File ‹~~/src/Provers/blast.ML›

(*  Title:      Provers/blast.ML
    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 unknown'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 an unsafe 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.
*)

signature BLAST_DATA =
sig
  structure Classical: CLASSICAL
  val Trueprop_const: string * typ
  val equality_name: string
  val not_name: string
  val notE: thm           (* [| ~P;  P |] ==> R *)
  val ccontr: thm
  val hyp_subst_tac: Proof.context -> bool -> int -> tactic
end;

signature BLAST =
sig
  exception TRANS of string    (*reports translation errors*)
  datatype term =
      Const of string * term list
    | Skolem of string * term option Unsynchronized.ref list
    | Free  of string
    | Var   of term option Unsynchronized.ref
    | Bound of int
    | Abs   of string*term
    | $  of term*term;
  val depth_tac: Proof.context -> int -> int -> tactic
  val depth_limit: int Config.T
  val trace: bool Config.T
  val stats: bool Config.T
  val blast_tac: Proof.context -> int -> tactic
  (*debugging tools*)
  type branch
  val tryIt: Proof.context -> int -> string ->
    {fullTrace: branch list list,
      result: ((int -> tactic) list * branch list list * (int * int * exn) list)}
end;

functor Blast(Data: BLAST_DATA): BLAST =
struct

structure Classical = Data.Classical;

(* options *)

val depth_limit = Attrib.setup_config_int \<^binding>‹blast_depth_limit› (K 20);
val (trace, _) = Attrib.config_bool \<^binding>‹blast_trace› (K false);
val (stats, _) = Attrib.config_bool \<^binding>‹blast_stats› (K false);


datatype term =
    Const  of string * term list  (*typargs constant--as a term!*)
  | Skolem of string * term option Unsynchronized.ref list
  | Free   of string
  | Var    of term option Unsynchronized.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,  (*unsafe formulae on this level*)
     lits:   term list,                 (*literals: irreducible formulae*)
     vars:   term option Unsynchronized.ref list,  (*variables occurring in branch*)
     lim:    int};                      (*resource limit*)


(* global state information *)

datatype state = State of
 {ctxt: Proof.context,
  names: Name.context Unsynchronized.ref,
  fullTrace: branch list list Unsynchronized.ref,
  trail: term option Unsynchronized.ref list Unsynchronized.ref,
  ntrail: int Unsynchronized.ref,
  nclosed: int Unsynchronized.ref,
  ntried: int Unsynchronized.ref}

fun reserved_const thy c =
  is_some (Sign.const_type thy c) andalso
    error ("blast: theory contains reserved constant " ^ quote c);

fun initialize ctxt =
  let
    val thy = Proof_Context.theory_of ctxt;
    val _ = reserved_const thy "*Goal*";
    val _ = reserved_const thy "*False*";
  in
    State
     {ctxt = ctxt,
      names = Unsynchronized.ref (Variable.names_of ctxt),
      fullTrace = Unsynchronized.ref [],
      trail = Unsynchronized.ref [],
      ntrail = Unsynchronized.ref 0,
      nclosed = Unsynchronized.ref 0, (*number of branches closed during the search*)
      ntried = Unsynchronized.ref 1} (*number of branches created by splitting (counting from 1)*)
  end;

fun gensym (State {names, ...}) x =
  Unsynchronized.change_result names (Name.variant x);


(** 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, TruepropT) = Data.Trueprop_const;

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 (Unsynchronized.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 (Unsynchronized.ref(SOME t))) aconv u = t aconv u
  | t aconv (Var (Unsynchronized.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 Unsynchronized.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 Unsynchronized.ref NONE), vars) = ins_var (v, vars)
  | add_term_vars (Var (Unsynchronized.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 (Unsynchronized.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 =
  let
    fun term lev (t as Bound i) = if i >= lev then Bound (i + inc) else t
      | term lev (Abs (a, t)) = Abs (a, term (lev + 1) t)
      | term lev (t $ u) = term lev t $ term lev u
      | term _ t = t;
  in term end;

