(*  Title:      Provers/blast.ML

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1997  University of Cambridge

 Generic tableau prover with proof reconstruction

   SKOLEMIZES ReplaceI WRONGLY: allow new vars in prems, or forbid such rules??

   Needs explicit instantiation of assumptions?

 Given the typeargs system, constructor Const could be eliminated, with

 TConst replaced by a constructor that takes the typargs list as an argument.

 However, Const is heavily used for logical connectives.

 Blast_tac is often more powerful than fast_tac, but has some limitations.

 Blast_tac...

   * ignores wrappers (addss, addbefore, addafter, addWrapper, ...);

     this restriction is intrinsic

   * ignores elimination rules that don't have the correct format

         (conclusion must be a formula variable)

   * rules must not require higher-order unification, e.g. apply_type in ZF

     + message "Function Var's argument not a bound variable" relates to this

   * its proof strategy is more general but can actually be slower

 Known problems:

   "Recursive" chains of rules can sometimes exclude other unsafe formulae

         from expansion.  This happens because newly-created formulae always

         have priority over existing ones.  But obviously recursive rules

         such as transitivity are treated specially to prevent this.  Sometimes

         the formulae get into the wrong order (see WRONG below).

   With substition for equalities (hyp_subst_tac):

         When substitution affects a haz formula or literal, it is moved

         back to the list of safe formulae.

         But there's no way of putting it in the right place.  A "moved" or

         "no DETERM" flag would prevent proofs failing here.

 *)

 (*Should be a type abbreviation?*)

 type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net;

 signature BLAST_DATA =

   sig

   type claset

   val equality_name: string

   val not_name: string

   val notE              : thm           (* [| ~P;  P |] ==> R *)

   val ccontr            : thm

   val contr_tac         : int -> tactic

   val dup_intr          : thm -> thm

   val hyp_subst_tac     : bool -> int -> tactic

   val claset            : unit -> claset

   val rep_cs    : (* dependent on classical.ML *)

       claset -> {safeIs: thm list, safeEs: thm list,

                  hazIs: thm list, hazEs: thm list,

                  swrappers: (string * wrapper) list,

                  uwrappers: (string * wrapper) list,

                  safe0_netpair: netpair, safep_netpair: netpair,

                  haz_netpair: netpair, dup_netpair: netpair, xtra_netpair: ContextRules.netpair}

   val cla_modifiers: (Args.T list -> (Method.modifier * Args.T list)) list

   val cla_meth': (claset -> int -> tactic) -> thm list -> Proof.context -> Proof.method

   end;

 signature BLAST =

   sig

   type claset

   exception TRANS of string    (*reports translation errors*)

   datatype term =

       Const of string * term list

     | Skolem of string * term option ref list

     | Free  of string

     | Var   of term option ref

     | Bound of int

     | Abs   of string*term

     | $  of term*term;

   type branch

   val depth_tac         : claset -> int -> int -> tactic

   val depth_limit       : int Config.T

   val blast_tac         : claset -> int -> tactic

   val Blast_tac         : int -> tactic

   val setup             : theory -> theory

   (*debugging tools*)

   val stats             : bool ref

   val trace             : bool ref

   val fullTrace         : branch list list ref

   val fromType          : (indexname * term) list ref -> Term.typ -> term

   val fromTerm          : theory -> Term.term -> term

   val fromSubgoal       : theory -> Term.term -> term

   val instVars          : term -> (unit -> unit)

   val toTerm            : int -> term -> Term.term

   val readGoal          : theory -> string -> term

   val tryInThy          : theory -> int -> string ->

                   (int->tactic) list * branch list list * (int*int*exn) list

   val normBr            : branch -> branch

   end;

 functor BlastFun(Data: BLAST_DATA) : BLAST =

 struct

 type claset = Data.claset;

 val trace = ref false

 and stats = ref false;   (*for runtime and search statistics*)

 datatype term =

     Const  of string * term list  (*typargs constant--as a terms!*)

   | Skolem of string * term option ref list

   | Free   of string

   | Var    of term option ref

   | Bound  of int

   | Abs    of string*term

   | op $   of term*term;

 (*Pending formulae carry md (may duplicate) flags*)

 type branch =

     {pairs: ((term*bool) list * (*safe formulae on this level*)

                (term*bool) list) list,  (*haz formulae  on this level*)

      lits:   term list,                 (*literals: irreducible formulae*)

      vars:   term option ref list,      (*variables occurring in branch*)

      lim:    int};                      (*resource limit*)

 (* global state information *)

 datatype state = State of

  {thy: theory,

   fullTrace: branch list list ref,

   trail: term option ref list ref,

   ntrail: int ref,

   nclosed: int ref,

   ntried: int ref}

 fun reject_const thy c =

   is_some (Sign.const_type thy c) andalso

     error ("blast: theory contains illegal constant " ^ quote c);

 fun initialize thy =

  (reject_const thy "*Goal*";

   reject_const thy "*False*";

   State

    {thy = thy,

     fullTrace = ref [],

     trail = ref [],

     ntrail = ref 0,

     nclosed = ref 0,  (*branches closed: number of branches closed during the search*)

     ntried = ref 1}); (*branches tried: number of branches created by splitting (counting from 1)*)

 (** Basic syntactic operations **)

 fun is_Var (Var _) = true

   | is_Var _ = false;

 fun dest_Var (Var x) =  x;

 fun rand (f$x) = x;

 (* maps   (f, [t1,...,tn])  to  f(t1,...,tn) *)

 val list_comb : term * term list -> term = Library.foldl (op $);

 (* maps   f(t1,...,tn)  to  (f, [t1,...,tn]) ; naturally tail-recursive*)

 fun strip_comb u : term * term list =

     let fun stripc (f$t, ts) = stripc (f, t::ts)

         |   stripc  x =  x

     in  stripc(u,[])  end;

 (* maps   f(t1,...,tn)  to  f , which is never a combination *)

 fun head_of (f$t) = head_of f

   | head_of u = u;

 (** Particular constants **)

 fun negate P = Const (Data.not_name, []) $ P;

 fun isNot (Const (c, _) $ _) = c = Data.not_name

   | isNot _ = false;

 fun mkGoal P = Const ("*Goal*", []) $ P;

 fun isGoal (Const ("*Goal*", _) $ _) = true

   | isGoal _ = false;

 val TruepropC = ObjectLogic.judgment_name (the_context ());

 val TruepropT = Sign.the_const_type (the_context ()) TruepropC;

 fun mk_Trueprop t = Term.$ (Term.Const (TruepropC, TruepropT), t);

 fun strip_Trueprop (tm as Const (c, _) $ t) = if c = TruepropC then t else tm

   | strip_Trueprop tm = tm;

 (*** Dealing with overloaded constants ***)

 (*alist is a map from TVar names to Vars.  We need to unify the TVars

   faithfully in order to track overloading*)

 fun fromType alist (Term.Type(a,Ts)) = list_comb (Const (a, []), map (fromType alist) Ts)

   | fromType alist (Term.TFree(a,_)) = Free a

   | fromType alist (Term.TVar (ixn,_)) =

               (case (AList.lookup (op =) (!alist) ixn) of

                    NONE => let val t' = Var(ref NONE)

