src/HOL/Tools/groebner.ML

include doc-src as component, and thus its sessions defined in ROOT;

(*  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 : 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
end

structure Groebner : GROEBNER =
struct

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 poly_eq p1 p2 =
  eq_list (fn ((c1, m1), (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(@{const_name HOL.disj},_)$l$r =>
   Drule.compose_single
    (refute_disj rfn (Thm.dest_arg tm), 2,
      Drule.compose_single (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},_)$p) => true
    | _  => false;
fun is_eq t =
 case term_of t of
 (Const(@{const_name HOL.eq},_)$_$_) => 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 (x, T, p) =>
        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 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 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);

val list_conj = fold1 (fn c => fn c' => Thm.apply (Thm.apply @{cterm HOL.conj} c) c') ;

fun mk_conj_tab th =
 let fun h acc th =
   case prop_of th of
   @{term "Trueprop"}$(@{term HOL.conj}$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 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 mkeq s t = Thm.apply @{cterm Trueprop} (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)
 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.apply (Thm.lambda 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 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

 val ring_mk_neg = fn tm => Thm.apply (ring_neg_tm) (tm);
 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_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 (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 (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 |> 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 ;
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 = Thm.combination (Drule.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 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(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,r) = 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(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 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 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,r) = 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 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(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 => (Thm.forall_intr v th) COMP allI) evs (th2b RS PFalse)
    val th3 =
      Thm.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 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  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 = 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 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 => 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 vars' = (remove op aconvc v vars) @ [v]
  val th4 = Conv.fconv_rule (Conv.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 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 ring_solve ctxt form =
  (case try (find_term 0 (* FIXME !? *)) form of
    NONE => Thm.reflexive form
  | SOME tm =>
      (case Semiring_Normalizer.match ctxt tm of
        NONE => Thm.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 => Thm.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(@{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
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 @{context} i
     THEN Object_Logic.full_atomize_tac i
     THEN asm_full_simp_tac (Simplifier.context ctxt (#poly_eq_ss thy)) i
     THEN clarify_tac @{context} 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

end;