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 (Unsynchronized.ref NONE)) => t
  | (Var (Unsynchronized.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 = Unsynchronized.ref []
      and alistTVar = Unsynchronized.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 (Unsynchronized.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 = Unsynchronized.ref "a"
      val updated = Unsynchronized.ref []
      fun inst (Const(a,ts)) = List.app inst ts
        | inst (Var(v as Unsynchronized.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 (\<^const_name>‹Pure.imp›, _) $ 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 (\<^const_name>‹Pure.imp›, _) $ 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 (Unsynchronized.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 state vars (Const (\<^const_name>‹Pure.all›, _) $ Abs (_, P)) =
        skoPrem state vars (subst_bound (Skolem (gensym state "S", vars), P))
  | skoPrem _ _ 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 state 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 state vars) Ps)
  end
  handle Bind => raise ElimBadConcl;


(*Like dup_elim, but puts the duplicated major premise FIRST*)
fun rev_dup_elim ctxt th = (th RSN (2, revcut_rl)) |> Thm.assumption (SOME ctxt) 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) (Thm.rotate_rule (~k) i st);
in
fun rot_subgoals_tac (rot, rl) =
     rot_tac (if rot then map nNewHyps rl else [])
end;


fun cond_tracing true msg = tracing (msg ())
  | cond_tracing false _ = ();

fun TRACE ctxt rl tac i st =
  (cond_tracing (Config.get ctxt trace) (fn () => Thm.string_of_thm ctxt rl); tac i st);

(*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 ctxt upd rl =
  TRACE ctxt rl (if upd then eresolve_tac ctxt [rl] else ematch_tac ctxt [rl]);

fun rmtac ctxt upd rl =
  TRACE ctxt rl (if upd then resolve_tac ctxt [rl] else match_tac ctxt [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 (state as State {ctxt, ...}) vars rl0 =
  let
    val thy = Proof_Context.theory_of ctxt
    val rl = Thm.transfer thy rl0
    val trl = rl |> Thm.prop_of |> fromTerm thy |> convertRule state vars
    fun tac (upd, dup,rot) i =
      emtac ctxt upd (if dup then rev_dup_elim ctxt rl else rl) i THEN
      rot_subgoals_tac (rot, #2 trl) i
  in SOME (trl, tac) end
  handle
    ElimBadPrem => (*reject: prems don't preserve conclusion*)
      (if Context_Position.is_visible ctxt then
        warning ("Ignoring weak elimination rule\n" ^ Thm.string_of_thm ctxt rl0)
       else ();
       Option.NONE)
  | ElimBadConcl => (*ignore: conclusion is not just a variable*)
      (cond_tracing (Config.get ctxt trace)
        (fn () => "Ignoring ill-formed elimination rule:\n" ^
          "conclusion should be a variable\n" ^ Thm.string_of_thm ctxt rl0);
       Option.NONE);


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

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

fun convertIntrRule state vars t =
  let val Ps = strip_imp_prems t
      val P  = strip_imp_concl t
  in  (mkGoal P, map (convertIntrPrem o skoPrem state 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 unsafe rules are now delayed, "dup" is always FALSE for
  introduction rules.*)
fun fromIntrRule (state as State {ctxt, ...}) vars rl0 =
  let
    val thy = Proof_Context.theory_of ctxt
    val rl = Thm.transfer thy rl0
    val trl = rl |> Thm.prop_of |> fromTerm thy |> convertIntrRule state vars
    fun tac (upd,dup,rot) i =
      rmtac ctxt upd (if dup then Classical.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);