                            in  alist := (ixn, t') :: !alist;  t'

                            end

                  | SOME v => v)

 fun fromConst thy alist (a, T) =

   Const (a, map (fromType alist) (Sign.const_typargs thy (a, T)));

 (*Tests whether 2 terms are alpha-convertible; chases instantiations*)

 fun (Const (a, ts)) aconv (Const (b, us)) = a=b andalso aconvs (ts, us)

   | (Skolem (a,_)) aconv (Skolem (b,_)) = a=b  (*arglists must then be equal*)

   | (Free a)       aconv (Free b)       = a=b

   | (Var(ref(SOME t))) aconv u          = t aconv u

   | t          aconv (Var(ref(SOME u))) = t aconv u

   | (Var v)        aconv (Var w)        = v=w   (*both Vars are un-assigned*)

   | (Bound i)      aconv (Bound j)      = i=j

   | (Abs(_,t))     aconv (Abs(_,u))     = t aconv u

   | (f$t)          aconv (g$u)          = (f aconv g) andalso (t aconv u)

   | _ aconv _  =  false

 and aconvs ([], []) = true

   | aconvs (t :: ts, u :: us) = t aconv u andalso aconvs (ts, us)

   | aconvs _ = false;

 fun mem_term (_, [])     = false

   | mem_term (t, t'::ts) = t aconv t' orelse mem_term(t,ts);

 fun ins_term(t,ts) = if mem_term(t,ts) then ts else t :: ts;

 fun mem_var (v: term option ref, []) = false

   | mem_var (v, v'::vs)              = v=v' orelse mem_var(v,vs);

 fun ins_var(v,vs) = if mem_var(v,vs) then vs else v :: vs;

 (** Vars **)

 (*Accumulates the Vars in the term, suppressing duplicates*)

 fun add_term_vars (Skolem(a,args),      vars) = add_vars_vars(args,vars)

   | add_term_vars (Var (v as ref NONE), vars) = ins_var (v, vars)

   | add_term_vars (Var (ref (SOME u)), vars)  = add_term_vars(u,vars)

   | add_term_vars (Const (_,ts),        vars) = add_terms_vars(ts,vars)

   | add_term_vars (Abs (_,body),        vars) = add_term_vars(body,vars)

   | add_term_vars (f$t, vars) =  add_term_vars (f, add_term_vars(t, vars))

   | add_term_vars (_,   vars) = vars

 (*Term list version.  [The fold functionals are slow]*)

 and add_terms_vars ([],    vars) = vars

   | add_terms_vars (t::ts, vars) = add_terms_vars (ts, add_term_vars(t,vars))

 (*Var list version.*)

 and add_vars_vars ([],    vars) = vars

   | add_vars_vars (ref (SOME u) :: vs, vars) =

         add_vars_vars (vs, add_term_vars(u,vars))

   | add_vars_vars (v::vs, vars) =   (*v must be a ref NONE*)

         add_vars_vars (vs, ins_var (v, vars));

 (*Chase assignments in "vars"; return a list of unassigned variables*)

 fun vars_in_vars vars = add_vars_vars(vars,[]);

 (*increment a term's non-local bound variables

      inc is  increment for bound variables

      lev is  level at which a bound variable is considered 'loose'*)

 fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u

   | incr_bv (inc, lev, Abs(a,body)) = Abs(a, incr_bv(inc,lev+1,body))

   | incr_bv (inc, lev, f$t) = incr_bv(inc,lev,f) $ incr_bv(inc,lev,t)

   | incr_bv (inc, lev, u) = u;

 fun incr_boundvars  0  t = t

   | incr_boundvars inc t = incr_bv(inc,0,t);

 (*Accumulate all 'loose' bound vars referring to level 'lev' or beyond.

    (Bound 0) is loose at level 0 *)

 fun add_loose_bnos (Bound i, lev, js)   = if i<lev then js

                                           else insert (op =) (i - lev) js

   | add_loose_bnos (Abs (_,t), lev, js) = add_loose_bnos (t, lev+1, js)

   | add_loose_bnos (f$t, lev, js)       =

                 add_loose_bnos (f, lev, add_loose_bnos (t, lev, js))

   | add_loose_bnos (_, _, js)           = js;

 fun loose_bnos t = add_loose_bnos (t, 0, []);

 fun subst_bound (arg, t) : term =

   let fun subst (t as Bound i, lev) =

             if i<lev then  t    (*var is locally bound*)

             else  if i=lev then incr_boundvars lev arg

                            else Bound(i-1)  (*loose: change it*)

         | subst (Abs(a,body), lev) = Abs(a, subst(body,lev+1))

         | subst (f$t, lev) =  subst(f,lev)  $  subst(t,lev)

         | subst (t,lev)    = t

   in  subst (t,0)  end;

 (*Normalize...but not the bodies of ABSTRACTIONS*)

 fun norm t = case t of

     Skolem (a,args)      => Skolem(a, vars_in_vars args)

   | Const(a,ts)          => Const(a, map norm ts)

   | (Var (ref NONE))     => t

   | (Var (ref (SOME u))) => norm u

   | (f $ u) => (case norm f of

                     Abs(_,body) => norm (subst_bound (u, body))

                   | nf => nf $ norm u)

   | _ => t;

 (*Weak (one-level) normalize for use in unification*)

 fun wkNormAux t = case t of

     (Var v) => (case !v of

                     SOME u => wkNorm u

                   | NONE   => t)

   | (f $ u) => (case wkNormAux f of

                     Abs(_,body) => wkNorm (subst_bound (u, body))

                   | nf          => nf $ u)

   | Abs (a,body) =>     (*eta-contract if possible*)

         (case wkNormAux body of

              nb as (f $ t) =>

                  if member (op =) (loose_bnos f) 0 orelse wkNorm t <> Bound 0

                  then Abs(a,nb)

                  else wkNorm (incr_boundvars ~1 f)

            | nb => Abs (a,nb))

   | _ => t

 and wkNorm t = case head_of t of

     Const _        => t

   | Skolem(a,args) => t

   | Free _         => t

   | _              => wkNormAux t;

 (*Does variable v occur in u?  For unification.

   Dangling bound vars are also forbidden.*)

 fun varOccur v =

   let fun occL lev [] = false   (*same as (exists occ), but faster*)

         | occL lev (u::us) = occ lev u orelse occL lev us

       and occ lev (Var w) =

               v=w orelse

               (case !w of NONE   => false

                         | SOME u => occ lev u)

         | occ lev (Skolem(_,args)) = occL lev (map Var args)

             (*ignore Const, since term variables can't occur in types (?) *)

         | occ lev (Bound i)  = lev <= i

         | occ lev (Abs(_,u)) = occ (lev+1) u

         | occ lev (f$u)      = occ lev u  orelse  occ lev f

         | occ lev _          = false;

   in  occ 0  end;

 exception UNIFY;

 (*Restore the trail to some previous state: for backtracking*)

 fun clearTo (State {ntrail, trail, ...}) n =

     while !ntrail<>n do

         (hd(!trail) := NONE;

          trail := tl (!trail);

          ntrail := !ntrail - 1);

 (*First-order unification with bound variables.

