(*  Title:      HOL/Tools/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: Proof.context -> 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: Proof.context -> 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

 end

 structure Groebner : GROEBNER =

 struct

 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'])

 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;

 (* 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 (K 0) vars)]

   else grob_mul l (grob_pow vars l (n - 1));

 (* 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 (_,m1) (_,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),_) => ([],h)

   | (_,([],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 ((_,m1),_) ((_,m2),_) = morder_lt m1 m2;

 fun poly_lt  p q =

   case (p,q) of

     (_,[]) => false

   | ([],_) => 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 poly_eq p1 p2 =

   eq_list (fn ((c1, m1), (c2, m2)) => c1 =/ c2 andalso (m1: int list) = m2) (p1, p2);

 fun memx ((p1,_),(p2,_)) 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,_)) = 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,_) => 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 (K 0) vars))]

         val grob_1 = [(rat_1,(map (K 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(@{const_name HOL.disj},_)$_$_ =>

    Drule.compose

     (refute_disj rfn (Thm.dest_arg tm), 2,

       Drule.compose (refute_disj rfn (Thm.dest_arg1 tm), 2, disjE))

   | _ => rfn tm ;

 val notnotD = @{thm notnotD};

 fun mk_binop ct x y = Thm.apply (Thm.apply ct x) y

 fun is_neg t =

     case term_of t of

       (Const(@{const_name Not},_)$_) => true

     | _  => false;

 fun is_eq t =

  case term_of t of

  (Const(@{const_name HOL.eq},_)$_$_) => 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 (@{const_name Ex}, _) $ Abs _ =>

         h (Thm.dest_abs NONE (Thm.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(@{const_name All},_)$Abs(_,_,_)) => true

 | _ => false;

 val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;

 val nnf_simps = @{thms nnf_simps};

 fun weak_dnf_conv ctxt =

   Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms weak_dnf_simps});

 val initial_ss =

   simpset_of (put_simpset HOL_basic_ss @{context}

     addsimps nnf_simps

     addsimps [not_all, not_ex]

     addsimps map (fn th => th RS sym) (@{thms ex_simps} @ @{thms all_simps}));

 fun initial_conv ctxt =

   Simplifier.rewrite (put_simpset initial_ss ctxt);

 val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));

 val cTrp = @{cterm "Trueprop"};

 val cConj = @{cterm HOL.conj};

 val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});

 val assume_Trueprop = Thm.apply cTrp #> Thm.assume;

 val list_mk_conj = list_mk_binop cConj;

 val conjs = list_dest_binop cConj;

 val mk_neg = Thm.apply 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.apply (Thm.apply 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 HOL.conj}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))

 | _ => [ct];

 fun fold1 f = foldr1 (uncurry f);

 fun mk_conj_tab th =

  let fun h acc th =

    case prop_of th of

    @{term "Trueprop"}$(@{term HOL.conj}$_$_) =>

      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 HOL.conj}$_$_) = 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 (Thm.assume (Thm.apply @{cterm Trueprop} l))

   val ctabr = mk_conj_tab (Thm.assume (Thm.apply @{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 Thm.implies_elim (Thm.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 = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}

  fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)

 fun choose v th th' = case concl_of th of

   @{term Trueprop} $ (Const(@{const_name 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 = Conv.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.apply @{cterm Trueprop} (Thm.apply p v))

     val th1 = Thm.forall_intr v (Thm.implies_intr pv th')

    in Thm.implies_elim (Thm.implies_elim th0 th) th1  end

 | _ => error ""  (* FIXME ? *)

 fun simple_choose v th =

    choose v (Thm.assume ((Thm.apply @{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.lambda v (Thm.dest_arg (Thm.cprop_of th))

   in Thm.implies_elim

     (Conv.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 = Thm.assume (Thm.apply @{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 Thm.implies_elim (Thm.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.apply (Thm.apply @{cterm "op == :: bool => _"} s) t

   fun mk_exists v th = Drule.arg_cong_rule (ext (ctyp_of_term v))

