rest else (c,m1)::rest end
-        else if morder_lt m2 m1 then (c1,m1)::(grob_add o1 l2)
-        else (c2,m2)::(grob_add l1 o2);
-
-fun grob_sub l1 l2 = grob_add l1 (grob_neg l2);
-
-fun grob_mmul (c1,m1) (c2,m2) = (c1*/c2, ListPair.map (op +) (m1, m2));
-
-fun grob_cmul cm pol = map (grob_mmul cm) pol;
-
-fun grob_mul l1 l2 =
-  case l1 of
-    [] => []
-  | (h1::t1) => grob_add (grob_cmul h1 l2) (grob_mul t1 l2);
-
-fun grob_inv l =
-  case l of
-    [(c,vs)] => if (forall (fn x => x = 0) vs) then
-                  if (c =/ rat_0) then error "grob_inv: division by zero"
-                  else [(rat_1 // c,vs)]
-              else error "grob_inv: non-constant divisor polynomial"
-  | _ => error "grob_inv: non-constant divisor polynomial";
-
-fun grob_div l1 l2 =
-  case l2 of
-    [(c,l)] => if (forall (fn x => x = 0) l) then
-                 if c =/ rat_0 then error "grob_div: division by zero"
-                 else grob_cmul (rat_1 // c,l) l1
-             else error "grob_div: non-constant divisor polynomial"
-  | _ => error "grob_div: non-constant divisor polynomial";
-
-fun grob_pow vars l n =
-  if n < 0 then error "grob_pow: negative power"
-  else if n = 0 then [(rat_1,map (fn v => 0) vars)]
-  else grob_mul l (grob_pow vars l (n - 1));
-
-fun degree vn p =
- case p of
-  [] => error "Zero polynomial"
-| [(c,ns)] => nth ns vn
-| (c,ns)::p' => Int.max (nth ns vn, degree vn p');
-
-fun head_deg vn p = let val d = degree vn p in
- (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;
-
-val is_zerop = forall (fn (c,ns) => c =/ rat_0 andalso forall (curry (op =) 0) ns);
-val grob_pdiv =
- let fun pdiv_aux vn (n,a) p k s =
-  if is_zerop s then (k,s) else
-  let val (m,b) = head_deg vn s
-  in if m < n then (k,s) else
-     let val p' = grob_mul p [(rat_1, map_index (fn (i,v) => if i = vn then m - n else 0)
-                                                (snd (hd s)))]
-     in if a = b then pdiv_aux vn (n,a) p k (grob_sub s p')
-        else pdiv_aux vn (n,a) p (k + 1) (grob_sub (grob_mul a s) (grob_mul b p'))
-     end
-  end
- in fn vn => fn s => fn p => pdiv_aux vn (head_deg vn p) p 0 s
- end;
-
-(* Monomial division operation. *)
-
-fun mdiv (c1,m1) (c2,m2) =
-  (c1//c2,
-   map2 (fn n1 => fn n2 => if n1 < n2 then error "mdiv" else n1 - n2) m1 m2);
-
-(* Lowest common multiple of two monomials. *)
-
-fun mlcm (c1,m1) (c2,m2) = (rat_1, ListPair.map Int.max (m1, m2));
-
-(* Reduce monomial cm by polynomial pol, returning replacement for cm.  *)
-
-fun reduce1 cm (pol,hpol) =
-  case pol of
-    [] => error "reduce1"
-  | cm1::cms => ((let val (c,m) = mdiv cm cm1 in
-                    (grob_cmul (minus_rat c,m) cms,
-                     Mmul((minus_rat c,m),hpol)) end)
-                handle  ERROR _ => error "reduce1");
-
-(* Try this for all polynomials in a basis.  *)
-fun tryfind f l =
-    case l of
-        [] => error "tryfind"
-      | (h::t) => ((f h) handle ERROR _ => tryfind f t);
-
-fun reduceb cm basis = tryfind (fn p => reduce1 cm p) basis;
-
-(* Reduction of a polynomial (always picking largest monomial possible).     *)
-
-fun reduce basis (pol,hist) =
-  case pol of
-    [] => (pol,hist)
-  | cm::ptl => ((let val (q,hnew) = reduceb cm basis in
-                   reduce basis (grob_add q ptl,Add(hnew,hist)) end)
-               handle (ERROR _) =>
-                   (let val (q,hist') = reduce basis (ptl,hist) in
-                       (cm::q,hist') end));
-
-(* Check for orthogonality w.r.t. LCM.                                       *)
-
-fun orthogonal l p1 p2 =
-  snd l = snd(grob_mmul (hd p1) (hd p2));
-
-(* Compute S-polynomial of two polynomials.                                  *)
-
-fun spoly cm ph1 ph2 =
-  case (ph1,ph2) of
-    (([],h),p) => ([],h)
-  | (p,([],h)) => ([],h)
-  | ((cm1::ptl1,his1),(cm2::ptl2,his2)) =>
-        (grob_sub (grob_cmul (mdiv cm cm1) ptl1)
-                  (grob_cmul (mdiv cm cm2) ptl2),
-         Add(Mmul(mdiv cm cm1,his1),
-             Mmul(mdiv (minus_rat(fst cm),snd cm) cm2,his2)));
-
-(* Make a polynomial monic.                                                  *)
-
-fun monic (pol,hist) =
-  if null pol then (pol,hist) else
-  let val (c',m') = hd pol in
-  (map (fn (c,m) => (c//c',m)) pol,
-   Mmul((rat_1 // c',map (K 0) m'),hist)) end;
-
-(* The most popular heuristic is to order critical pairs by LCM monomial.    *)
-
-fun forder ((c1,m1),_) ((c2,m2),_) = morder_lt m1 m2;
-
-fun poly_lt  p q =
-  case (p,q) of
-    (p,[]) => false
-  | ([],q) => true
-  | ((c1,m1)::o1,(c2,m2)::o2) =>
-        c1 </ c2 orelse
-        c1 =/ c2 andalso ((morder_lt m1 m2) orelse m1 = m2 andalso poly_lt o1 o2);
-
-fun align  ((p,hp),(q,hq)) =
-  if poly_lt p q then ((p,hp),(q,hq)) else ((q,hq),(p,hp));
-fun forall2 p l1 l2 =
-  case (l1,l2) of
-    ([],[]) => true
-  | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
-  | _ => false;
-
-fun poly_eq p1 p2 =
-  forall2 (fn (c1,m1) => fn (c2,m2) => c1 =/ c2 andalso (m1: int list) = m2) p1 p2;
-
-fun memx ((p1,h1),(p2,h2)) ppairs =
-  not (exists (fn ((q1,_),(q2,_)) => poly_eq p1 q1 andalso poly_eq p2 q2) ppairs);
-
-(* Buchberger's second criterion.                                            *)
-
-fun criterion2 basis (lcm,((p1,h1),(p2,h2))) opairs =
-  exists (fn g => not(poly_eq (fst g) p1) andalso not(poly_eq (fst g) p2) andalso
-                   can (mdiv lcm) (hd(fst g)) andalso
-                   not(memx (align (g,(p1,h1))) (map snd opairs)) andalso
-                   not(memx (align (g,(p2,h2))) (map snd opairs))) basis;
-
-(* Test for hitting constant polynomial.                                     *)
-
-fun constant_poly p =
-  length p = 1 andalso forall (fn x => x = 0) (snd(hd p));
-
-(* Grobner basis algorithm.                                                  *)
-
-(* FIXME: try to get rid of mergesort? *)
-fun merge ord l1 l2 =
- case l1 of
-  [] => l2
- | h1::t1 =>
-   case l2 of
-    [] => l1
-   | h2::t2 => if ord h1 h2 then h1::(merge ord t1 l2)
-               else h2::(merge ord l1 t2);
-fun mergesort ord l =
- let
- fun mergepairs l1 l2 =
-  case (l1,l2) of
-   ([s],[]) => s
- | (l,[]) => mergepairs [] l
- | (l,[s1]) => mergepairs (s1::l) []
- | (l,(s1::s2::ss)) => mergepairs ((merge ord s1 s2)::l) ss
- in if null l  then []  else mergepairs [] (map (fn x => [x]) l)
- end;
-
-
-fun grobner_basis basis pairs =
- case pairs of
-   [] => basis
- | (l,(p1,p2))::opairs =>
-   let val (sph as (sp,hist)) = monic (reduce basis (spoly l p1 p2))
-   in 
-    if null sp orelse criterion2 basis (l,(p1,p2)) opairs
-    then grobner_basis basis opairs
-    else if constant_poly sp then grobner_basis (sph::basis) []
-    else 
-     let 
-      val rawcps = map (fn p => (mlcm (hd(fst p)) (hd sp),align(p,sph)))
-                              basis
-      val newcps = filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q)))
-                        rawcps
-     in grobner_basis (sph::basis)
-                 (merge forder opairs (mergesort forder newcps))
-     end
-   end;
-
-(* Interreduce initial polynomials.                                          *)
-
-fun grobner_interreduce rpols ipols =
-  case ipols of
-    [] => map monic (rev rpols)
-  | p::ps => let val p' = reduce (rpols @ ps) p in
-             if null (fst p') then grobner_interreduce rpols ps
-             else grobner_interreduce (p'::rpols) ps end;
-
-(* Overall function.                                                         *)
-
-fun grobner pols =
-    let val npols = map_index (fn (n, p) => (p, Start n)) pols
-        val phists = filter (fn (p,_) => not (null p)) npols
-        val bas = grobner_interreduce [] (map monic phists)
-        val prs0 = map_product pair bas bas
-        val prs1 = filter (fn ((x,_),(y,_)) => poly_lt x y) prs0
-        val prs2 = map (fn (p,q) => (mlcm (hd(fst p)) (hd(fst q)),(p,q))) prs1
-        val prs3 =
-            filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q))) prs2 in
-        grobner_basis bas (mergesort forder prs3) end;
-
-(* Get proof of contradiction from Grobner basis.                            *)
-
-fun find p l =
-  case l of
-      [] => error "find"
-    | (h::t) => if p(h) then h else find p t;
-
-fun grobner_refute pols =
-  let val gb = grobner pols in
-  snd(find (fn (p,h) => length p = 1 andalso forall (fn x=> x=0) (snd(hd p))) gb)
-  end;
-
-(* Turn proof into a certificate as sum of multipliers.                      *)
-(* In principle this is very inefficient: in a heavily shared proof it may   *)
-(* make the same calculation many times. Could put in a cache or something.  *)
-
-fun resolve_proof vars prf =
-  case prf of
-    Start(~1) => []
-  | Start m => [(m,[(rat_1,map (K 0) vars)])]
-  | Mmul(pol,lin) =>
-        let val lis = resolve_proof vars lin in
-            map (fn (n,p) => (n,grob_cmul pol p)) lis end
-  | Add(lin1,lin2) =>
-        let val lis1 = resolve_proof vars lin1
-            val lis2 = resolve_proof vars lin2
-            val dom = distinct (op =) (union (op =) (map fst lis1) (map fst lis2))