   "vars" is a list of variables local to the rule and NOT to be put

         on the trail (no point in doing so)

 *)

 fun unify state (vars,t,u) =

     let val State {ntrail, trail, ...} = state

         val n = !ntrail

         fun update (t as Var v, u) =

             if t aconv u then ()

             else if varOccur v u then raise UNIFY

             else if mem_var(v, vars) then v := SOME u

                  else (*avoid updating Vars in the branch if possible!*)

                       if is_Var u andalso mem_var(dest_Var u, vars)

                       then dest_Var u := SOME t

                       else (v := SOME u;

                             trail := v :: !trail;  ntrail := !ntrail + 1)

         fun unifyAux (t,u) =

             case (wkNorm t,  wkNorm u) of

                 (nt as Var v,  nu) => update(nt,nu)

               | (nu,  nt as Var v) => update(nt,nu)

               | (Const(a,ats), Const(b,bts)) => if a=b then unifysAux(ats,bts)

                                                 else raise UNIFY

               | (Abs(_,t'),  Abs(_,u')) => unifyAux(t',u')

                     (*NB: can yield unifiers having dangling Bound vars!*)

               | (f$t',  g$u') => (unifyAux(f,g); unifyAux(t',u'))

               | (nt,  nu)    => if nt aconv nu then () else raise UNIFY

         and unifysAux ([], []) = ()

           | unifysAux (t :: ts, u :: us) = (unifyAux (t, u); unifysAux (ts, us))

           | unifysAux _ = raise UNIFY;

     in  (unifyAux(t,u); true) handle UNIFY => (clearTo state n; false)

     end;

 (*Convert from "real" terms to prototerms; eta-contract.

   Code is similar to fromSubgoal.*)

 fun fromTerm thy t =

   let val alistVar = ref []

       and alistTVar = ref []

       fun from (Term.Const aT) = fromConst thy alistTVar aT

         | from (Term.Free  (a,_)) = Free a

         | from (Term.Bound i)     = Bound i

         | from (Term.Var (ixn,T)) =

               (case (AList.lookup (op =) (!alistVar) ixn) of

                    NONE => let val t' = Var(ref NONE)

                            in  alistVar := (ixn, t') :: !alistVar;  t'

                            end

                  | SOME v => v)

         | from (Term.Abs (a,_,u)) =

               (case  from u  of

                 u' as (f $ Bound 0) =>

                   if member (op =) (loose_bnos f) 0 then Abs(a,u')

                   else incr_boundvars ~1 f

               | u' => Abs(a,u'))

         | from (Term.$ (f,u)) = from f $ from u

   in  from t  end;

 (*A debugging function: replaces all Vars by dummy Frees for visual inspection

   of whether they are distinct.  Function revert undoes the assignments.*)

 fun instVars t =

   let val name = ref "a"

       val updated = ref []

       fun inst (Const(a,ts)) = List.app inst ts

         | inst (Var(v as ref NONE)) = (updated := v :: (!updated);

                                        v       := SOME (Free ("?" ^ !name));

                                        name    := Symbol.bump_string (!name))

         | inst (Abs(a,t))    = inst t

         | inst (f $ u)       = (inst f; inst u)

         | inst _             = ()

       fun revert() = List.app (fn v => v:=NONE) (!updated)

   in  inst t; revert  end;

 (* A1==>...An==>B  goes to  [A1,...,An], where B is not an implication *)

 fun strip_imp_prems (Const ("==>", _) $ A $ B) = strip_Trueprop A :: strip_imp_prems B

   | strip_imp_prems _ = [];

 (* A1==>...An==>B  goes to B, where B is not an implication *)

 fun strip_imp_concl (Const ("==>", _) $ A $ B) = strip_imp_concl B

   | strip_imp_concl A = strip_Trueprop A;

 (*** Conversion of Elimination Rules to Tableau Operations ***)

 exception ElimBadConcl and ElimBadPrem;

 (*The conclusion becomes the goal/negated assumption *False*: delete it!

   If we don't find it then the premise is ill-formed and could cause

   PROOF FAILED*)

 fun delete_concl [] = raise ElimBadPrem

   | delete_concl (P :: Ps) =

       (case P of

         Const (c, _) $ Var (ref (SOME (Const ("*False*", _)))) =>

           if c = "*Goal*" orelse c = Data.not_name then Ps

           else P :: delete_concl Ps

       | _ => P :: delete_concl Ps);

 fun skoPrem vars (Const ("all", _) $ Abs (_, P)) =

         skoPrem vars (subst_bound (Skolem (gensym "S_", vars), P))

   | skoPrem vars P = P;

 fun convertPrem t =

     delete_concl (mkGoal (strip_imp_concl t) :: strip_imp_prems t);

 (*Expects elimination rules to have a formula variable as conclusion*)

 fun convertRule vars t =

   let val (P::Ps) = strip_imp_prems t

       val Var v   = strip_imp_concl t

   in  v := SOME (Const ("*False*", []));

       (P, map (convertPrem o skoPrem vars) Ps)

   end

   handle Bind => raise ElimBadConcl;

 (*Like dup_elim, but puts the duplicated major premise FIRST*)

 fun rev_dup_elim th = (th RSN (2, revcut_rl)) |> assumption 2 |> Seq.hd;

 (*Rotate the assumptions in all new subgoals for the LIFO discipline*)

 local

   (*Count new hyps so that they can be rotated*)

   fun nNewHyps []                         = 0

     | nNewHyps (Const ("*Goal*", _) $ _ :: Ps) = nNewHyps Ps

     | nNewHyps (P::Ps)                    = 1 + nNewHyps Ps;

   fun rot_tac [] i st      = Seq.single st

     | rot_tac (0::ks) i st = rot_tac ks (i+1) st

     | rot_tac (k::ks) i st = rot_tac ks (i+1) (rotate_rule (~k) i st);

 in

 fun rot_subgoals_tac (rot, rl) =

      rot_tac (if rot then map nNewHyps rl else [])

 end;

 fun TRACE rl tac st i = if !trace then (Display.prth rl; tac st i) else tac st i;

 (*Resolution/matching tactics: if upd then the proof state may be updated.

   Matching makes the tactics more deterministic in the presence of Vars.*)

 fun emtac upd rl = TRACE rl (if upd then etac rl else ematch_tac [rl]);

 fun rmtac upd rl = TRACE rl (if upd then rtac rl else match_tac [rl]);

 (*Tableau rule from elimination rule.

   Flag "upd" says that the inference updated the branch.

