(*  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 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') = 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 !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 (#message (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' = 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 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 =>