-        in
-            map (fn n => let val a = these (AList.lookup (op =) lis1 n)
-                             val b = these (AList.lookup (op =) lis2 n)
-                         in (n,grob_add a b) end) dom end;
-
-(* Run the procedure and produce Weak Nullstellensatz certificate.           *)
-
-fun grobner_weak vars pols =
-    let val cert = resolve_proof vars (grobner_refute pols)
-        val l =
-            fold_rev (fold_rev (lcm_rat o denominator_rat o fst) o snd) cert (rat_1) in
-        (l,map (fn (i,p) => (i,map (fn (d,m) => (l*/d,m)) p)) cert) end;
-
-(* Prove a polynomial is in ideal generated by others, using Grobner basis.  *)
-
-fun grobner_ideal vars pols pol =
-  let val (pol',h) = reduce (grobner pols) (grob_neg pol,Start(~1)) in
-  if not (null pol') then error "grobner_ideal: not in the ideal" else
-  resolve_proof vars h end;
-
-(* Produce Strong Nullstellensatz certificate for a power of pol.            *)
-
-fun grobner_strong vars pols pol =
-    let val vars' = @{cterm "True"}::vars
-        val grob_z = [(rat_1,1::(map (fn x => 0) vars))]
-        val grob_1 = [(rat_1,(map (fn x => 0) vars'))]
-        fun augment p= map (fn (c,m) => (c,0::m)) p
-        val pols' = map augment pols
-        val pol' = augment pol
-        val allpols = (grob_sub (grob_mul grob_z pol') grob_1)::pols'
-        val (l,cert) = grobner_weak vars' allpols
-        val d = fold (fold (Integer.max o hd o snd) o snd) cert 0
-        fun transform_monomial (c,m) =
-            grob_cmul (c,tl m) (grob_pow vars pol (d - hd m))
-        fun transform_polynomial q = fold_rev (grob_add o transform_monomial) q []
-        val cert' = map (fn (c,q) => (c-1,transform_polynomial q))
-                        (filter (fn (k,_) => k <> 0) cert) in
-        (d,l,cert') end;
-
-
-(* Overall parametrized universal procedure for (semi)rings.                 *)
-(* We return an ideal_conv and the actual ring prover.                       *)
-
-fun refute_disj rfn tm =
- case term_of tm of
-  Const("op |",_)$l$r =>
-   compose_single(refute_disj rfn (dest_arg tm),2,compose_single(refute_disj rfn (dest_arg1 tm),2,disjE))
-  | _ => rfn tm ;
-
-val notnotD = @{thm notnotD};
-fun mk_binop ct x y = capply (capply ct x) y
-
-val mk_comb = capply;
-fun is_neg t =
-    case term_of t of
-      (Const("Not",_)$p) => true
-    | _  => false;
-fun is_eq t =
- case term_of t of
- (Const("op =",_)$_$_) => true
-| _  => false;
-
-fun end_itlist f l =
-  case l of
-        []     => error "end_itlist"
-      | [x]    => x
-      | (h::t) => f h (end_itlist f t);
-
-val list_mk_binop = fn b => end_itlist (mk_binop b);
-
-val list_dest_binop = fn b =>
- let fun h acc t =
-  ((let val (l,r) = dest_binary b t in h (h acc r) l end)
-   handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *)
- in h []
- end;
-
-val strip_exists =
- let fun h (acc, t) =
-      case (term_of t) of
-       Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
-     | _ => (acc,t)
- in fn t => h ([],t)
- end;
-
-fun is_forall t =
- case term_of t of
-  (Const("All",_)$Abs(_,_,_)) => true
-| _ => false;
-
-val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
-val bool_simps = @{thms bool_simps};
-val nnf_simps = @{thms nnf_simps};
-val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
-val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms weak_dnf_simps});
-val initial_conv =
-    Simplifier.rewrite
-     (HOL_basic_ss addsimps nnf_simps
-       addsimps [not_all, not_ex]
-       addsimps map (fn th => th RS sym) (@{thms ex_simps} @ @{thms all_simps}));
-
-val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
-
-val cTrp = @{cterm "Trueprop"};
-val cConj = @{cterm "op &"};
-val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
-val assume_Trueprop = mk_comb cTrp #> assume;
-val list_mk_conj = list_mk_binop cConj;
-val conjs = list_dest_binop cConj;
-val mk_neg = mk_comb cNot;
-
-fun striplist dest = 
- let
-  fun h acc x = case try dest x of
-    SOME (a,b) => h (h acc b) a
-  | NONE => x::acc
- in h [] end;
-fun list_mk_binop b = foldr1 (fn (s,t) => Thm.capply (Thm.capply b s) t);
-
-val eq_commute = mk_meta_eq @{thm eq_commute};
-
-fun sym_conv eq = 
- let val (l,r) = Thm.dest_binop eq
- in instantiate' [SOME (ctyp_of_term l)] [SOME l, SOME r] eq_commute
- end;
-
-  (* FIXME : copied from cqe.ML -- complex QE*)
-fun conjuncts ct =
- case term_of ct of
-  @{term "op &"}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
-| _ => [ct];
-
-fun fold1 f = foldr1 (uncurry f);
-
-val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm "op &"} c) c') ;
-
-fun mk_conj_tab th = 
- let fun h acc th = 
-   case prop_of th of
-   @{term "Trueprop"}$(@{term "op &"}$p$q) => 
-     h (h acc (th RS conjunct2)) (th RS conjunct1)
-  | @{term "Trueprop"}$p => (p,th)::acc
-in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
-
-fun is_conj (@{term "op &"}$_$_) = true
-  | is_conj _ = false;
-
-fun prove_conj tab cjs = 
- case cjs of 
-   [c] => if is_conj (term_of c) then prove_conj tab (conjuncts c) else tab c
- | c::cs => conjI OF [prove_conj tab [c], prove_conj tab cs];
-
-fun conj_ac_rule eq = 
- let 
-  val (l,r) = Thm.dest_equals eq
-  val ctabl = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} l))
-  val ctabr = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} r))
-  fun tabl c = the (Termtab.lookup ctabl (term_of c))
-  fun tabr c = the (Termtab.lookup ctabr (term_of c))
-  val thl  = prove_conj tabl (conjuncts r) |> implies_intr_hyps
-  val thr  = prove_conj tabr (conjuncts l) |> implies_intr_hyps
-  val eqI = instantiate' [] [SOME l, SOME r] @{thm iffI}
- in implies_elim (implies_elim eqI thl) thr |> mk_meta_eq end;
-
- (* END FIXME.*)
-
-   (* Conversion for the equivalence of existential statements where 
-      EX quantifiers are rearranged differently *)
- fun ext T = cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
- fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
-
-fun choose v th th' = case concl_of th of 
-  @{term Trueprop} $ (Const("Ex",_)$_) => 
-   let
-    val p = (funpow 2 Thm.dest_arg o cprop_of) th
-    val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
-    val th0 = fconv_rule (Thm.beta_conversion true)
-        (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
-    val pv = (Thm.rhs_of o Thm.beta_conversion true) 
-          (Thm.capply @{cterm Trueprop} (Thm.capply p v))
-    val th1 = forall_intr v (implies_intr pv th')
-   in implies_elim (implies_elim th0 th) th1  end
-| _ => error ""
-
-fun simple_choose v th = 
-   choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
-
-
- fun mkexi v th = 
-  let 
-   val p = Thm.cabs v (Thm.dest_arg (Thm.cprop_of th))
-  in implies_elim 
-    (fconv_rule (Thm.beta_conversion true) (instantiate' [SOME (ctyp_of_term v)] [SOME p, SOME v] @{thm exI}))
-      th
-  end
- fun ex_eq_conv t = 
-  let 
-  val (p0,q0) = Thm.dest_binop t
-  val (vs',P) = strip_exists p0 
-  val (vs,_) = strip_exists q0 
-   val th = assume (Thm.capply @{cterm Trueprop} P)
-   val th1 =  implies_intr_hyps (fold simple_choose vs' (fold mkexi vs th))
-   val th2 =  implies_intr_hyps (fold simple_choose vs (fold mkexi vs' th))
-   val p = (Thm.dest_arg o Thm.dest_arg1 o cprop_of) th1
-   val q = (Thm.dest_arg o Thm.dest_arg o cprop_of) th1
-  in implies_elim (implies_elim (instantiate' [] [SOME p, SOME q] iffI) th1) th2
-     |> mk_meta_eq
-  end;
-
-
- fun getname v = case term_of v of 
-  Free(s,_) => s
- | Var ((s,_),_) => s
- | _ => "x"
- fun mk_eq s t = Thm.capply (Thm.capply @{cterm "op == :: bool => _"} s) t
- fun mkeq s t = Thm.capply @{cterm Trueprop} (Thm.capply (Thm.capply @{cterm "op = :: bool => _"} s) t)
- fun mk_exists v th = arg_cong_rule (ext (ctyp_of_term v))
-   (Thm.abstract_rule (getname v) v th)
- val simp_ex_conv = 
-     Simplifier.rewrite (HOL_basic_ss addsimps @{thms simp_thms(39)})
-
-fun frees t = Thm.add_cterm_frees t [];
-fun free_in v t = member op aconvc (frees t) v;
-
-val vsubst = let
- fun vsubst (t,v) tm =  
-   (Thm.rhs_of o Thm.beta_conversion false) (Thm.capply (Thm.cabs v tm) t)
-in fold vsubst end;
-
-
-(** main **)
-
-fun ring_and_ideal_conv
-  {vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules), 
-   field = (f_ops, f_rules), idom, ideal}
-  dest_const mk_const ring_eq_conv ring_normalize_conv =
-let
-  val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
-  val [ring_add_tm, ring_mul_tm, ring_pow_tm] =
-    map dest_fun2 [add_pat, mul_pat, pow_pat];
-
-  val (ring_sub_tm, ring_neg_tm) =
-    (case r_ops of
-     [sub_pat, neg_pat] => (dest_fun2 sub_pat, dest_fun neg_pat)
-    |_  => (@{cterm "True"}, @{cterm "True"}));
-
-  val (field_div_tm, field_inv_tm) =
-    (case f_ops of
-       [div_pat, inv_pat] => (dest_fun2 div_pat, dest_fun inv_pat)
-     | _ => (@{cterm "True"}, @{cterm "True"}));
-
-  val [idom_thm, neq_thm] = idom;
-  val [idl_sub, idl_add0] = 
-     if length ideal = 2 then ideal else [eq_commute, eq_commute]
-  fun ring_dest_neg t =
-    let val (l,r) = dest_comb t 