   Flag "dup" requests duplication of the affected formula.*)

 fun fromRule thy vars rl =

   let val trl = rl |> Thm.prop_of |> fromTerm thy |> convertRule vars

       fun tac (upd, dup,rot) i =

         emtac upd (if dup then rev_dup_elim rl else rl) i

         THEN

         rot_subgoals_tac (rot, #2 trl) i

   in Option.SOME (trl, tac) end

   handle ElimBadPrem => (*reject: prems don't preserve conclusion*)

             (warning("Ignoring weak elimination rule\n" ^ Display.string_of_thm rl);

              Option.NONE)

        | ElimBadConcl => (*ignore: conclusion is not just a variable*)

            (if !trace then (warning("Ignoring ill-formed elimination rule:\n" ^

                        "conclusion should be a variable\n" ^ Display.string_of_thm rl))

             else ();

             Option.NONE);

 (*** Conversion of Introduction Rules ***)

 fun convertIntrPrem t = mkGoal (strip_imp_concl t) :: strip_imp_prems t;

 fun convertIntrRule vars t =

   let val Ps = strip_imp_prems t

       val P  = strip_imp_concl t

   in  (mkGoal P, map (convertIntrPrem o skoPrem vars) Ps)

   end;

 (*Tableau rule from introduction rule.

   Flag "upd" says that the inference updated the branch.

   Flag "dup" requests duplication of the affected formula.

   Since haz rules are now delayed, "dup" is always FALSE for

   introduction rules.*)

 fun fromIntrRule thy vars rl =

   let val trl = rl |> Thm.prop_of |> fromTerm thy |> convertIntrRule vars

       fun tac (upd,dup,rot) i =

          rmtac upd (if dup then Data.dup_intr rl else rl) i

          THEN

          rot_subgoals_tac (rot, #2 trl) i

   in (trl, tac) end;

 val dummyVar = Term.Var (("etc",0), dummyT);

 (*Convert from prototerms to ordinary terms with dummy types

   Ignore abstractions; identify all Vars; STOP at given depth*)

 fun toTerm 0 _             = dummyVar

   | toTerm d (Const(a,_))  = Term.Const (a,dummyT)  (*no need to convert typargs*)

   | toTerm d (Skolem(a,_)) = Term.Const (a,dummyT)

   | toTerm d (Free a)      = Term.Free  (a,dummyT)

   | toTerm d (Bound i)     = Term.Bound i

   | toTerm d (Var _)       = dummyVar

   | toTerm d (Abs(a,_))    = dummyVar

   | toTerm d (f $ u)       = Term.$ (toTerm d f, toTerm (d-1) u);

 fun netMkRules thy P vars (nps: netpair list) =

   case P of

       (Const ("*Goal*", _) $ G) =>

         let val pG = mk_Trueprop (toTerm 2 G)

             val intrs = maps (fn (inet,_) => Net.unify_term inet pG) nps

         in  map (fromIntrRule thy vars o #2) (Tactic.orderlist intrs)  end

     | _ =>

         let val pP = mk_Trueprop (toTerm 3 P)

             val elims = maps (fn (_,enet) => Net.unify_term enet pP) nps

         in  map_filter (fromRule thy vars o #2) (Tactic.orderlist elims)  end;

 (*Normalize a branch--for tracing*)

 fun norm2 (G,md) = (norm G, md);

 fun normLev (Gs,Hs) = (map norm2 Gs, map norm2 Hs);

 fun normBr {pairs, lits, vars, lim} =

      {pairs = map normLev pairs,

       lits  = map norm lits,

       vars  = vars,

       lim   = lim};

 val dummyTVar = Term.TVar(("a",0), []);

 val dummyVar2 = Term.Var(("var",0), dummyT);

 (*convert blast_tac's type representation to real types for tracing*)

 fun showType (Free a)  = Term.TFree (a,[])

   | showType (Var _)   = dummyTVar

   | showType t         =

       (case strip_comb t of

            (Const (a, _), us) => Term.Type(a, map showType us)

          | _ => dummyT);

 (*Display top-level overloading if any*)

 fun topType thy (Const (c, ts)) = SOME (Sign.const_instance thy (c, map showType ts))

   | topType thy (Abs(a,t)) = topType thy t

   | topType thy (f $ u) = (case topType thy f of NONE => topType thy u | some => some)

   | topType _ _ = NONE;

 (*Convert from prototerms to ordinary terms with dummy types for tracing*)

 fun showTerm d (Const (a,_)) = Term.Const (a,dummyT)

   | showTerm d (Skolem(a,_)) = Term.Const (a,dummyT)

   | showTerm d (Free a)      = Term.Free  (a,dummyT)

   | showTerm d (Bound i)     = Term.Bound i

   | showTerm d (Var(ref(SOME u))) = showTerm d u

   | showTerm d (Var(ref NONE))    = dummyVar2

   | showTerm d (Abs(a,t))    = if d=0 then dummyVar

                                else Term.Abs(a, dummyT, showTerm (d-1) t)

   | showTerm d (f $ u)       = if d=0 then dummyVar

                                else Term.$ (showTerm d f, showTerm (d-1) u);

 fun string_of thy d t = Syntax.string_of_term_global thy (showTerm d t);

 (*Convert a Goal to an ordinary Not.  Used also in dup_intr, where a goal like

   Ex(P) is duplicated as the assumption ~Ex(P). *)

 fun negOfGoal (Const ("*Goal*", _) $ G) = negate G

   | negOfGoal G = G;

 fun negOfGoal2 (G,md) = (negOfGoal G, md);

 (*Converts all Goals to Nots in the safe parts of a branch.  They could

   have been moved there from the literals list after substitution (equalSubst).

   There can be at most one--this function could be made more efficient.*)

 fun negOfGoals pairs = map (fn (Gs,haz) => (map negOfGoal2 Gs, haz)) pairs;

 (*Tactic.  Convert *Goal* to negated assumption in FIRST position*)

 fun negOfGoal_tac i = TRACE Data.ccontr (rtac Data.ccontr) i THEN

                       rotate_tac ~1 i;

 fun traceTerm thy t =

   let val t' = norm (negOfGoal t)

       val stm = string_of thy 8 t'

   in

       case topType thy t' of

           NONE   => stm   (*no type to attach*)

         | SOME T => stm ^ "\t:: " ^ Syntax.string_of_typ_global thy T

   end;

 (*Print tracing information at each iteration of prover*)

 fun tracing (State {thy, fullTrace, ...}) brs =

   let fun printPairs (((G,_)::_,_)::_)  = Output.immediate_output(traceTerm thy G)

         | printPairs (([],(H,_)::_)::_) = Output.immediate_output(traceTerm thy H ^ "\t (Unsafe)")

         | printPairs _                 = ()

       fun printBrs (brs0 as {pairs, lits, lim, ...} :: brs) =

             (fullTrace := brs0 :: !fullTrace;

              List.app (fn _ => Output.immediate_output "+") brs;

              Output.immediate_output (" [" ^ Int.toString lim ^ "] ");

              printPairs pairs;

              writeln"")

   in if !trace then printBrs (map normBr brs) else ()

   end;

 fun traceMsg s = if !trace then writeln s else ();