(*Too flexible assertions or goals. Motivated by examples such as "(⋀P. ~P) ⟹ 0==1"*)
fun isVarForm (Var _) = true
  | isVarForm (Const (c, _) $ Var _) = (c = Data.not_name)
  | isVarForm _ = false;

fun netMkRules state P vars (nps: Classical.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 state vars o #2) (order_list intrs)  end
    | _ =>
        if isVarForm P then [] (*The formula is too flexible, reject*)
        else
        let val pP = mk_Trueprop (toTerm 3 P)
            val elims = maps (fn (_,enet) => Net.unify_term enet pP) nps
        in  map_filter (fromRule state vars o #2) (order_list 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 (Unsynchronized.ref(SOME u))) = showTerm d u
  | showTerm d (Var (Unsynchronized.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 ctxt d t = Syntax.string_of_term ctxt (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, unsafe) => (map negOfGoal2 Gs, unsafe)) pairs;

(*Tactic.  Convert *Goal* to negated assumption in FIRST position*)
fun negOfGoal_tac ctxt i =
  TRACE ctxt Data.ccontr (resolve_tac ctxt [Data.ccontr]) i THEN rotate_tac ~1 i;

fun traceTerm ctxt t =
  let val thy = Proof_Context.theory_of ctxt
      val t' = norm (negOfGoal t)
      val stm = string_of ctxt 8 t'
  in
      case topType thy t' of
          NONE   => stm   (*no type to attach*)
        | SOME T => stm ^ " :: " ^ Syntax.string_of_typ ctxt T
  end;


(*Print tracing information at each iteration of prover*)
fun trace_prover (State {ctxt, fullTrace, ...}) brs =
  let fun printPairs (((G,_)::_,_)::_)  = tracing (traceTerm ctxt G)
        | printPairs (([],(H,_)::_)::_) = tracing (traceTerm ctxt H ^ "  (Unsafe)")
        | printPairs _                 = ()
      fun printBrs (brs0 as {pairs, lits, lim, ...} :: brs) =
            (fullTrace := brs0 :: !fullTrace;
             List.app (fn _ => tracing "+") brs;
             tracing (" [" ^ string_of_int lim ^ "] ");
             printPairs pairs;
             tracing "")
  in if Config.get ctxt trace then printBrs (map normBr brs) else () end;

(*Tracing: variables updated in the last branch operation?*)
fun traceVars (State {ctxt, ntrail, trail, ...}) ntrl =
  if Config.get ctxt trace then
      (case !ntrail-ntrl of
            0 => ()
          | 1 => tracing " 1 variable UPDATED:"
          | n => tracing (" " ^ string_of_int n ^ " variables UPDATED:");
       (*display the instantiations themselves, though no variable names*)
       List.app (fn v => tracing ("   " ^ string_of ctxt 4 (the (!v))))
           (List.take(!trail, !ntrail-ntrl));
       tracing "")
    else ();

(*Tracing: how many new branches are created?*)
fun traceNew true prems =
      (case length prems of
        0 => tracing "branch closed by rule"
      | 1 => tracing "branch extended (1 new subgoal)"
      | n => tracing ("branch split: "^ string_of_int n ^ " new subgoals"))
  | traceNew _ _ = ();



(*** 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(Unsynchronized.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(Unsynchronized.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 ctxt (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') = List.foldr subForm (changed, []) Gs
                val (changed'', Hs') = List.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') = List.foldr subLit ([], []) lits
      val (changed', pairs') = List.foldr subFrame (changed, []) pairs
  in  if Config.get ctxt trace then tracing ("Substituting " ^ traceTerm ctxt u ^
                              " for " ^ traceTerm ctxt 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*)
fun contr_tac ctxt =
  ematch_tac ctxt [Data.notE] THEN' (eq_assume_tac ORELSE' assume_tac ctxt);