    (Thm.abstract_rule (getname v) v th)

  fun simp_ex_conv ctxt =

    Simplifier.rewrite (put_simpset HOL_basic_ss ctxt 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.apply (Thm.lambda v tm) t)

 in fold vsubst end;

 (** main **)

 fun ring_and_ideal_conv

   {vars = _, semiring = (sr_ops, _), ring = (r_ops, _),

    field = (f_ops, _), 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 Thm.dest_fun2 [add_pat, mul_pat, pow_pat];

   val (ring_sub_tm, ring_neg_tm) =

     (case r_ops of

      [sub_pat, neg_pat] => (Thm.dest_fun2 sub_pat, Thm.dest_fun neg_pat)

     |_  => (@{cterm "True"}, @{cterm "True"}));

   val (field_div_tm, field_inv_tm) =

     (case f_ops of

        [div_pat, inv_pat] => (Thm.dest_fun2 div_pat, Thm.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) = Thm.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

  fun field_dest_inv t =

     let val (l,r) = Thm.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_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 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 (Thm.dest_arg tm) acc

     else if can ring_dest_pow tm then grobvars (Thm.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 (Thm.dest_arg1 tm) (grobvars (Thm.dest_arg tm) acc)

     else if can field_dest_inv tm

          then

           let val gvs = grobvars (Thm.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 (Thm.dest_arg1 tm) acc

              val gvs = grobvars (Thm.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 (K 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 |> Thm.dest_arg |> Thm.dest_arg |> Thm.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 (Thm.dest_arg1 eq) (grobvars (Thm.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 ;

 fun idom_rule ctxt = simplify (put_simpset HOL_basic_ss ctxt 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 = Thm.combination (Drule.arg_cong_rule ring_add_tm th1);

 fun refute ctxt 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 Thm.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 ctxt (HOLogic.conj_intr th1 th2)) nths

       val th2 =

         Conv.fconv_rule

           ((Conv.arg_conv #> Conv.arg_conv) (Conv.binop_conv ring_normalize_conv)) th1

       val conc = th2 |> concl |> Thm.dest_arg

       val (l,_) = conc |> dest_eq

     in Thm.implies_intr (Thm.apply cTrp tm)

                     (Thm.equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th2))

                            (Thm.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 ctxt (HOLogic.conj_intr th1 th2)) nths

        val (vars,pol::pols) =

           grobify_equations(list_mk_conj(Thm.dest_arg(concl nth)::map concl eths))

        val (deg,l,cert) = grobner_strong vars pols pol

        val th1 =

         Conv.fconv_rule ((Conv.arg_conv o Conv.arg_conv) (Conv.binop_conv ring_normalize_conv)) nth

        val th2 = funpow deg (idom_rule ctxt o HOLogic.conj_intr th1) neq_01

       in (vars,l,cert,th2)

       end)

     val cert_pos = map (fn (i,p) => (i,filter (fn (c,_) => c >/ rat_0) p)) cert

     val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (minus_rat c,m))

                                             (filter (fn (c,_) => 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 Thm.reflexive(mk_const rat_0) else

         end_itlist mk_add

             (map (fn (i,p) => Drule.arg_cong_rule (Thm.apply 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 (Thm.symmetric th1) th2 |> mk_object_eq) noteqth

     val th4 =

       Conv.fconv_rule ((Conv.arg_conv o Conv.arg_conv o Conv.binop_conv) ring_normalize_conv)

         (neq_rule l th3)

     val (l, _) = dest_eq(Thm.dest_arg(concl th4))

    in Thm.implies_intr (Thm.apply cTrp tm)

                         (Thm.equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th4))

                    (Thm.reflexive l |> mk_object_eq))

    end

   end) handle ERROR _ => raise CTERM ("Groebner-refute: unable to refute",[tm]))

 fun ring ctxt tm =

  let

   fun mk_forall x p =

     Thm.apply

       (Drule.cterm_rule (instantiate' [SOME (ctyp_of_term x)] [])