 (*Tracing: variables updated in the last branch operation?*)

 fun traceVars (State {thy, ntrail, trail, ...}) ntrl =

   if !trace then

       (case !ntrail-ntrl of

             0 => ()

           | 1 => Output.immediate_output"\t1 variable UPDATED:"

           | n => Output.immediate_output("\t" ^ Int.toString n ^ " variables UPDATED:");

        (*display the instantiations themselves, though no variable names*)

        List.app (fn v => Output.immediate_output("   " ^ string_of thy 4 (the (!v))))

            (List.take(!trail, !ntrail-ntrl));

        writeln"")

     else ();

 (*Tracing: how many new branches are created?*)

 fun traceNew prems =

     if !trace then

         case length prems of

             0 => Output.immediate_output"branch closed by rule"

           | 1 => Output.immediate_output"branch extended (1 new subgoal)"

           | n => Output.immediate_output("branch split: "^ Int.toString n ^ " new subgoals")

     else ();

 (*** Code for handling equality: naive substitution, like hyp_subst_tac ***)

 (*Replace the ATOMIC term "old" by "new" in t*)

 fun subst_atomic (old,new) t =

     let fun subst (Var(ref(SOME u))) = subst u

           | subst (Abs(a,body))      = Abs(a, subst body)

           | subst (f$t)              = subst f $ subst t

           | subst t                  = if t aconv old then new else t

     in  subst t  end;

 (*Eta-contract a term from outside: just enough to reduce it to an atom*)

 fun eta_contract_atom (t0 as Abs(a, body)) =

       (case  eta_contract2 body  of

         f $ Bound 0 => if member (op =) (loose_bnos f) 0 then t0

                        else eta_contract_atom (incr_boundvars ~1 f)

       | _ => t0)

   | eta_contract_atom t = t

 and eta_contract2 (f$t) = f $ eta_contract_atom t

   | eta_contract2 t     = eta_contract_atom t;

 (*When can we safely delete the equality?

     Not if it equates two constants; consider 0=1.

     Not if it resembles x=t[x], since substitution does not eliminate x.

     Not if it resembles ?x=0; another goal could instantiate ?x to Suc(i)

   Prefer to eliminate Bound variables if possible.

   Result:  true = use as is,  false = reorient first *)

 (*Can t occur in u?  For substitution.

   Does NOT examine the args of Skolem terms: substitution does not affect them.

   REFLEXIVE because hyp_subst_tac fails on x=x.*)

 fun substOccur t =

   let (*NO vars are permitted in u except the arguments of t, if it is

         a Skolem term.  This ensures that no equations are deleted that could

         be instantiated to a cycle.  For example, x=?a is rejected because ?a

         could be instantiated to Suc(x).*)

       val vars = case t of

                      Skolem(_,vars) => vars

                    | _ => []

       fun occEq u = (t aconv u) orelse occ u

       and occ (Var(ref(SOME u))) = occEq u

         | occ (Var v)            = not (mem_var (v, vars))

         | occ (Abs(_,u))         = occEq u

         | occ (f$u)              = occEq u  orelse  occEq f

         | occ (_)                = false;

   in  occEq  end;

 exception DEST_EQ;

 (*Take apart an equality.  NO constant Trueprop*)

 fun dest_eq (Const (c, _) $ t $ u) =

       if c = Data.equality_name then (eta_contract_atom t, eta_contract_atom u)

       else raise DEST_EQ

   | dest_eq _ = raise DEST_EQ;

 (*Reject the equality if u occurs in (or equals!) t*)

 fun check (t,u,v) = if substOccur t u then raise DEST_EQ else v;

 (*IF the goal is an equality with a substitutable variable

   THEN orient that equality ELSE raise exception DEST_EQ*)

 fun orientGoal (t,u) = case (t,u) of

        (Skolem _, _) => check(t,u,(t,u))        (*eliminates t*)

      | (_, Skolem _) => check(u,t,(u,t))        (*eliminates u*)

      | (Free _, _)   => check(t,u,(t,u))        (*eliminates t*)

      | (_, Free _)   => check(u,t,(u,t))        (*eliminates u*)

      | _             => raise DEST_EQ;

 (*Substitute through the branch if an equality goal (else raise DEST_EQ).

   Moves affected literals back into the branch, but it is not clear where

   they should go: this could make proofs fail.*)

 fun equalSubst thy (G, {pairs, lits, vars, lim}) =

   let val (t,u) = orientGoal(dest_eq G)

       val subst = subst_atomic (t,u)

       fun subst2(G,md) = (subst G, md)

       (*substitute throughout list; extract affected formulae*)

       fun subForm ((G,md), (changed, pairs)) =

             let val nG = subst G

             in  if nG aconv G then (changed, (G,md)::pairs)

                               else ((nG,md)::changed, pairs)

             end

       (*substitute throughout "stack frame"; extract affected formulae*)

       fun subFrame ((Gs,Hs), (changed, frames)) =

             let val (changed', Gs') = foldr subForm (changed, []) Gs

                 val (changed'', Hs') = foldr subForm (changed', []) Hs

             in  (changed'', (Gs',Hs')::frames)  end

       (*substitute throughout literals; extract affected ones*)

       fun subLit (lit, (changed, nlits)) =

             let val nlit = subst lit

             in  if nlit aconv lit then (changed, nlit::nlits)

                                   else ((nlit,true)::changed, nlits)

             end

       val (changed, lits') = foldr subLit ([], []) lits

       val (changed', pairs') = foldr subFrame (changed, []) pairs

   in  if !trace then writeln ("Substituting " ^ traceTerm thy u ^

                               " for " ^ traceTerm thy t ^ " in branch" )

       else ();

       {pairs = (changed',[])::pairs',   (*affected formulas, and others*)

        lits  = lits',                   (*unaffected literals*)

        vars  = vars,

        lim   = lim}

   end;

 exception NEWBRANCHES and CLOSEF;

 exception PROVE;

 (*Trying eq_contr_tac first INCREASES the effort, slowing reconstruction*)

 val contr_tac = ematch_tac [Data.notE] THEN'

                 (eq_assume_tac ORELSE' assume_tac);

 val eContr_tac  = TRACE Data.notE contr_tac;

 val eAssume_tac = TRACE asm_rl   (eq_assume_tac ORELSE' assume_tac);

 (*Try to unify complementary literals and return the corresponding tactic. *)

 fun tryClose state (G, L) =

   let

     fun close t u tac = if unify state ([], t, u) then SOME tac else NONE;

     fun arg (_ $ t) = t;

   in

     if isGoal G then close (arg G) L eAssume_tac

     else if isGoal L then close G (arg L) eAssume_tac

     else if isNot G then close (arg G) L eContr_tac

     else if isNot L then close G (arg L) eContr_tac

     else NONE

   end;