(*Try to unify complementary literals and return the corresponding tactic. *)
fun tryClose state (G, L) =
  let
    val State {ctxt, ...} = state;
    val eContr_tac = TRACE ctxt Data.notE contr_tac;
    val eAssume_tac = TRACE ctxt asm_rl (eq_assume_tac ORELSE' assume_tac ctxt);
    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 ctxt)
    else if isNot L then close G (arg L) (eContr_tac ctxt)
    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 {ctxt, ntrail, trail, ...}) (nbrs: int, nxtVars, choices) =
      let fun traceIt last =
                let val ll = length last
                    and lc = length choices
                in if Config.get ctxt trace andalso ll<lc then
                    (tracing ("Pruning " ^ string_of_int(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 trace (choices as (ntrl, nbrs, exn)::_) =
      (cond_tracing trace
        (fn () => "Backtracking; now there are " ^ string_of_int nbrs ^ " branches");
       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
    tracing (Timing.message (Timing.result start) ^ " for search.  Closed: "
             ^ string_of_int (!nclosed) ^
             " tried: " ^ string_of_int (!ntried) ^
             " tactics: " ^ string_of_int (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, brs, cont) =
 let val State {ctxt, ntrail, nclosed, ntried, ...} = state;
     val trace = Config.get ctxt trace;
     val stats = Config.get ctxt stats;
     val {safe0_netpair, safep_netpair, unsafe_netpair, ...} =
       Classical.rep_cs (Classical.claset_of ctxt);
     val safeList = [safe0_netpair, safep_netpair];
     val unsafeList = [unsafe_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, unsafe)::pairs,
                       lits, vars, lim} :: brs) =
             (*apply a safe rule only (possibly allowing instantiation);
               defer any unsafe 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 state 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, unsafe)::pairs)),
                             lits  = map negOfGoal lits,
                             vars  = vars',
                             lim   = lim'}
                       else {pairs = ((joinMd md prem, []) ::
                                      (br, unsafe) :: 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' = List.foldr add_terms_vars vars prems
                         val choices' = (ntrl, nbrs, PRV) :: choices
                         val tacs' = (tac(updated,false,true))
                                     :: tacs  (*no duplication; rotate*)
                     in
                         traceNew trace 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 (cond_tracing trace (fn () => "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 trace 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 (tracing "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 unsafe) with head goal*)
              fun closeFl [] = raise CLOSEF
                | closeFl ((br, unsafe)::pairs) =
                    closeF (map fst br)
                      handle CLOSEF => closeF (map fst unsafe)
                        handle CLOSEF => closeFl pairs
          in
             trace_prover state brs0;
             if lim<0 then (cond_tracing trace (fn () => "Limit reached."); backtrack trace choices)
             else
             prv (Data.hyp_subst_tac ctxt trace :: tacs,
                  brs0::trs,  choices,
                  equalSubst ctxt
                    (G, {pairs = (br,unsafe)::pairs,
                         lits  = lits, vars  = vars, lim   = lim})
                    :: brs)
             handle DEST_EQ =>   closeF lits
              handle CLOSEF =>   closeFl ((br,unsafe)::pairs)
                handle CLOSEF => deeper rules
                  handle NEWBRANCHES =>
                   (case netMkRules state G vars unsafeList of
                       [] => (*there are no plausible unsafe rules*)
                             (cond_tracing trace (fn () => "moving formula to literals");
                              prv (tacs, brs0::trs, choices,
                                   {pairs = (br,unsafe)::pairs,
                                    lits  = addLit(G,lits),
                                    vars  = vars,
                                    lim   = lim}  :: brs))
                    | _ => (*G admits some unsafe rules: try later*)
                           (cond_tracing trace (fn () => "moving formula to unsafe list");
                            prv (if isGoal G then negOfGoal_tac ctxt :: tacs
                                             else tacs,
                                 brs0::trs,
                                 choices,
                                 {pairs = (br, unsafe@[(negOfGoal G, md)])::pairs,
                                  lits  = lits,
                                  vars  = vars,
                                  lim   = lim}  :: brs)))
          end
       | prv (tacs, trs, choices,
              {pairs = ([],unsafe)::(Gs,unsafe')::pairs, lits, vars, lim} :: brs) =
             (*no more "safe" formulae: transfer unsafe down a level*)
           prv (tacs, trs, choices,
                {pairs = (Gs,unsafe@unsafe')::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 unsafe rule*)
          let exception PRV (*backtrack to precisely this recursion!*)
              val H = norm H
              val ntrl = !ntrail
              val rules = netMkRules state H vars unsafeList
              (*new premises of unsafe 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 unsafe 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' = List.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*)
                           (cond_tracing trace (fn () => "Excessive branching: KILLED");
                            clearTo state ntrl;  raise NEWBRANCHES)
                       else
                         traceNew trace prems;
                         cond_tracing (trace andalso dup) (fn () => " (duplicating)");
                         cond_tracing (trace andalso recur) (fn () => " (recursive)");
                         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 trace choices
                     end
                    else deeper grls
          in
             trace_prover state brs0;
             if lim<1 then (cond_tracing trace (fn () => "Limit reached."); backtrack trace 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 trace 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(\<^const_name>‹Pure.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 (state as State {ctxt, ...}) t =
  let val thy = Proof_Context.theory_of ctxt
      val alistVar = Unsynchronized.ref []
      and alistTVar = Unsynchronized.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, (Unsynchronized.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 state "T", getVars (!alistVar) i), t))
          else t