-    in if Term.could_unify(term_of l,term_of ring_neg_tm) then r 
-       else raise CTERM ("ring_dest_neg", [t])
-    end
-
- val ring_mk_neg = fn tm => mk_comb (ring_neg_tm) (tm);
- fun field_dest_inv t =
-    let val (l,r) = dest_comb t in
-        if Term.could_unify(term_of l, term_of field_inv_tm) then r 
-        else raise CTERM ("field_dest_inv", [t])
-    end
- val ring_dest_add = dest_binary ring_add_tm;
- val ring_mk_add = mk_binop ring_add_tm;
- val ring_dest_sub = dest_binary ring_sub_tm;
- val ring_mk_sub = mk_binop ring_sub_tm;
- val ring_dest_mul = dest_binary ring_mul_tm;
- val ring_mk_mul = mk_binop ring_mul_tm;
- val field_dest_div = dest_binary field_div_tm;
- val field_mk_div = mk_binop field_div_tm;
- val ring_dest_pow = dest_binary ring_pow_tm;
- val ring_mk_pow = mk_binop ring_pow_tm ;
- fun grobvars tm acc =
-    if can dest_const tm then acc
-    else if can ring_dest_neg tm then grobvars (dest_arg tm) acc
-    else if can ring_dest_pow tm then grobvars (dest_arg1 tm) acc
-    else if can ring_dest_add tm orelse can ring_dest_sub tm
-            orelse can ring_dest_mul tm
-    then grobvars (dest_arg1 tm) (grobvars (dest_arg tm) acc)
-    else if can field_dest_inv tm
-         then
-          let val gvs = grobvars (dest_arg tm) [] 
-          in if null gvs then acc else tm::acc
-          end
-    else if can field_dest_div tm then
-         let val lvs = grobvars (dest_arg1 tm) acc
-             val gvs = grobvars (dest_arg tm) []
-          in if null gvs then lvs else tm::acc
-          end 
-    else tm::acc ;
-
-fun grobify_term vars tm =
-((if not (member (op aconvc) vars tm) then raise CTERM ("Not a variable", [tm]) else
-     [(rat_1,map (fn i => if i aconvc tm then 1 else 0) vars)])
-handle  CTERM _ =>
- ((let val x = dest_const tm
- in if x =/ rat_0 then [] else [(x,map (fn v => 0) vars)]
- end)
- handle ERROR _ =>
-  ((grob_neg(grobify_term vars (ring_dest_neg tm)))
-  handle CTERM _ =>
-   (
-   (grob_inv(grobify_term vars (field_dest_inv tm)))
-   handle CTERM _ => 
-    ((let val (l,r) = ring_dest_add tm
-    in grob_add (grobify_term vars l) (grobify_term vars r)
-    end)
-    handle CTERM _ =>
-     ((let val (l,r) = ring_dest_sub tm
-     in grob_sub (grobify_term vars l) (grobify_term vars r)
-     end)
-     handle  CTERM _ =>
-      ((let val (l,r) = ring_dest_mul tm
-      in grob_mul (grobify_term vars l) (grobify_term vars r)
-      end)
-       handle CTERM _ =>
-        (  (let val (l,r) = field_dest_div tm
-          in grob_div (grobify_term vars l) (grobify_term vars r)
-          end)
-         handle CTERM _ =>
-          ((let val (l,r) = ring_dest_pow tm
-          in grob_pow vars (grobify_term vars l) ((term_of #> HOLogic.dest_number #> snd) r)
-          end)
-           handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
-val eq_tm = idom_thm |> concl |> dest_arg |> dest_arg |> dest_fun2;
-val dest_eq = dest_binary eq_tm;
-
-fun grobify_equation vars tm =
-    let val (l,r) = dest_binary eq_tm tm
-    in grob_sub (grobify_term vars l) (grobify_term vars r)
-    end;
-
-fun grobify_equations tm =
- let
-  val cjs = conjs tm
-  val  rawvars = fold_rev (fn eq => fn a =>
-                                       grobvars (dest_arg1 eq) (grobvars (dest_arg eq) a)) cjs []
-  val vars = sort (fn (x, y) => Term_Ord.term_ord(term_of x,term_of y))
-                  (distinct (op aconvc) rawvars)
- in (vars,map (grobify_equation vars) cjs)
- end;
-
-val holify_polynomial =
- let fun holify_varpow (v,n) =
-  if n = 1 then v else ring_mk_pow v (Numeral.mk_cnumber @{ctyp "nat"} n)  (* FIXME *)
- fun holify_monomial vars (c,m) =
-  let val xps = map holify_varpow (filter (fn (_,n) => n <> 0) (vars ~~ m))
-   in end_itlist ring_mk_mul (mk_const c :: xps)
-  end
- fun holify_polynomial vars p =
-     if null p then mk_const (rat_0)
-     else end_itlist ring_mk_add (map (holify_monomial vars) p)
- in holify_polynomial
- end ;
-val idom_rule = simplify (HOL_basic_ss addsimps [idom_thm]);
-fun prove_nz n = eqF_elim
-                 (ring_eq_conv(mk_binop eq_tm (mk_const n) (mk_const(rat_0))));
-val neq_01 = prove_nz (rat_1);
-fun neq_rule n th = [prove_nz n, th] MRS neq_thm;
-fun mk_add th1 = combination(arg_cong_rule ring_add_tm th1);
-
-fun refute tm =
- if tm aconvc false_tm then assume_Trueprop tm else
- ((let
-   val (nths0,eths0) = List.partition (is_neg o concl) (HOLogic.conj_elims (assume_Trueprop tm))
-   val  nths = filter (is_eq o dest_arg o concl) nths0
-   val eths = filter (is_eq o concl) eths0
-  in
-   if null eths then
-    let
-      val th1 = end_itlist (fn th1 => fn th2 => idom_rule(HOLogic.conj_intr th1 th2)) nths
-      val th2 = Conv.fconv_rule
-                ((arg_conv #> arg_conv)
-                     (binop_conv ring_normalize_conv)) th1
-      val conc = th2 |> concl |> dest_arg
-      val (l,r) = conc |> dest_eq
-    in implies_intr (mk_comb cTrp tm)
-                    (equal_elim (arg_cong_rule cTrp (eqF_intr th2))
-                           (reflexive l |> mk_object_eq))
-    end
-   else
-   let
-    val (vars,l,cert,noteqth) =(
-     if null nths then
-      let val (vars,pols) = grobify_equations(list_mk_conj(map concl eths))
-          val (l,cert) = grobner_weak vars pols
-      in (vars,l,cert,neq_01)
-      end
-     else
-      let
-       val nth = end_itlist (fn th1 => fn th2 => idom_rule(HOLogic.conj_intr th1 th2)) nths
-       val (vars,pol::pols) =
-          grobify_equations(list_mk_conj(dest_arg(concl nth)::map concl eths))
-       val (deg,l,cert) = grobner_strong vars pols pol
-       val th1 = Conv.fconv_rule((arg_conv o arg_conv)(binop_conv ring_normalize_conv)) nth
-       val th2 = funpow deg (idom_rule o HOLogic.conj_intr th1) neq_01
-      in (vars,l,cert,th2)
-      end)
-    val cert_pos = map (fn (i,p) => (i,filter (fn (c,m) => c >/ rat_0) p)) cert
-    val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (minus_rat c,m))
-                                            (filter (fn (c,m) => c </ rat_0) p))) cert
-    val  herts_pos = map (fn (i,p) => (i,holify_polynomial vars p)) cert_pos
-    val  herts_neg = map (fn (i,p) => (i,holify_polynomial vars p)) cert_neg
-    fun thm_fn pols =
-        if null pols then reflexive(mk_const rat_0) else
-        end_itlist mk_add
-            (map (fn (i,p) => arg_cong_rule (mk_comb ring_mul_tm p)
-              (nth eths i |> mk_meta_eq)) pols)
-    val th1 = thm_fn herts_pos
-    val th2 = thm_fn herts_neg
-    val th3 = HOLogic.conj_intr(mk_add (symmetric th1) th2 |> mk_object_eq) noteqth
-    val th4 = Conv.fconv_rule ((arg_conv o arg_conv o binop_conv) ring_normalize_conv)
-                               (neq_rule l th3)
-    val (l,r) = dest_eq(dest_arg(concl th4))
-   in implies_intr (mk_comb cTrp tm)
-                        (equal_elim (arg_cong_rule cTrp (eqF_intr th4))
-                   (reflexive l |> mk_object_eq))
-   end
-  end) handle ERROR _ => raise CTERM ("Gorbner-refute: unable to refute",[tm]))
-
-fun ring tm =
- let
-  fun mk_forall x p =
-      mk_comb (cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (cabs x p)
-  val avs = add_cterm_frees tm []
-  val P' = fold mk_forall avs tm
-  val th1 = initial_conv(mk_neg P')
-  val (evs,bod) = strip_exists(concl th1) in
-   if is_forall bod then raise CTERM("ring: non-universal formula",[tm])
-   else
-   let
-    val th1a = weak_dnf_conv bod
-    val boda = concl th1a
-    val th2a = refute_disj refute boda
-    val th2b = [mk_object_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans
-    val th2 = fold (fn v => fn th => (forall_intr v th) COMP allI) evs (th2b RS PFalse)
-    val th3 = equal_elim
-                (Simplifier.rewrite (HOL_basic_ss addsimps [not_ex RS sym])
-                          (th2 |> cprop_of)) th2
-    in specl avs
-             ([[[mk_object_eq th1, th3 RS PFalse'] MRS trans] MRS PFalse] MRS notnotD)
-   end
- end
-fun ideal tms tm ord =
- let
-  val rawvars = fold_rev grobvars (tm::tms) []
-  val vars = sort ord (distinct (fn (x,y) => (term_of x) aconv (term_of y)) rawvars)
-  val pols = map (grobify_term vars) tms
-  val pol = grobify_term vars tm
-  val cert = grobner_ideal vars pols pol
- in map_range (fn n => these (AList.lookup (op =) cert n) |> holify_polynomial vars)
-   (length pols)
- end
-
-fun poly_eq_conv t = 
- let val (a,b) = Thm.dest_binop t
- in fconv_rule (arg_conv (arg1_conv ring_normalize_conv)) 
-     (instantiate' [] [SOME a, SOME b] idl_sub)
- end
- val poly_eq_simproc = 
-  let 
-   fun proc phi  ss t = 
-    let val th = poly_eq_conv t
-    in if Thm.is_reflexive th then NONE else SOME th
-    end
-   in make_simproc {lhss = [Thm.lhs_of idl_sub], 
-                name = "poly_eq_simproc", proc = proc, identifier = []}
-   end;
-  val poly_eq_ss = HOL_basic_ss addsimps @{thms simp_thms}
-                        addsimprocs [poly_eq_simproc]
-
- local
-  fun is_defined v t =
-  let 
-   val mons = striplist(dest_binary ring_add_tm) t 
-  in member (op aconvc) mons v andalso 
-    forall (fn m => v aconvc m 