 (*Were there Skolem terms in the premise?  Must NOT chase Vars*)

 fun hasSkolem (Skolem _)     = true

   | hasSkolem (Abs (_,body)) = hasSkolem body

   | hasSkolem (f$t)          =  hasSkolem f orelse hasSkolem t

   | hasSkolem _              = false;

 (*Attach the right "may duplicate" flag to new formulae: if they contain

   Skolem terms then allow duplication.*)

 fun joinMd md [] = []

   | joinMd md (G::Gs) = (G, hasSkolem G orelse md) :: joinMd md Gs;

 (** Backtracking and Pruning **)

 (*clashVar vars (n,trail) determines whether any of the last n elements

   of "trail" occur in "vars" OR in their instantiations*)

 fun clashVar [] = (fn _ => false)

   | clashVar vars =

       let fun clash (0, _)     = false

             | clash (_, [])    = false

             | clash (n, v::vs) = exists (varOccur v) vars orelse clash(n-1,vs)

       in  clash  end;

 (*nbrs = # of branches just prior to closing this one.  Delete choice points

   for goals proved by the latest inference, provided NO variables in the

   next branch have been updated.*)

 fun prune _ (1, nxtVars, choices) = choices  (*DON'T prune at very end: allow

                                              backtracking over bad proofs*)

   | prune (State {ntrail, trail, ...}) (nbrs: int, nxtVars, choices) =

       let fun traceIt last =

                 let val ll = length last

                     and lc = length choices

                 in if !trace andalso ll<lc then

                     (writeln("Pruning " ^ Int.toString(lc-ll) ^ " levels");

                      last)

                    else last

                 end

           fun pruneAux (last, _, _, []) = last

             | pruneAux (last, ntrl, trl, (ntrl',nbrs',exn) :: choices) =

                 if nbrs' < nbrs

                 then last  (*don't backtrack beyond first solution of goal*)

                 else if nbrs' > nbrs then pruneAux (last, ntrl, trl, choices)

                 else (* nbrs'=nbrs *)

                      if clashVar nxtVars (ntrl-ntrl', trl) then last

                      else (*no clashes: can go back at least this far!*)

                           pruneAux(choices, ntrl', List.drop(trl, ntrl-ntrl'),

                                    choices)

   in  traceIt (pruneAux (choices, !ntrail, !trail, choices))  end;

 fun nextVars ({pairs, lits, vars, lim} :: _) = map Var vars

   | nextVars []                              = [];

 fun backtrack (choices as (ntrl, nbrs, exn)::_) =

       (if !trace then (writeln ("Backtracking; now there are " ^

                                 Int.toString nbrs ^ " branches"))

                  else ();

        raise exn)

   | backtrack _ = raise PROVE;

 (*Add the literal G, handling *Goal* and detecting duplicates.*)

 fun addLit (Const ("*Goal*", _) $ G, lits) =

       (*New literal is a *Goal*, so change all other Goals to Nots*)

       let fun bad (Const ("*Goal*", _) $ _) = true

             | bad (Const (c, _) $ G')   = c = Data.not_name andalso G aconv G'

             | bad _                   = false;

           fun change [] = []

             | change (lit :: lits) =

                 (case lit of

                   Const (c, _) $ G' =>

                     if c = "*Goal*" orelse c = Data.not_name then

                       if G aconv G' then change lits

                       else negate G' :: change lits

                     else lit :: change lits

                 | _ => lit :: change lits)

       in

         Const ("*Goal*", []) $ G :: (if exists bad lits then change lits else lits)

       end

   | addLit (G,lits) = ins_term(G, lits)

 (*For calculating the "penalty" to assess on a branching factor of n

   log2 seems a little too severe*)

 fun log n = if n<4 then 0 else 1 + log(n div 4);

 (*match(t,u) says whether the term u might be an instance of the pattern t

   Used to detect "recursive" rules such as transitivity*)

 fun match (Var _) u   = true

   | match (Const (a,tas)) (Const (b,tbs)) =

       a = "*Goal*" andalso b = Data.not_name orelse

       a = Data.not_name andalso b = "*Goal*" orelse

       a = b andalso matchs tas tbs

   | match (Free a)        (Free b)        = (a=b)

   | match (Bound i)       (Bound j)       = (i=j)

   | match (Abs(_,t))      (Abs(_,u))      = match t u

   | match (f$t)           (g$u)           = match f g andalso match t u

   | match t               u   = false

 and matchs [] [] = true

   | matchs (t :: ts) (u :: us) = match t u andalso matchs ts us;

 fun printStats (State {ntried, nclosed, ...}) (b, start, tacs) =

   if b then

     writeln (end_timing start ^ " for search.  Closed: "

              ^ Int.toString (!nclosed) ^

              " tried: " ^ Int.toString (!ntried) ^

              " tactics: " ^ Int.toString (length tacs))

   else ();

 (*Tableau prover based on leanTaP.  Argument is a list of branches.  Each

   branch contains a list of unexpanded formulae, a list of literals, and a

   bound on unsafe expansions.

  "start" is CPU time at start, for printing search time

 *)

 fun prove (state, start, cs, brs, cont) =

  let val State {thy, ntrail, nclosed, ntried, ...} = state;

      val {safe0_netpair, safep_netpair, haz_netpair, ...} = Data.rep_cs cs

      val safeList = [safe0_netpair, safep_netpair]

      and hazList  = [haz_netpair]

      fun prv (tacs, trs, choices, []) =

                 (printStats state (!trace orelse !stats, start, tacs);

                  cont (tacs, trs, choices))   (*all branches closed!*)

        | prv (tacs, trs, choices,

               brs0 as {pairs = ((G,md)::br, haz)::pairs,

                        lits, vars, lim} :: brs) =

              (*apply a safe rule only (possibly allowing instantiation);

                defer any haz formulae*)

           let exception PRV (*backtrack to precisely this recursion!*)

               val ntrl = !ntrail

               val nbrs = length brs0

               val nxtVars = nextVars brs

               val G = norm G

               val rules = netMkRules thy G vars safeList

               (*Make a new branch, decrementing "lim" if instantiations occur*)

               fun newBr (vars',lim') prems =

                   map (fn prem =>

                        if (exists isGoal prem)

                        then {pairs = ((joinMd md prem, []) ::

                                       negOfGoals ((br, haz)::pairs)),

                              lits  = map negOfGoal lits,

                              vars  = vars',

                              lim   = lim'}

                        else {pairs = ((joinMd md prem, []) ::

                                       (br, haz) :: pairs),

                              lits = lits,

                              vars = vars',

                              lim  = lim'})

                   prems @

                   brs

               (*Seek a matching rule.  If unifiable then add new premises

                 to branch.*)

               fun deeper [] = raise NEWBRANCHES

                 | deeper (((P,prems),tac)::grls) =

                     if unify state (add_term_vars(P,[]), P, G)

                     then  (*P comes from the rule; G comes from the branch.*)

                      let val updated = ntrl < !ntrail (*branch updated*)

                          val lim' = if updated

                                     then lim - (1+log(length rules))

                                     else lim   (*discourage branching updates*)

                          val vars  = vars_in_vars vars

                          val vars' = foldr add_terms_vars vars prems

                          val choices' = (ntrl, nbrs, PRV) :: choices

                          val tacs' = (tac(updated,false,true))

                                      :: tacs  (*no duplication; rotate*)

                      in

                          traceNew prems;  traceVars state ntrl;

                          (if null prems then (*closed the branch: prune!*)

                             (nclosed := !nclosed + 1;

                              prv(tacs',  brs0::trs,

                                  prune state (nbrs, nxtVars, choices'),

                                  brs))

                           else (*prems non-null*)

                           if lim'<0 (*faster to kill ALL the alternatives*)

                           then (traceMsg"Excessive branching: KILLED";

                                 clearTo state ntrl;  raise NEWBRANCHES)

                           else

                             (ntried := !ntried + length prems - 1;

                              prv(tacs',  brs0::trs, choices',

                                  newBr (vars',lim') prems)))

                          handle PRV =>

                            if updated then

                                 (*Backtrack at this level.