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


(*Tableau engine and proof reconstruction operating on subgoal 1.
 "start" is CPU time at start, for printing SEARCH time (also prints reconstruction time)
 "lim" is depth limit.*)
fun raw_blast start ctxt lim st =
  let val state = initialize ctxt
      val trace = Config.get ctxt trace;
      val stats = Config.get ctxt stats;
      val skoprem = fromSubgoal state (#1 (Logic.dest_implies (Thm.prop_of st)))
      val hyps  = strip_imp_prems skoprem
      and concl = strip_imp_concl skoprem
      fun cont (tacs,_,choices) =
          let val start = Timing.start ()
          in
          case Seq.pull(EVERY' (rev tacs) 1 st) of
              NONE => (cond_tracing trace (fn () => "PROOF FAILED for depth " ^ string_of_int lim);
                       backtrack trace choices)
            | cell => (cond_tracing (trace orelse stats)
                        (fn () => Timing.message (Timing.result start) ^ " for reconstruction");
                       Seq.make(fn()=> cell))
          end handle TERM _ =>
            (cond_tracing trace (fn () => "PROOF RAISED EXN TERM for depth " ^ string_of_int lim);
                       backtrack trace choices)
  in
    prove (state, start, [initBranch (mkGoal concl :: hyps, lim)], cont)
  end
  handle PROVE => Seq.empty
    | TRANS s => (cond_tracing (Config.get ctxt trace) (fn () => "Blast: " ^ s); Seq.empty);

fun depth_tac ctxt lim i st =
  SELECT_GOAL
    (Object_Logic.atomize_prems_tac ctxt 1 THEN
      raw_blast (Timing.start ()) ctxt lim) i st;

fun blast_tac ctxt i st =
  let
    val start = Timing.start ();
    val lim = Config.get ctxt depth_limit;
  in
    SELECT_GOAL
     (Object_Logic.atomize_prems_tac ctxt 1 THEN
      DEEPEN (1, lim) (fn m => fn _ => raw_blast start ctxt m) 0 1) i st
  end;



(*** For debugging: these apply the prover to a subgoal and return
     the resulting tactics, trace, etc.                            ***)

(*Read a string to make an initial, singleton branch*)
fun readGoal ctxt s =
  Syntax.read_prop ctxt s |> fromTerm (Proof_Context.theory_of ctxt) |> rand |> mkGoal;

fun tryIt ctxt lim s =
  let
    val state as State {fullTrace, ...} = initialize ctxt;
    val res = timeap prove (state, Timing.start (), [initBranch ([readGoal ctxt s], lim)], I);
  in {fullTrace = !fullTrace, result = res} end;



(** method setup **)

val _ =
  Theory.setup
    (Method.setup \<^binding>‹blast›
      (Scan.lift (Scan.option Parse.nat) --| Method.sections Classical.cla_modifiers >>
        (fn NONE => SIMPLE_METHOD' o blast_tac
          | SOME lim => (fn ctxt => SIMPLE_METHOD' (depth_tac ctxt lim))))
      "classical tableau prover");

end;