-          orelse not(member (op aconvc) (Thm.add_cterm_frees m []) v)) mons
-  end
-
-  fun isolate_variable vars tm =
-  let 
-   val th = poly_eq_conv tm
-   val th' = (sym_conv then_conv poly_eq_conv) tm
-   val (v,th1) = 
-   case find_first(fn v=> is_defined v (Thm.dest_arg1 (Thm.rhs_of th))) vars of
-    SOME v => (v,th')
-   | NONE => (the (find_first 
-          (fn v => is_defined v (Thm.dest_arg1 (Thm.rhs_of th'))) vars) ,th)
-   val th2 = transitive th1 
-        (instantiate' []  [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v] 
-          idl_add0)
-   in fconv_rule(funpow 2 arg_conv ring_normalize_conv) th2
-   end
- in
- fun unwind_polys_conv tm =
- let 
-  val (vars,bod) = strip_exists tm
-  val cjs = striplist (dest_binary @{cterm "op &"}) bod
-  val th1 = (the (get_first (try (isolate_variable vars)) cjs) 
-             handle Option => raise CTERM ("unwind_polys_conv",[tm]))
-  val eq = Thm.lhs_of th1
-  val bod' = list_mk_binop @{cterm "op &"} (eq::(remove op aconvc eq cjs))
-  val th2 = conj_ac_rule (mk_eq bod bod')
-  val th3 = transitive th2 
-         (Drule.binop_cong_rule @{cterm "op &"} th1 
-                (reflexive (Thm.dest_arg (Thm.rhs_of th2))))
-  val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3))
-  val vars' = (remove op aconvc v vars) @ [v]
-  val th4 = fconv_rule (arg_conv simp_ex_conv) (mk_exists v th3)
-  val th5 = ex_eq_conv (mk_eq tm (fold mk_ex (remove op aconvc v vars) (Thm.lhs_of th4)))
- in transitive th5 (fold mk_exists (remove op aconvc v vars) th4)
- end;
-end
-
-local
- fun scrub_var v m =
-  let 
-   val ps = striplist ring_dest_mul m 
-   val ps' = remove op aconvc v ps
-  in if null ps' then one_tm else fold1 ring_mk_mul ps'
-  end
- fun find_multipliers v mons =
-  let 
-   val mons1 = filter (fn m => free_in v m) mons 
-   val mons2 = map (scrub_var v) mons1 
-   in  if null mons2 then zero_tm else fold1 ring_mk_add mons2
-  end
-
- fun isolate_monomials vars tm =
- let 
-  val (cmons,vmons) =
-    List.partition (fn m => null (inter (op aconvc) vars (frees m)))
-                   (striplist ring_dest_add tm)
-  val cofactors = map (fn v => find_multipliers v vmons) vars
-  val cnc = if null cmons then zero_tm
-             else Thm.capply ring_neg_tm
-                    (list_mk_binop ring_add_tm cmons) 
-  in (cofactors,cnc)
-  end;
-
-fun isolate_variables evs ps eq =
- let 
-  val vars = filter (fn v => free_in v eq) evs
-  val (qs,p) = isolate_monomials vars eq
-  val rs = ideal (qs @ ps) p 
-              (fn (s,t) => Term_Ord.term_ord (term_of s, term_of t))
- in (eq, take (length qs) rs ~~ vars)
- end;
- fun subst_in_poly i p = Thm.rhs_of (ring_normalize_conv (vsubst i p));
-in
- fun solve_idealism evs ps eqs =
-  if null evs then [] else
-  let 
-   val (eq,cfs) = get_first (try (isolate_variables evs ps)) eqs |> the
-   val evs' = subtract op aconvc evs (map snd cfs)
-   val eqs' = map (subst_in_poly cfs) (remove op aconvc eq eqs)
-  in cfs @ solve_idealism evs' ps eqs'
-  end;
-end;
-
-
-in {ring_conv = ring, simple_ideal = ideal, multi_ideal = solve_idealism, 
-    poly_eq_ss = poly_eq_ss, unwind_conv = unwind_polys_conv}
-end;
-
-
-fun find_term bounds tm =
-  (case term_of tm of
-    Const ("op =", T) $ _ $ _ =>
-      if domain_type T = HOLogic.boolT then find_args bounds tm
-      else dest_arg tm
-  | Const ("Not", _) $ _ => find_term bounds (dest_arg tm)
-  | Const ("All", _) $ _ => find_body bounds (dest_arg tm)
-  | Const ("Ex", _) $ _ => find_body bounds (dest_arg tm)
-  | Const ("op &", _) $ _ $ _ => find_args bounds tm
-  | Const ("op |", _) $ _ $ _ => find_args bounds tm
-  | Const ("op -->", _) $ _ $ _ => find_args bounds tm
-  | @{term "op ==>"} $_$_ => find_args bounds tm
-  | Const("op ==",_)$_$_ => find_args bounds tm
-  | @{term Trueprop}$_ => find_term bounds (dest_arg tm)
-  | _ => raise TERM ("find_term", []))
-and find_args bounds tm =
-  let val (t, u) = Thm.dest_binop tm
-  in (find_term bounds t handle TERM _ => find_term bounds u) end
-and find_body bounds b =
-  let val (_, b') = dest_abs (SOME (Name.bound bounds)) b
-  in find_term (bounds + 1) b' end;
-
-
-fun get_ring_ideal_convs ctxt form = 
- case try (find_term 0) form of
-  NONE => NONE
-| SOME tm =>
-  (case Normalizer.match ctxt tm of
-    NONE => NONE
-  | SOME (res as (theory, {is_const, dest_const, 
-          mk_const, conv = ring_eq_conv})) =>
-     SOME (ring_and_ideal_conv theory
-          dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
-          (Normalizer.semiring_normalize_wrapper ctxt res)))
-
-fun ring_solve ctxt form =
-  (case try (find_term 0 (* FIXME !? *)) form of
-    NONE => reflexive form
-  | SOME tm =>
-      (case Normalizer.match ctxt tm of
-        NONE => reflexive form
-      | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
-        #ring_conv (ring_and_ideal_conv theory
-          dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
-          (Normalizer.semiring_normalize_wrapper ctxt res)) form));
-
-fun presimplify ctxt add_thms del_thms = asm_full_simp_tac (Simplifier.context ctxt
-  (HOL_basic_ss addsimps (Algebra_Simplification.get ctxt) delsimps del_thms addsimps add_thms));
-
-fun ring_tac add_ths del_ths ctxt =
-  Object_Logic.full_atomize_tac
-  THEN' presimplify ctxt add_ths del_ths
-  THEN' CSUBGOAL (fn (p, i) =>
-    rtac (let val form = Object_Logic.dest_judgment p
-          in case get_ring_ideal_convs ctxt form of
-           NONE => reflexive form
-          | SOME thy => #ring_conv thy form
-          end) i
-      handle TERM _ => no_tac
-        | CTERM _ => no_tac
-        | THM _ => no_tac);
-
-local
- fun lhs t = case term_of t of
-  Const("op =",_)$_$_ => Thm.dest_arg1 t
- | _=> raise CTERM ("ideal_tac - lhs",[t])
- fun exitac NONE = no_tac
-   | exitac (SOME y) = rtac (instantiate' [SOME (ctyp_of_term y)] [NONE,SOME y] exI) 1
-in 
-fun ideal_tac add_ths del_ths ctxt = 
-  presimplify ctxt add_ths del_ths
- THEN'
- CSUBGOAL (fn (p, i) =>
-  case get_ring_ideal_convs ctxt p of
-   NONE => no_tac
- | SOME thy => 
-  let
-   fun poly_exists_tac {asms = asms, concl = concl, prems = prems,
-            params = params, context = ctxt, schematics = scs} = 
-    let
-     val (evs,bod) = strip_exists (Thm.dest_arg concl)
-     val ps = map_filter (try (lhs o Thm.dest_arg)) asms 
-     val cfs = (map swap o #multi_ideal thy evs ps) 
-                   (map Thm.dest_arg1 (conjuncts bod))
-     val ws = map (exitac o AList.lookup op aconvc cfs) evs
-    in EVERY (rev ws) THEN Method.insert_tac prems 1 
-        THEN ring_tac add_ths del_ths ctxt 1
-   end
-  in  
-     clarify_tac @{claset} i 
-     THEN Object_Logic.full_atomize_tac i 
-     THEN asm_full_simp_tac (Simplifier.context ctxt (#poly_eq_ss thy)) i 
-     THEN clarify_tac @{claset} i 
-     THEN (REPEAT (CONVERSION (#unwind_conv thy) i))
-     THEN SUBPROOF poly_exists_tac ctxt i
-  end
- handle TERM _ => no_tac
-     | CTERM _ => no_tac
-     | THM _ => no_tac); 
-end;
-
-fun algebra_tac add_ths del_ths ctxt i = 
- ring_tac add_ths del_ths ctxt i ORELSE ideal_tac add_ths del_ths ctxt i
- 
-local
-
-fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
-val addN = "add"
-val delN = "del"
-val any_keyword = keyword addN || keyword delN
-val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-
-in
-
-val algebra_method = ((Scan.optional (keyword addN |-- thms) []) -- 
-   (Scan.optional (keyword delN |-- thms) [])) >>
-  (fn (add_ths, del_ths) => fn ctxt =>
-       SIMPLE_METHOD' (algebra_tac add_ths del_ths ctxt))
-
-end;
-
-end;
diff -r 7f1da69cacb3 -r cf558aeb35b0 src/HOL/Tools/groebner.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/groebner.ML	Fri May 07 16:12:25 2010 +0200
@@ -0,0 +1,1045 @@
+(*  Title:      HOL/Tools/Groebner_Basis/groebner.ML
+    Author:     Amine Chaieb, TU Muenchen
+*)
+
+signature GROEBNER =
+sig
+  val ring_and_ideal_conv :
+    {idom: thm list, ring: cterm list * thm list, field: cterm list * thm list,
+     vars: cterm list, semiring: cterm list * thm list, ideal : thm list} ->
+    (cterm -> Rat.rat) -> (Rat.rat -> cterm) ->
+    conv ->  conv ->
+     {ring_conv : conv, 
+     simple_ideal: (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list),
+     multi_ideal: cterm list -> cterm list -> cterm list -> (cterm * cterm) list,
+     poly_eq_ss: simpset, unwind_conv : conv}
+  val ring_tac: thm list -> thm list -> Proof.context -> int -> tactic
+  val ideal_tac: thm list -> thm list -> Proof.context -> int -> tactic
+  val algebra_tac: thm list -> thm list -> Proof.context -> int -> tactic
+  val algebra_method: (Proof.context -> Method.method) context_parser
+end
+
+structure Groebner : GROEBNER =
+struct
+
+open Conv Drule Thm;
+
+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_binary ct ct' =
+  if is_binop ct ct' then Thm.dest_binop ct'
+  else raise CTERM ("dest_binary: bad binop", [ct, ct'])
+
+fun inst_thm inst = Thm.instantiate ([], inst);
+
+val rat_0 = Rat.zero;
+val rat_1 = Rat.one;
+val minus_rat = Rat.neg;
+val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