                                   Reset Vars and try another rule*)

                                 (clearTo state ntrl;  deeper grls)

                            else (*backtrack to previous level*)

                                 backtrack choices

                      end

                     else deeper grls

               (*Try to close branch by unifying with head goal*)

               fun closeF [] = raise CLOSEF

                 | closeF (L::Ls) =

                     case tryClose state (G,L) of

                         NONE     => closeF Ls

                       | SOME tac =>

                             let val choices' =

                                     (if !trace then (Output.immediate_output"branch closed";

                                                      traceVars state ntrl)

                                                else ();

                                      prune state (nbrs, nxtVars,

                                             (ntrl, nbrs, PRV) :: choices))

                             in  nclosed := !nclosed + 1;

                                 prv (tac::tacs, brs0::trs, choices', brs)

                                 handle PRV =>

                                     (*reset Vars and try another literal

                                       [this handler is pruned if possible!]*)

                                  (clearTo state ntrl;  closeF Ls)

                             end

               (*Try to unify a queued formula (safe or haz) with head goal*)

               fun closeFl [] = raise CLOSEF

                 | closeFl ((br, haz)::pairs) =

                     closeF (map fst br)

                       handle CLOSEF => closeF (map fst haz)

                         handle CLOSEF => closeFl pairs

           in tracing state brs0;

              if lim<0 then (traceMsg "Limit reached.  "; backtrack choices)

              else

              prv (Data.hyp_subst_tac (!trace) :: tacs,

                   brs0::trs,  choices,

                   equalSubst thy

                     (G, {pairs = (br,haz)::pairs,

                          lits  = lits, vars  = vars, lim   = lim})

                     :: brs)

              handle DEST_EQ =>   closeF lits

               handle CLOSEF =>   closeFl ((br,haz)::pairs)

                 handle CLOSEF => deeper rules

                   handle NEWBRANCHES =>

                    (case netMkRules thy G vars hazList of

                        [] => (*there are no plausible haz rules*)

                              (traceMsg "moving formula to literals";

                               prv (tacs, brs0::trs, choices,

                                    {pairs = (br,haz)::pairs,

                                     lits  = addLit(G,lits),

                                     vars  = vars,

                                     lim   = lim}  :: brs))

                     | _ => (*G admits some haz rules: try later*)

                            (traceMsg "moving formula to haz list";

                             prv (if isGoal G then negOfGoal_tac :: tacs

                                              else tacs,

                                  brs0::trs,

                                  choices,

                                  {pairs = (br, haz@[(negOfGoal G, md)])::pairs,

                                   lits  = lits,

                                   vars  = vars,

                                   lim   = lim}  :: brs)))

           end

        | prv (tacs, trs, choices,

               {pairs = ([],haz)::(Gs,haz')::pairs, lits, vars, lim} :: brs) =

              (*no more "safe" formulae: transfer haz down a level*)

            prv (tacs, trs, choices,

                 {pairs = (Gs,haz@haz')::pairs,

                  lits  = lits,

                  vars  = vars,

                  lim    = lim} :: brs)

        | prv (tacs, trs, choices,

               brs0 as {pairs = [([], (H,md)::Hs)],

                        lits, vars, lim} :: brs) =

              (*no safe steps possible at any level: apply a haz rule*)

           let exception PRV (*backtrack to precisely this recursion!*)

               val H = norm H

               val ntrl = !ntrail

               val rules = netMkRules thy H vars hazList

               (*new premises of haz rules may NOT be duplicated*)

               fun newPrem (vars,P,dup,lim') prem =

                   let val Gs' = map (fn Q => (Q,false)) prem

                       and Hs' = if dup then Hs @ [(negOfGoal H, md)] else Hs

                       and lits' = if (exists isGoal prem)

                                   then map negOfGoal lits

                                   else lits

                   in  {pairs = if exists (match P) prem then [(Gs',Hs')]

                                (*Recursive in this premise.  Don't make new

                                  "stack frame".  New haz premises will end up

                                  at the BACK of the queue, preventing

                                  exclusion of others*)

                             else [(Gs',[]), ([],Hs')],

                        lits = lits',

                        vars = vars,

                        lim  = lim'}

                   end

               fun newBr x prems = map (newPrem x) prems  @  brs

               (*Seek a matching rule.  If unifiable then add new premises

                 to branch.*)

               fun deeper [] = raise NEWBRANCHES

                 | deeper (((P,prems),tac)::grls) =

                     if unify state (add_term_vars(P,[]), P, H)

                     then

                      let val updated = ntrl < !ntrail (*branch updated*)

                          val vars  = vars_in_vars vars

                          val vars' = foldr add_terms_vars vars prems

                             (*duplicate H if md permits*)

                          val dup = md (*earlier had "andalso vars' <> vars":

                                   duplicate only if the subgoal has new vars*)

                              (*any instances of P in the subgoals?

                                NB: boolean "recur" affects tracing only!*)

                          and recur = exists (exists (match P)) prems

                          val lim' = (*Decrement "lim" extra if updates occur*)

                              if updated then lim - (1+log(length rules))

                              else lim-1

                                  (*It is tempting to leave "lim" UNCHANGED if

                                    both dup and recur are false.  Proofs are

                                    found at shallower depths, but looping

                                    occurs too often...*)

                          val mayUndo =

                              (*Allowing backtracking from a rule application

                                if other matching rules exist, if the rule

                                updated variables, or if the rule did not

                                introduce new variables.  This latter condition

                                means it is not a standard "gamma-rule" but

                                some other form of unsafe rule.  Aim is to

                                emulate Fast_tac, which allows all unsafe steps

                                to be undone.*)

                              not(null grls)   (*other rules to try?*)

                              orelse updated

                              orelse vars=vars'   (*no new Vars?*)

                          val tac' = tac(updated, dup, true)

                        (*if recur then perhaps shouldn't call rotate_tac: new

                          formulae should be last, but that's WRONG if the new

                          formulae are Goals, since they remain in the first

                          position*)

                      in

                        if lim'<0 andalso not (null prems)