         @{cpat "All:: (?'a => bool) => _"}) (Thm.lambda x p)

   val avs = Thm.add_cterm_frees tm []

   val P' = fold mk_forall avs tm

   val th1 = initial_conv ctxt (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 ctxt bod

     val boda = concl th1a

     val th2a = refute_disj (refute ctxt) boda

     val th2b = [mk_object_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans

     val th2 = fold (fn v => fn th => (Thm.forall_intr v th) COMP allI) evs (th2b RS PFalse)

     val th3 =

       Thm.equal_elim

         (Simplifier.rewrite (put_simpset HOL_basic_ss ctxt 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 Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv ring_normalize_conv))

      (instantiate' [] [SOME a, SOME b] idl_sub)

  end

  val poly_eq_simproc =

   let

    fun proc phi ctxt 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 =

    simpset_of (put_simpset HOL_basic_ss @{context}

     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 = Thm.transitive th1

         (instantiate' []  [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v]

           idl_add0)

    in Conv.fconv_rule(funpow 2 Conv.arg_conv ring_normalize_conv) th2

    end

  in

  fun unwind_polys_conv ctxt tm =

  let

   val (vars,bod) = strip_exists tm

   val cjs = striplist (dest_binary @{cterm HOL.conj}) bod

   val th1 = (the (get_first (try (isolate_variable vars)) cjs)

              handle Option.Option => raise CTERM ("unwind_polys_conv",[tm]))

   val eq = Thm.lhs_of th1

   val bod' = list_mk_binop @{cterm HOL.conj} (eq::(remove op aconvc eq cjs))

   val th2 = conj_ac_rule (mk_eq bod bod')

   val th3 =

     Thm.transitive th2

       (Drule.binop_cong_rule @{cterm HOL.conj} th1

         (Thm.reflexive (Thm.dest_arg (Thm.rhs_of th2))))

   val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3))

   val th4 = Conv.fconv_rule (Conv.arg_conv (simp_ex_conv ctxt)) (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 Thm.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.apply 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 (@{const_name HOL.eq}, T) $ _ $ _ =>

       if domain_type T = HOLogic.boolT then find_args bounds tm

       else Thm.dest_arg tm

   | Const (@{const_name Not}, _) $ _ => find_term bounds (Thm.dest_arg tm)

   | Const (@{const_name All}, _) $ _ => find_body bounds (Thm.dest_arg tm)

   | Const (@{const_name Ex}, _) $ _ => find_body bounds (Thm.dest_arg tm)

   | Const (@{const_name HOL.conj}, _) $ _ $ _ => find_args bounds tm

   | Const (@{const_name HOL.disj}, _) $ _ $ _ => find_args bounds tm

   | Const (@{const_name HOL.implies}, _) $ _ $ _ => find_args bounds tm

   | @{term "op ==>"} $_$_ => find_args bounds tm

   | Const("op ==",_)$_$_ => find_args bounds tm

   | @{term Trueprop}$_ => find_term bounds (Thm.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') = Thm.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 presimplify ctxt add_thms del_thms =

   asm_full_simp_tac (put_simpset HOL_basic_ss ctxt

     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 => Thm.reflexive form

           | SOME thy => #ring_conv thy ctxt form

           end) i

       handle TERM _ => no_tac

         | CTERM _ => no_tac

         | THM _ => no_tac);

 local

  fun lhs t = case term_of t of

   Const(@{const_name HOL.eq},_)$_$_ => 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

  val claset = claset_of @{context}

 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 = _, context = ctxt, schematics = _} =

     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 (put_claset claset ctxt) i

      THEN Object_Logic.full_atomize_tac i

      THEN asm_full_simp_tac (put_simpset (#poly_eq_ss thy) ctxt) i

      THEN clarify_tac (put_claset claset ctxt) i

      THEN (REPEAT (CONVERSION (#unwind_conv thy ctxt) 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

 end;