+fun int_of_rat a =
+    case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
+val lcm_rat = fn x => fn y => Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
+
+val (eqF_intr, eqF_elim) =
+  let val [th1,th2] = @{thms PFalse}
+  in (fn th => th COMP th2, fn th => th COMP th1) end;
+
+val (PFalse, PFalse') =
+ let val PFalse_eq = nth @{thms simp_thms} 13
+ in (PFalse_eq RS iffD1, PFalse_eq RS iffD2) end;
+
+
+(* Type for recording history, i.e. how a polynomial was obtained. *)
+
+datatype history =
+   Start of int
+ | Mmul of (Rat.rat * int list) * history
+ | Add of history * history;
+
+
+(* Monomial ordering. *)
+
+fun morder_lt m1 m2=
+    let fun lexorder l1 l2 =
+            case (l1,l2) of
+                ([],[]) => false
+              | (x1::o1,x2::o2) => x1 > x2 orelse x1 = x2 andalso lexorder o1 o2
+              | _ => error "morder: inconsistent monomial lengths"
+        val n1 = Integer.sum m1
+        val n2 = Integer.sum m2 in
+    n1 < n2 orelse n1 = n2 andalso lexorder m1 m2
+    end;
+
+fun morder_le m1 m2 = morder_lt m1 m2 orelse (m1 = m2);
+
+fun morder_gt m1 m2 = morder_lt m2 m1;
+
+(* Arithmetic on canonical polynomials. *)
+
+fun grob_neg l = map (fn (c,m) => (minus_rat c,m)) l;
+
+fun grob_add l1 l2 =
+  case (l1,l2) of
+    ([],l2) => l2
+  | (l1,[]) => l1
+  | ((c1,m1)::o1,(c2,m2)::o2) =>
+        if m1 = m2 then
+          let val c = c1+/c2 val rest = grob_add o1 o2 in
+          if c =/ rat_0 then rest else (c,m1)::rest end
+        else if morder_lt m2 m1 then (c1,m1)::(grob_add o1 l2)
+        else (c2,m2)::(grob_add l1 o2);
+
+fun grob_sub l1 l2 = grob_add l1 (grob_neg l2);
+
+fun grob_mmul (c1,m1) (c2,m2) = (c1*/c2, ListPair.map (op +) (m1, m2));
+
+fun grob_cmul cm pol = map (grob_mmul cm) pol;
+
+fun grob_mul l1 l2 =
+  case l1 of
+    [] => []
+  | (h1::t1) => grob_add (grob_cmul h1 l2) (grob_mul t1 l2);
+
+fun grob_inv l =
+  case l of
+    [(c,vs)] => if (forall (fn x => x = 0) vs) then
+                  if (c =/ rat_0) then error "grob_inv: division by zero"
+                  else [(rat_1 // c,vs)]
+              else error "grob_inv: non-constant divisor polynomial"
+  | _ => error "grob_inv: non-constant divisor polynomial";
+
+fun grob_div l1 l2 =
+  case l2 of
+    [(c,l)] => if (forall (fn x => x = 0) l) then
+                 if c =/ rat_0 then error "grob_div: division by zero"
+                 else grob_cmul (rat_1 // c,l) l1
+             else error "grob_div: non-constant divisor polynomial"
+  | _ => error "grob_div: non-constant divisor polynomial";
+
+fun grob_pow vars l n =
+  if n < 0 then error "grob_pow: negative power"
+  else if n = 0 then [(rat_1,map (fn v => 0) vars)]
+  else grob_mul l (grob_pow vars l (n - 1));
+
+fun degree vn p =
+ case p of
+  [] => error "Zero polynomial"
+| [(c,ns)] => nth ns vn
+| (c,ns)::p' => Int.max (nth ns vn, degree vn p');
+
+fun head_deg vn p = let val d = degree vn p in
+ (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;
+
+val is_zerop = forall (fn (c,ns) => c =/ rat_0 andalso forall (curry (op =) 0) ns);
+val grob_pdiv =
+ let fun pdiv_aux vn (n,a) p k s =
+  if is_zerop s then (k,s) else
+  let val (m,b) = head_deg vn s
+  in if m < n then (k,s) else
+     let val p' = grob_mul p [(rat_1, map_index (fn (i,v) => if i = vn then m - n else 0)
+                                                (snd (hd s)))]
+     in if a = b then pdiv_aux vn (n,a) p k (grob_sub s p')
+        else pdiv_aux vn (n,a) p (k + 1) (grob_sub (grob_mul a s) (grob_mul b p'))
+     end
+  end
+ in fn vn => fn s => fn p => pdiv_aux vn (head_deg vn p) p 0 s
+ end;
+
+(* Monomial division operation. *)
+
+fun mdiv (c1,m1) (c2,m2) =
+  (c1//c2,
+   map2 (fn n1 => fn n2 => if n1 < n2 then error "mdiv" else n1 - n2) m1 m2);
+
+(* Lowest common multiple of two monomials. *)
+
+fun mlcm (c1,m1) (c2,m2) = (rat_1, ListPair.map Int.max (m1, m2));
+
+(* Reduce monomial cm by polynomial pol, returning replacement for cm.  *)
+
+fun reduce1 cm (pol,hpol) =
+  case pol of
+    [] => error "reduce1"
+  | cm1::cms => ((let val (c,m) = mdiv cm cm1 in
+                    (grob_cmul (minus_rat c,m) cms,
+                     Mmul((minus_rat c,m),hpol)) end)
+                handle  ERROR _ => error "reduce1");
+
+(* Try this for all polynomials in a basis.  *)
+fun tryfind f l =
+    case l of
+        [] => error "tryfind"
+      | (h::t) => ((f h) handle ERROR _ => tryfind f t);
+
+fun reduceb cm basis = tryfind (fn p => reduce1 cm p) basis;
+
+(* Reduction of a polynomial (always picking largest monomial possible).     *)
+
+fun reduce basis (pol,hist) =
+  case pol of
+    [] => (pol,hist)
+  | cm::ptl => ((let val (q,hnew) = reduceb cm basis in
+                   reduce basis (grob_add q ptl,Add(hnew,hist)) end)
+               handle (ERROR _) =>
+                   (let val (q,hist') = reduce basis (ptl,hist) in
+                       (cm::q,hist') end));
+
+(* Check for orthogonality w.r.t. LCM.                                       *)
+
+fun orthogonal l p1 p2 =
+  snd l = snd(grob_mmul (hd p1) (hd p2));
+
+(* Compute S-polynomial of two polynomials.                                  *)
+
+fun spoly cm ph1 ph2 =
+  case (ph1,ph2) of
+    (([],h),p) => ([],h)
+  | (p,([],h)) => ([],h)
+  | ((cm1::ptl1,his1),(cm2::ptl2,his2)) =>
+        (grob_sub (grob_cmul (mdiv cm cm1) ptl1)
+                  (grob_cmul (mdiv cm cm2) ptl2),
+         Add(Mmul(mdiv cm cm1,his1),
+             Mmul(mdiv (minus_rat(fst cm),snd cm) cm2,his2)));
+
+(* Make a polynomial monic.                                                  *)
+
+fun monic (pol,hist) =
+  if null pol then (pol,hist) else
+  let val (c',m') = hd pol in
+  (map (fn (c,m) => (c//c',m)) pol,
+   Mmul((rat_1 // c',map (K 0) m'),hist)) end;
+
+(* The most popular heuristic is to order critical pairs by LCM monomial.    *)
+
+fun forder ((c1,m1),_) ((c2,m2),_) = morder_lt m1 m2;
+
+fun poly_lt  p q =
+  case (p,q) of
+    (p,[]) => false
+  | ([],q) => true
+  | ((c1,m1)::o1,(c2,m2)::o2) =>
+        c1 </ c2 orelse
+        c1 =/ c2 andalso ((morder_lt m1 m2) orelse m1 = m2 andalso poly_lt o1 o2);
+
+fun align  ((p,hp),(q,hq)) =
+  if poly_lt p q then ((p,hp),(q,hq)) else ((q,hq),(p,hp));
+fun forall2 p l1 l2 =
+  case (l1,l2) of
+    ([],[]) => true
+  | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
+  | _ => false;
+
+fun poly_eq p1 p2 =
+  forall2 (fn (c1,m1) => fn (c2,m2) => c1 =/ c2 andalso (m1: int list) = m2) p1 p2;
+
+fun memx ((p1,h1),(p2,h2)) ppairs =
+  not (exists (fn ((q1,_),(q2,_)) => poly_eq p1 q1 andalso poly_eq p2 q2) ppairs);
+
+(* Buchberger's second criterion.                                            *)
+
+fun criterion2 basis (lcm,((p1,h1),(p2,h2))) opairs =
+  exists (fn g => not(poly_eq (fst g) p1) andalso not(poly_eq (fst g) p2) andalso
+                   can (mdiv lcm) (hd(fst g)) andalso
+                   not(memx (align (g,(p1,h1))) (map snd opairs)) andalso
+                   not(memx (align (g,(p2,h2))) (map snd opairs))) basis;
+
+(* Test for hitting constant polynomial.                                     *)
+
+fun constant_poly p =
+  length p = 1 andalso forall (fn x => x = 0) (snd(hd p));
+
+(* Grobner basis algorithm.                                                  *)
+
+(* FIXME: try to get rid of mergesort? *)
+fun merge ord l1 l2 =
+ case l1 of
+  [] => l2
+ | h1::t1 =>
+   case l2 of
+    [] => l1
+   | h2::t2 => if ord h1 h2 then h1::(merge ord t1 l2)
+               else h2::(merge ord l1 t2);
+fun mergesort ord l =
+ let
+ fun mergepairs l1 l2 =
+  case (l1,l2) of
+   ([s],[]) => s
+ | (l,[]) => mergepairs [] l
+ | (l,[s1]) => mergepairs (s1::l) []
+ | (l,(s1::s2::ss)) => mergepairs ((merge ord s1 s2)::l) ss
+ in if null l  then []  else mergepairs [] (map (fn x => [x]) l)
+ end;
+
+
+fun grobner_basis basis pairs =
+ case pairs of
+   [] => basis
+ | (l,(p1,p2))::opairs =>
+   let val (sph as (sp,hist)) = monic (reduce basis (spoly l p1 p2))
+   in 
+    if null sp orelse criterion2 basis (l,(p1,p2)) opairs
+    then grobner_basis basis opairs
+    else if constant_poly sp then grobner_basis (sph::basis) []
+    else 
+     let 
+      val rawcps = map (fn p => (mlcm (hd(fst p)) (hd sp),align(p,sph)))
+                              basis
+      val newcps = filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q)))
+                        rawcps
+     in grobner_basis (sph::basis)
+                 (merge forder opairs (mergesort forder newcps))
+     end
+   end;
+
+(* Interreduce initial polynomials.                                          *)
+
+fun grobner_interreduce rpols ipols =
+  case ipols of
+    [] => map monic (rev rpols)
+  | p::ps => let val p' = reduce (rpols @ ps) p in
+             if null (fst p') then grobner_interreduce rpols ps
+             else grobner_interreduce (p'::rpols) ps end;
+
+(* Overall function.                                                         *)
+
+fun grobner pols =
+    let val npols = map_index (fn (n, p) => (p, Start n)) pols