                        then (*it's faster to kill ALL the alternatives*)

                            (traceMsg"Excessive branching: KILLED";

                             clearTo state ntrl;  raise NEWBRANCHES)

                        else

                          traceNew prems;

                          if !trace andalso dup then Output.immediate_output" (duplicating)"

                                                  else ();

                          if !trace andalso recur then Output.immediate_output" (recursive)"

                                                  else ();

                          traceVars state ntrl;

                          if null prems then nclosed := !nclosed + 1

                          else ntried := !ntried + length prems - 1;

                          prv(tac' :: tacs,

                              brs0::trs,

                              (ntrl, length brs0, PRV) :: choices,

                              newBr (vars', P, dup, lim') prems)

                           handle PRV =>

                               if mayUndo

                               then (*reset Vars and try another rule*)

                                    (clearTo state ntrl;  deeper grls)

                               else (*backtrack to previous level*)

                                    backtrack choices

                      end

                     else deeper grls

           in tracing state brs0;

              if lim<1 then (traceMsg "Limit reached.  "; backtrack choices)

              else deeper rules

              handle NEWBRANCHES =>

                  (*cannot close branch: move H to literals*)

                  prv (tacs,  brs0::trs,  choices,

                       {pairs = [([], Hs)],

                        lits  = H::lits,

                        vars  = vars,

                        lim   = lim}  :: brs)

           end

        | prv (tacs, trs, choices, _ :: brs) = backtrack choices

  in prv ([], [], [(!ntrail, length brs, PROVE)], brs) end;

 (*Construct an initial branch.*)

 fun initBranch (ts,lim) =

     {pairs = [(map (fn t => (t,true)) ts, [])],

      lits  = [],

      vars  = add_terms_vars (ts,[]),

      lim   = lim};

 (*** Conversion & Skolemization of the Isabelle proof state ***)

 (*Make a list of all the parameters in a subgoal, even if nested*)

 local open Term

 in

 fun discard_foralls (Const("all",_)$Abs(a,T,t)) = discard_foralls t

   | discard_foralls t = t;

 end;

 (*List of variables not appearing as arguments to the given parameter*)

 fun getVars []                  i = []

   | getVars ((_,(v,is))::alist) (i: int) =

         if member (op =) is i then getVars alist i

         else v :: getVars alist i;

 exception TRANS of string;

 (*Translation of a subgoal: Skolemize all parameters*)

 fun fromSubgoal thy t =

   let val alistVar = ref []

       and alistTVar = ref []

       fun hdvar ((ix,(v,is))::_) = v

       fun from lev t =

         let val (ht,ts) = Term.strip_comb t

             fun apply u = list_comb (u, map (from lev) ts)

             fun bounds [] = []

               | bounds (Term.Bound i::ts) =

                   if i<lev then raise TRANS

                       "Function unknown's argument not a parameter"

                   else i-lev :: bounds ts

               | bounds ts = raise TRANS

                       "Function unknown's argument not a bound variable"

         in

           case ht of

               Term.Const aT    => apply (fromConst thy alistTVar aT)

             | Term.Free  (a,_) => apply (Free a)

             | Term.Bound i     => apply (Bound i)

             | Term.Var (ix,_) =>

                   (case (AList.lookup (op =) (!alistVar) ix) of

                        NONE => (alistVar := (ix, (ref NONE, bounds ts))

                                           :: !alistVar;

                                 Var (hdvar(!alistVar)))

                      | SOME(v,is) => if is=bounds ts then Var v

                             else raise TRANS

                                 ("Discrepancy among occurrences of "

                                  ^ Term.string_of_vname ix))

             | Term.Abs (a,_,body) =>

                   if null ts then Abs(a, from (lev+1) body)

                   else raise TRANS "argument not in normal form"

         end

       val npars = length (Logic.strip_params t)

       (*Skolemize a subgoal from a proof state*)

       fun skoSubgoal i t =

           if i<npars then

               skoSubgoal (i+1)

                 (subst_bound (Skolem (gensym "T_", getVars (!alistVar) i),

                               t))

           else t

   in  skoSubgoal 0 (from 0 (discard_foralls t))  end;

 (*Tactic using tableau engine and proof reconstruction.

  "start" is CPU time at start, for printing SEARCH time

         (also prints reconstruction time)

  "lim" is depth limit.*)

 fun timing_depth_tac start cs lim i st0 =

   let val thy = Thm.theory_of_thm st0

       val state = initialize thy

       val st = Conv.gconv_rule ObjectLogic.atomize_prems i st0

       val skoprem = fromSubgoal thy (List.nth(prems_of st, i-1))

       val hyps  = strip_imp_prems skoprem

       and concl = strip_imp_concl skoprem

       fun cont (tacs,_,choices) =

           let val start = start_timing ()

           in

           case Seq.pull(EVERY' (rev tacs) i st) of

               NONE => (writeln ("PROOF FAILED for depth " ^

                                 Int.toString lim);

                        if !trace then error "************************\n"

                        else ();

                        backtrack choices)

             | cell => (if (!trace orelse !stats)

                        then writeln (end_timing start ^ " for reconstruction")

                        else ();

                        Seq.make(fn()=> cell))

           end

   in prove (state, start, cs, [initBranch (mkGoal concl :: hyps, lim)], cont) end

   handle PROVE     => Seq.empty

 (*Public version with fixed depth*)

 fun depth_tac cs lim i st = timing_depth_tac (start_timing ()) cs lim i st;

 val (depth_limit, setup_depth_limit) = Attrib.config_int_global "blast_depth_limit" 20;

 fun blast_tac cs i st =

     ((DEEPEN (1, Config.get_thy (Thm.theory_of_thm st) depth_limit)

         (timing_depth_tac (start_timing ()) cs) 0) i

      THEN flexflex_tac) st

     handle TRANS s =>

       ((if !trace then warning ("blast: " ^ s) else ());

        Seq.empty);

 fun Blast_tac i = blast_tac (Data.claset()) i;

 (*** For debugging: these apply the prover to a subgoal and return

      the resulting tactics, trace, etc.                            ***)

 val fullTrace = ref ([]: branch list list);

 (*Read a string to make an initial, singleton branch*)

 fun readGoal thy s = Syntax.read_prop_global thy s |> fromTerm thy |> rand |> mkGoal;

 fun tryInThy thy lim s =

   let

     val state as State {fullTrace = ft, ...} = initialize thy;

     val res = timeap prove

       (state, start_timing(), Data.claset(), [initBranch ([readGoal thy s], lim)], I);

     val _ = fullTrace := !ft;

   in res end;

 (** method setup **)

 val blast_method =

   Method.bang_sectioned_args' Data.cla_modifiers (Scan.lift (Scan.option OuterParse.nat))

     (fn NONE => Data.cla_meth' blast_tac

       | SOME lim => Data.cla_meth' (fn cs => depth_tac cs lim));

 val setup =

   setup_depth_limit #>

   Method.add_methods [("blast", blast_method, "tableau prover")];

 end;