+        val phists = filter (fn (p,_) => not (null p)) npols
+        val bas = grobner_interreduce [] (map monic phists)
+        val prs0 = map_product pair bas bas
+        val prs1 = filter (fn ((x,_),(y,_)) => poly_lt x y) prs0
+        val prs2 = map (fn (p,q) => (mlcm (hd(fst p)) (hd(fst q)),(p,q))) prs1
+        val prs3 =
+            filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q))) prs2 in
+        grobner_basis bas (mergesort forder prs3) end;
+
+(* Get proof of contradiction from Grobner basis.                            *)
+
+fun find p l =
+  case l of
+      [] => error "find"
+    | (h::t) => if p(h) then h else find p t;
+
+fun grobner_refute pols =
+  let val gb = grobner pols in
+  snd(find (fn (p,h) => length p = 1 andalso forall (fn x=> x=0) (snd(hd p))) gb)
+  end;
+
+(* Turn proof into a certificate as sum of multipliers.                      *)
+(* In principle this is very inefficient: in a heavily shared proof it may   *)
+(* make the same calculation many times. Could put in a cache or something.  *)
+
+fun resolve_proof vars prf =
+  case prf of
+    Start(~1) => []
+  | Start m => [(m,[(rat_1,map (K 0) vars)])]
+  | Mmul(pol,lin) =>
+        let val lis = resolve_proof vars lin in
+            map (fn (n,p) => (n,grob_cmul pol p)) lis end
+  | Add(lin1,lin2) =>
+        let val lis1 = resolve_proof vars lin1
+            val lis2 = resolve_proof vars lin2
+            val dom = distinct (op =) (union (op =) (map fst lis1) (map fst lis2))
+        in
+            map (fn n => let val a = these (AList.lookup (op =) lis1 n)
+                             val b = these (AList.lookup (op =) lis2 n)
+                         in (n,grob_add a b) end) dom end;
+
+(* Run the procedure and produce Weak Nullstellensatz certificate.           *)
+
+fun grobner_weak vars pols =
+    let val cert = resolve_proof vars (grobner_refute pols)
+        val l =
+            fold_rev (fold_rev (lcm_rat o denominator_rat o fst) o snd) cert (rat_1) in
+        (l,map (fn (i,p) => (i,map (fn (d,m) => (l*/d,m)) p)) cert) end;
+
+(* Prove a polynomial is in ideal generated by others, using Grobner basis.  *)
+
+fun grobner_ideal vars pols pol =
+  let val (pol',h) = reduce (grobner pols) (grob_neg pol,Start(~1)) in
+  if not (null pol') then error "grobner_ideal: not in the ideal" else
+  resolve_proof vars h end;
+
+(* Produce Strong Nullstellensatz certificate for a power of pol.            *)
+
+fun grobner_strong vars pols pol =
+    let val vars' = @{cterm "True"}::vars
+        val grob_z = [(rat_1,1::(map (fn x => 0) vars))]
+        val grob_1 = [(rat_1,(map (fn x => 0) vars'))]
+        fun augment p= map (fn (c,m) => (c,0::m)) p
+        val pols' = map augment pols
+        val pol' = augment pol
+        val allpols = (grob_sub (grob_mul grob_z pol') grob_1)::pols'
+        val (l,cert) = grobner_weak vars' allpols
+        val d = fold (fold (Integer.max o hd o snd) o snd) cert 0
+        fun transform_monomial (c,m) =
+            grob_cmul (c,tl m) (grob_pow vars pol (d - hd m))
+        fun transform_polynomial q = fold_rev (grob_add o transform_monomial) q []
+        val cert' = map (fn (c,q) => (c-1,transform_polynomial q))
+                        (filter (fn (k,_) => k <> 0) cert) in
+        (d,l,cert') end;
+
+
+(* Overall parametrized universal procedure for (semi)rings.                 *)
+(* We return an ideal_conv and the actual ring prover.                       *)
+
+fun refute_disj rfn tm =
+ case term_of tm of
+  Const("op |",_)$l$r =>
+   compose_single(refute_disj rfn (dest_arg tm),2,compose_single(refute_disj rfn (dest_arg1 tm),2,disjE))
+  | _ => rfn tm ;
+
+val notnotD = @{thm notnotD};
+fun mk_binop ct x y = capply (capply ct x) y
+
+val mk_comb = capply;
+fun is_neg t =
+    case term_of t of
+      (Const("Not",_)$p) => true
+    | _  => false;
+fun is_eq t =
+ case term_of t of
+ (Const("op =",_)$_$_) => true
+| _  => false;
+
+fun end_itlist f l =
+  case l of
+        []     => error "end_itlist"
+      | [x]    => x
+      | (h::t) => f h (end_itlist f t);
+
+val list_mk_binop = fn b => end_itlist (mk_binop b);
+
+val list_dest_binop = fn b =>
+ let fun h acc t =
+  ((let val (l,r) = dest_binary b t in h (h acc r) l end)
+   handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *)
+ in h []
+ end;
+
+val strip_exists =
+ let fun h (acc, t) =
+      case (term_of t) of
+       Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
+     | _ => (acc,t)
+ in fn t => h ([],t)
+ end;
+
+fun is_forall t =
+ case term_of t of
+  (Const("All",_)$Abs(_,_,_)) => true
+| _ => false;
+
+val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
+val bool_simps = @{thms bool_simps};
+val nnf_simps = @{thms nnf_simps};
+val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
+val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms weak_dnf_simps});
+val initial_conv =
+    Simplifier.rewrite
+     (HOL_basic_ss addsimps nnf_simps
+       addsimps [not_all, not_ex]
+       addsimps map (fn th => th RS sym) (@{thms ex_simps} @ @{thms all_simps}));
+
+val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
+
+val cTrp = @{cterm "Trueprop"};
+val cConj = @{cterm "op &"};
+val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
+val assume_Trueprop = mk_comb cTrp #> assume;
+val list_mk_conj = list_mk_binop cConj;
+val conjs = list_dest_binop cConj;
+val mk_neg = mk_comb cNot;
+
+fun striplist dest = 
+ let
+  fun h acc x = case try dest x of
+    SOME (a,b) => h (h acc b) a
+  | NONE => x::acc
+ in h [] end;
+fun list_mk_binop b = foldr1 (fn (s,t) => Thm.capply (Thm.capply b s) t);
+
+val eq_commute = mk_meta_eq @{thm eq_commute};
+
+fun sym_conv eq = 
+ let val (l,r) = Thm.dest_binop eq
+ in instantiate' [SOME (ctyp_of_term l)] [SOME l, SOME r] eq_commute
+ end;
+
+  (* FIXME : copied from cqe.ML -- complex QE*)
+fun conjuncts ct =
+ case term_of ct of
+  @{term "op &"}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
+| _ => [ct];
+
+fun fold1 f = foldr1 (uncurry f);
+
+val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm "op &"} c) c') ;
+
+fun mk_conj_tab th = 
+ let fun h acc th = 
+   case prop_of th of
+   @{term "Trueprop"}$(@{term "op &"}$p$q) => 
+     h (h acc (th RS conjunct2)) (th RS conjunct1)
+  | @{term "Trueprop"}$p => (p,th)::acc
+in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
+
+fun is_conj (@{term "op &"}$_$_) = true
+  | is_conj _ = false;
+
+fun prove_conj tab cjs = 
+ case cjs of 
+   [c] => if is_conj (term_of c) then prove_conj tab (conjuncts c) else tab c
+ | c::cs => conjI OF [prove_conj tab [c], prove_conj tab cs];
+
+fun conj_ac_rule eq = 
+ let 
+  val (l,r) = Thm.dest_equals eq
+  val ctabl = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} l))
+  val ctabr = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} r))
+  fun tabl c = the (Termtab.lookup ctabl (term_of c))
+  fun tabr c = the (Termtab.lookup ctabr (term_of c))
+  val thl  = prove_conj tabl (conjuncts r) |> implies_intr_hyps
+  val thr  = prove_conj tabr (conjuncts l) |> implies_intr_hyps
+  val eqI = instantiate' [] [SOME l, SOME r] @{thm iffI}
+ in implies_elim (implies_elim eqI thl) thr |> mk_meta_eq end;
+
+ (* END FIXME.*)
+
+   (* Conversion for the equivalence of existential statements where 
+      EX quantifiers are rearranged differently *)
+ fun ext T = cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
+ fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
+
+fun choose v th th' = case concl_of th of 
+  @{term Trueprop} $ (Const("Ex",_)$_) => 
+   let
+    val p = (funpow 2 Thm.dest_arg o cprop_of) th
+    val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
+    val th0 = fconv_rule (Thm.beta_conversion true)
+        (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
+    val pv = (Thm.rhs_of o Thm.beta_conversion true) 
+          (Thm.capply @{cterm Trueprop} (Thm.capply p v))
+    val th1 = forall_intr v (implies_intr pv th')
+   in implies_elim (implies_elim th0 th) th1  end
+| _ => error ""
+
+fun simple_choose v th = 
+   choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
+
+
+ fun mkexi v th = 
+  let 
+   val p = Thm.cabs v (Thm.dest_arg (Thm.cprop_of th))
+  in implies_elim 
+    (fconv_rule (Thm.beta_conversion true) (instantiate' [SOME (ctyp_of_term v)] [SOME p, SOME v] @{thm exI}))
+      th
+  end
+ fun ex_eq_conv t = 
+  let 
+  val (p0,q0) = Thm.dest_binop t
+  val (vs',P) = strip_exists p0 
+  val (vs,_) = strip_exists q0 
+   val th = assume (Thm.capply @{cterm Trueprop} P)
+   val th1 =  implies_intr_hyps (fold simple_choose vs' (fold mkexi vs th))
+   val th2 =  implies_intr_hyps (fold simple_choose vs (fold mkexi vs' th))
+   val p = (Thm.dest_arg o Thm.dest_arg1 o cprop_of) th1
+   val q = (Thm.dest_arg o Thm.dest_arg o cprop_of) th1
+  in implies_elim (implies_elim (instantiate' [] [SOME p, SOME q] iffI) th1) th2
+     |> mk_meta_eq
+  end;
+
+
+ fun getname v = case term_of v of 
+  Free(s,_) => s
+ | Var ((s,_),_) => s
+ | _ => "x"
+ fun mk_eq s t = Thm.capply (Thm.capply @{cterm "op == :: bool => _"} s) t
+ fun mkeq s t = Thm.capply @{cterm Trueprop} (Thm.capply (Thm.capply @{cterm "op = :: bool => _"} s) t)
+ fun mk_exists v th = arg_cong_rule (ext (ctyp_of_term v))
+   (Thm.abstract_rule (getname v) v th)
+ val simp_ex_conv = 
+     Simplifier.rewrite (HOL_basic_ss addsimps @{thms simp_thms(39)})
+
+fun frees t = Thm.add_cterm_frees t [];
+fun free_in v t = member op aconvc (frees t) v;
+
+val vsubst = let
+ fun vsubst (t,v) tm =  
+   (Thm.rhs_of o Thm.beta_conversion false) (Thm.capply (Thm.cabs v tm) t)
+in fold vsubst end;
+
+
+(** main **)
+
+fun ring_and_ideal_conv
+  {vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules), 
+   field = (f_ops, f_rules), idom, ideal}
+  dest_const mk_const ring_eq_conv ring_normalize_conv =
+let
+  val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
+  val [ring_add_tm, ring_mul_tm, ring_pow_tm] =
+    map dest_fun2 [add_pat, mul_pat, pow_pat];
+
+  val (ring_sub_tm, ring_neg_tm) =
+    (case r_ops of
+     [sub_pat, neg_pat] => (dest_fun2 sub_pat, dest_fun neg_pat)
+    |_  => (@{cterm "True"}, @{cterm "True"}));
+
+  val (field_div_tm, field_inv_tm) =
+    (case f_ops of
+       [div_pat, inv_pat] => (dest_fun2 div_pat, dest_fun inv_pat)
+     | _ => (@{cterm "True"}, @{cterm "True"}));
+
+  val [idom_thm, neq_thm] = idom;
+  val [idl_sub, idl_add0] = 
+     if length ideal = 2 then ideal else [eq_commute, eq_commute]
+  fun ring_dest_neg t =
+    let val (l,r) = dest_comb t 
+    in if Term.could_unify(term_of l,term_of ring_neg_tm) then r 
+       else raise CTERM ("ring_dest_neg", [t])
+    end
+
+ val ring_mk_neg = fn tm => mk_comb (ring_neg_tm) (tm);
+ fun field_dest_inv t =
+    let val (l,r) = dest_comb t in
+        if Term.could_unify(term_of l, term_of field_inv_tm) then r 
+        else raise CTERM ("field_dest_inv", [t])
+    end
+ val ring_dest_add = dest_binary ring_add_tm;
+ val ring_mk_add = mk_binop ring_add_tm;
+ val ring_dest_sub = dest_binary ring_sub_tm;
+ val ring_mk_sub = mk_binop ring_sub_tm;
+ val ring_dest_mul = dest_binary ring_mul_tm;
+ val ring_mk_mul = mk_binop ring_mul_tm;
+ val field_dest_div = dest_binary field_div_tm;
+ val field_mk_div = mk_binop field_div_tm;
+ val ring_dest_pow = dest_binary ring_pow_tm;
+ val ring_mk_pow = mk_binop ring_pow_tm ;
+ fun grobvars tm acc =
+    if can dest_const tm then acc
+    else if can ring_dest_neg tm then grobvars (dest_arg tm) acc
+    else if can ring_dest_pow tm then grobvars (dest_arg1 tm) acc
+    else if can ring_dest_add tm orelse can ring_dest_sub tm
+            orelse can ring_dest_mul tm
+    then grobvars (dest_arg1 tm) (grobvars (dest_arg tm) acc)
+    else if can field_dest_inv tm
+         then
+          let val gvs = grobvars (dest_arg tm) [] 
+          in if null gvs then acc else tm::acc
+          end
+    else if can field_dest_div tm then
+         let val lvs = grobvars (dest_arg1 tm) acc
+             val gvs = grobvars (dest_arg tm) []
+          in if null gvs then lvs else tm::acc
+          end 
+    else tm::acc ;
+
+fun grobify_term vars tm =
+((if not (member (op aconvc) vars tm) then raise CTERM ("Not a variable", [tm]) else
+     [(rat_1,map (fn i => if i aconvc tm then 1 else 0) vars)])
+handle  CTERM _ =>
+ ((let val x = dest_const tm
+ in if x =/ rat_0 then [] else [(x,map (fn v => 0) vars)]
+ end)
+ handle ERROR _ =>
+  ((grob_neg(grobify_term vars (ring_dest_neg tm)))
+  handle CTERM _ =>
+   (
+   (grob_inv(grobify_term vars (field_dest_inv tm)))
+   handle CTERM _ => 
+    ((let val (l,r) = ring_dest_add tm
+    in grob_add (grobify_term vars l) (grobify_term vars r)
+    end)
+    handle CTERM _ =>
+     ((let val (l,r) = ring_dest_sub tm
+     in grob_sub (grobify_term vars l) (grobify_term vars r)
+     end)
+     handle  CTERM _ =>
+      ((let val (l,r) = ring_dest_mul tm
+      in grob_mul (grobify_term vars l) (grobify_term vars r)
+      end)
+       handle CTERM _ =>
+        (  (let val (l,r) = field_dest_div tm
+          in grob_div (grobify_term vars l) (grobify_term vars r)
+          end)
+         handle CTERM _ =>
+          ((let val (l,r) = ring_dest_pow tm
+          in grob_pow vars (grobify_term vars l) ((term_of #> HOLogic.dest_number #> snd) r)
+          end)
+           handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
+val eq_tm = idom_thm |> concl |> dest_arg |> dest_arg |> dest_fun2;
+val dest_eq = dest_binary eq_tm;
+
+fun grobify_equation vars tm =
+    let val (l,r) = dest_binary eq_tm tm
+    in grob_sub (grobify_term vars l) (grobify_term vars r)
+    end;
+
+fun grobify_equations tm =
+ let
+  val cjs = conjs tm
+  val  rawvars = fold_rev (fn eq => fn a =>
+                                       grobvars (dest_arg1 eq) (grobvars (dest_arg eq) a)) cjs []
+  val vars = sort (fn (x, y) => Term_Ord.term_ord(term_of x,term_of y))
+                  (distinct (op aconvc) rawvars)
+ in (vars,map (grobify_equation vars) cjs)
+ end;
+
+val holify_polynomial =
+ let fun holify_varpow (v,n) =
+  if n = 1 then v else ring_mk_pow v (Numeral.mk_cnumber @{ctyp "nat"} n)  (* FIXME *)
+ fun holify_monomial vars (c,m) =
+  let val xps = map holify_varpow (filter (fn (_,n) => n <> 0) (vars ~~ m))
+   in end_itlist ring_mk_mul (mk_const c :: xps)
+  end
+ fun holify_polynomial vars p =
+     if null p then mk_const (rat_0)
+     else end_itlist ring_mk_add (map (holify_monomial vars) p)
+ in holify_polynomial
+ end ;
+val idom_rule = simplify (HOL_basic_ss addsimps [idom_thm]);
+fun prove_nz n = eqF_elim
+                 (ring_eq_conv(mk_binop eq_tm (mk_const n) (mk_const(rat_0))));
+val neq_01 = prove_nz (rat_1);
+fun neq_rule n th = [prove_nz n, th] MRS neq_thm;
+fun mk_add th1 = combination(arg_cong_rule ring_add_tm th1);
+
+fun refute tm =
+ if tm aconvc false_tm then assume_Trueprop tm else
+ ((let
+   val (nths0,eths0) = List.partition (is_neg o concl) (HOLogic.conj_elims (assume_Trueprop tm))
+   val  nths = filter (is_eq o dest_arg o concl) nths0
+   val eths = filter (is_eq o concl) eths0
+  in
+   if null eths then
+    let
+      val th1 = end_itlist (fn th1 => fn th2 => idom_rule(HOLogic.conj_intr th1 th2)) nths
+      val th2 = Conv.fconv_rule
+                ((arg_conv #> arg_conv)
+                     (binop_conv ring_normalize_conv)) th1
+      val conc = th2 |> concl |> dest_arg
+      val (l,r) = conc |> dest_eq
+    in implies_intr (mk_comb cTrp tm)
+                    (equal_elim (arg_cong_rule cTrp (eqF_intr th2))
+                           (reflexive l |> mk_object_eq))
+    end
+   else
+   let
+    val (vars,l,cert,noteqth) =(
+     if null nths then
+      let val (vars,pols) = grobify_equations(list_mk_conj(map concl eths))
+          val (l,cert) = grobner_weak vars pols
+      in (vars,l,cert,neq_01)
+      end
+     else
+      let
+       val nth = end_itlist (fn th1 => fn th2 => idom_rule(HOLogic.conj_intr th1 th2)) nths
+       val (vars,pol::pols) =
+          grobify_equations(list_mk_conj(dest_arg(concl nth)::map concl eths))
+       val (deg,l,cert) = grobner_strong vars pols pol
+       val th1 = Conv.fconv_rule((arg_conv o arg_conv)(binop_conv ring_normalize_conv)) nth
+       val th2 = funpow deg (idom_rule o HOLogic.conj_intr th1) neq_01
+      in (vars,l,cert,th2)
+      end)
+    val cert_pos = map (fn (i,p) => (i,filter (fn (c,m) => c >/ rat_0) p)) cert
+    val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (minus_rat c,m))
+                                            (filter (fn (c,m) => c </ rat_0) p))) cert
+    val  herts_pos = map (fn (i,p) => (i,holify_polynomial vars p)) cert_pos
+    val  herts_neg = map (fn (i,p) => (i,holify_polynomial vars p)) cert_neg
+    fun thm_fn pols =
+        if null pols then reflexive(mk_const rat_0) else
+        end_itlist mk_add
+            (map (fn (i,p) => arg_cong_rule (mk_comb ring_mul_tm p)
+              (nth eths i |> mk_meta_eq)) pols)
+    val th1 = thm_fn herts_pos
+    val th2 = thm_fn herts_neg
+    val th3 = HOLogic.conj_intr(mk_add (symmetric th1) th2 |> mk_object_eq) noteqth
+    val th4 = Conv.fconv_rule ((arg_conv o arg_conv o binop_conv) ring_normalize_conv)
+                               (neq_rule l th3)
+    val (l,r) = dest_eq(dest_arg(concl th4))
+   in implies_intr (mk_comb cTrp tm)
+                        (equal_elim (arg_cong_rule cTrp (eqF_intr th4))
+                   (reflexive l |> mk_object_eq))
+   end
+  end) handle ERROR _ => raise CTERM ("Gorbner-refute: unable to refute",[tm]))
+
+fun ring tm =
+ let
+  fun mk_forall x p =
+      mk_comb (cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (cabs x p)
+  val avs = add_cterm_frees tm []
+  val P' = fold mk_forall avs tm
+  val th1 = initial_conv(mk_neg P')
+  val (evs,bod) = strip_exists(concl th1) in
+   if is_forall bod then raise CTERM("ring: non-universal formula",[tm])
+   else
+   let
+    val th1a = weak_dnf_conv bod
+    val boda = concl th1a
+    val th2a = refute_disj refute boda
+    val th2b = [mk_object_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans
+    val th2 = fold (fn v => fn th => (forall_intr v th) COMP allI) evs (th2b RS PFalse)
+    val th3 = equal_elim
+                (Simplifier.rewrite (HOL_basic_ss addsimps [not_ex RS sym])
+                          (th2 |> cprop_of)) th2
+    in specl avs
+             ([[[mk_object_eq th1, th3 RS PFalse'] MRS trans] MRS PFalse] MRS notnotD)
+   end
+ end
+fun ideal tms tm ord =
+ let
+  val rawvars = fold_rev grobvars (tm::tms) []
+  val vars = sort ord (distinct (fn (x,y) => (term_of x) aconv (term_of y)) rawvars)
+  val pols = map (grobify_term vars) tms
+  val pol = grobify_term vars tm
+  val cert = grobner_ideal vars pols pol
+ in map_range (fn n => these (AList.lookup (op =) cert n) |> holify_polynomial vars)
+   (length pols)
+ end
+
+fun poly_eq_conv t = 
+ let val (a,b) = Thm.dest_binop t
+ in fconv_rule (arg_conv (arg1_conv ring_normalize_conv)) 
+     (instantiate' [] [SOME a, SOME b] idl_sub)
+ end
+ val poly_eq_simproc = 
+  let 
+   fun proc phi  ss t = 
+    let val th = poly_eq_conv t
+    in if Thm.is_reflexive th then NONE else SOME th
+    end
+   in make_simproc {lhss = [Thm.lhs_of idl_sub], 
+                name = "poly_eq_simproc", proc = proc, identifier = []}
+   end;
+  val poly_eq_ss = HOL_basic_ss addsimps @{thms simp_thms}
+                        addsimprocs [poly_eq_simproc]
+
+ local
+  fun is_defined v t =
+  let 
+   val mons = striplist(dest_binary ring_add_tm) t 
+  in member (op aconvc) mons v andalso 
+    forall (fn m => v aconvc m 
+          orelse not(member (op aconvc) (Thm.add_cterm_frees m []) v)) mons
+  end
+
+  fun isolate_variable vars tm =
+  let 
+   val th = poly_eq_conv tm
+   val th' = (sym_conv then_conv poly_eq_conv) tm
+   val (v,th1) = 
+   case find_first(fn v=> is_defined v (Thm.dest_arg1 (Thm.rhs_of th))) vars of
+    SOME v => (v,th')
+   | NONE => (the (find_first 
+          (fn v => is_defined v (Thm.dest_arg1 (Thm.rhs_of th'))) vars) ,th)
+   val th2 = transitive th1 
+        (instantiate' []  [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v] 
+          idl_add0)
+   in fconv_rule(funpow 2 arg_conv ring_normalize_conv) th2
+   end
+ in
+ fun unwind_polys_conv tm =
+ let 
+  val (vars,bod) = strip_exists tm
+  val cjs = striplist (dest_binary @{cterm "op &"}) bod
+  val th1 = (the (get_first (try (isolate_variable vars)) cjs) 
+             handle Option => raise CTERM ("unwind_polys_conv",[tm]))
+  val eq = Thm.lhs_of th1
+  val bod' = list_mk_binop @{cterm "op &"} (eq::(remove op aconvc eq cjs))
+  val th2 = conj_ac_rule (mk_eq bod bod')
+  val th3 = transitive th2 
+         (Drule.binop_cong_rule @{cterm "op &"} th1 
+                (reflexive (Thm.dest_arg (Thm.rhs_of th2))))
+  val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3))
+  val vars' = (remove op aconvc v vars) @ [v]
+  val th4 = fconv_rule (arg_conv simp_ex_conv) (mk_exists v th3)
+  val th5 = ex_eq_conv (mk_eq tm (fold mk_ex (remove op aconvc v vars) (Thm.lhs_of th4)))
+ in transitive th5 (fold mk_exists (remove op aconvc v vars) th4)
+ end;
+end
+
+local
+ fun scrub_var v m =
+  let 
+   val ps = striplist ring_dest_mul m 
+   val ps' = remove op aconvc v ps
+  in if null ps' then one_tm else fold1 ring_mk_mul ps'
+  end
+ fun find_multipliers v mons =
+  let 
+   val mons1 = filter (fn m => free_in v m) mons 
+   val mons2 = map (scrub_var v) mons1 
+   in  if null mons2 then zero_tm else fold1 ring_mk_add mons2
+  end
+
+ fun isolate_monomials vars tm =
+ let 
+  val (cmons,vmons) =
+    List.partition (fn m => null (inter (op aconvc) vars (frees m)))
+                   (striplist ring_dest_add tm)
+  val cofactors = map (fn v => find_multipliers v vmons) vars
+  val cnc = if null cmons then zero_tm
+             else Thm.capply ring_neg_tm
+                    (list_mk_binop ring_add_tm cmons) 
+  in (cofactors,cnc)
+  end;
+
+fun isolate_variables evs ps eq =
+ let 
+  val vars = filter (fn v => free_in v eq) evs
+  val (qs,p) = isolate_monomials vars eq
+  val rs = ideal (qs @ ps) p 
+              (fn (s,t) => Term_Ord.term_ord (term_of s, term_of t))
+ in (eq, take (length qs) rs ~~ vars)
+ end;
+ fun subst_in_poly i p = Thm.rhs_of (ring_normalize_conv (vsubst i p));
+in
+ fun solve_idealism evs ps eqs =
+  if null evs then [] else
+  let 
+   val (eq,cfs) = get_first (try (isolate_variables evs ps)) eqs |> the
+   val evs' = subtract op aconvc evs (map snd cfs)
+   val eqs' = map (subst_in_poly cfs) (remove op aconvc eq eqs)
+  in cfs @ solve_idealism evs' ps eqs'
+  end;
+end;
+
+
+in {ring_conv = ring, simple_ideal = ideal, multi_ideal = solve_idealism, 
+    poly_eq_ss = poly_eq_ss, unwind_conv = unwind_polys_conv}
+end;
+
+
+fun find_term bounds tm =
+  (case term_of tm of
+    Const ("op =", T) $ _ $ _ =>
+      if domain_type T = HOLogic.boolT then find_args bounds tm
+      else dest_arg tm
+  | Const ("Not", _) $ _ => find_term bounds (dest_arg tm)
+  | Const ("All", _) $ _ => find_body bounds (dest_arg tm)
+  | Const ("Ex", _) $ _ => find_body bounds (dest_arg tm)
+  | Const ("op &", _) $ _ $ _ => find_args bounds tm
+  | Const ("op |", _) $ _ $ _ => find_args bounds tm
+  | Const ("op -->", _) $ _ $ _ => find_args bounds tm
+  | @{term "op ==>"} $_$_ => find_args bounds tm
+  | Const("op ==",_)$_$_ => find_args bounds tm
+  | @{term Trueprop}$_ => find_term bounds (dest_arg tm)
+  | _ => raise TERM ("find_term", []))
+and find_args bounds tm =
+  let val (t, u) = Thm.dest_binop tm
+  in (find_term bounds t handle TERM _ => find_term bounds u) end
+and find_body bounds b =
+  let val (_, b') = dest_abs (SOME (Name.bound bounds)) b
+  in find_term (bounds + 1) b' end;
+
+
+fun get_ring_ideal_convs ctxt form = 
+ case try (find_term 0) form of
+  NONE => NONE
+| SOME tm =>
+  (case Semiring_Normalizer.match ctxt tm of
+    NONE => NONE
+  | SOME (res as (theory, {is_const, dest_const, 
+          mk_const, conv = ring_eq_conv})) =>
+     SOME (ring_and_ideal_conv theory
+          dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
+          (Semiring_Normalizer.semiring_normalize_wrapper ctxt res)))
+
+fun ring_solve ctxt form =
+  (case try (find_term 0 (* FIXME !? *)) form of
+    NONE => reflexive form
+  | SOME tm =>
+      (case Semiring_Normalizer.match ctxt tm of
+        NONE => reflexive form
+      | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
+        #ring_conv (ring_and_ideal_conv theory
+          dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
+          (Semiring_Normalizer.semiring_normalize_wrapper ctxt res)) form));
+
+fun presimplify ctxt add_thms del_thms = asm_full_simp_tac (Simplifier.context ctxt
+  (HOL_basic_ss addsimps (Algebra_Simplification.get ctxt) delsimps del_thms addsimps add_thms));
+
+fun ring_tac add_ths del_ths ctxt =
+  Object_Logic.full_atomize_tac
+  THEN' presimplify ctxt add_ths del_ths
+  THEN' CSUBGOAL (fn (p, i) =>
+    rtac (let val form = Object_Logic.dest_judgment p
+          in case get_ring_ideal_convs ctxt form of
+           NONE => reflexive form
+          | SOME thy => #ring_conv thy form
+          end) i
+      handle TERM _ => no_tac
+        | CTERM _ => no_tac
+        | THM _ => no_tac);
+
+local
+ fun lhs t = case term_of t of
+  Const("op =",_)$_$_ => Thm.dest_arg1 t
+ | _=> raise CTERM ("ideal_tac - lhs",[t])
+ fun exitac NONE = no_tac
+   | exitac (SOME y) = rtac (instantiate' [SOME (ctyp_of_term y)] [NONE,SOME y] exI) 1
+in 
+fun ideal_tac add_ths del_ths ctxt = 
+  presimplify ctxt add_ths del_ths
+ THEN'
+ CSUBGOAL (fn (p, i) =>
+  case get_ring_ideal_convs ctxt p of
+   NONE => no_tac
+ | SOME thy => 
+  let
+   fun poly_exists_tac {asms = asms, concl = concl, prems = prems,
+            params = params, context = ctxt, schematics = scs} = 
+    let
+     val (evs,bod) = strip_exists (Thm.dest_arg concl)
+     val ps = map_filter (try (lhs o Thm.dest_arg)) asms 
+     val cfs = (map swap o #multi_ideal thy evs ps) 
+                   (map Thm.dest_arg1 (conjuncts bod))
+     val ws = map (exitac o AList.lookup op aconvc cfs) evs
+    in EVERY (rev ws) THEN Method.insert_tac prems 1 
+        THEN ring_tac add_ths del_ths ctxt 1
+   end
+  in  
+     clarify_tac @{claset} i 
+     THEN Object_Logic.full_atomize_tac i 
+     THEN asm_full_simp_tac (Simplifier.context ctxt (#poly_eq_ss thy)) i 
+     THEN clarify_tac @{claset} i 
+     THEN (REPEAT (CONVERSION (#unwind_conv thy) i))
+     THEN SUBPROOF poly_exists_tac ctxt i
+  end
+ handle TERM _ => no_tac
+     | CTERM _ => no_tac
+     | THM _ => no_tac); 
+end;
+
+fun algebra_tac add_ths del_ths ctxt i = 
+ ring_tac add_ths del_ths ctxt i ORELSE ideal_tac add_ths del_ths ctxt i
+ 
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
+val addN = "add"
+val delN = "del"
+val any_keyword = keyword addN || keyword delN
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+
+in
+
+val algebra_method = ((Scan.optional (keyword addN |-- thms) []) -- 
+   (Scan.optional (keyword delN |-- thms) [])) >>
+  (fn (add_ths, del_ths) => fn ctxt =>
+       SIMPLE_METHOD' (algebra_tac add_ths del_ths ctxt))
+
+end;
+
+end;