(*  Title:      HOL/Tools/sat_solver.ML

     Author:     Tjark Weber

     Copyright   2004-2009

 Interface to external SAT solvers, and (simple) built-in SAT solvers.

 *)

 signature SAT_SOLVER =

 sig

   exception NOT_CONFIGURED

   type assignment = int -> bool option

   type proof      = int list Inttab.table * int

   datatype result = SATISFIABLE of assignment

                   | UNSATISFIABLE of proof option

                   | UNKNOWN

   type solver     = PropLogic.prop_formula -> result

   (* auxiliary functions to create external SAT solvers *)

   val write_dimacs_cnf_file : Path.T -> PropLogic.prop_formula -> unit

   val write_dimacs_sat_file : Path.T -> PropLogic.prop_formula -> unit

   val read_std_result_file  : Path.T -> string * string * string -> result

   val make_external_solver  : string -> (PropLogic.prop_formula -> unit) ->

                                 (unit -> result) -> solver

   val read_dimacs_cnf_file : Path.T -> PropLogic.prop_formula

   (* generic solver interface *)

   val solvers       : (string * solver) list ref

   val add_solver    : string * solver -> unit

   val invoke_solver : string -> solver  (* exception Option *)

 end;

 structure SatSolver : SAT_SOLVER =

 struct

   open PropLogic;

 (* ------------------------------------------------------------------------- *)

 (* should be raised by an external SAT solver to indicate that the solver is *)

 (* not configured properly                                                   *)

 (* ------------------------------------------------------------------------- *)

   exception NOT_CONFIGURED;

 (* ------------------------------------------------------------------------- *)

 (* type of partial (satisfying) assignments: 'a i = NONE' means that 'a' is  *)

 (*      a satisfying assignment regardless of the value of variable 'i'      *)

 (* ------------------------------------------------------------------------- *)

   type assignment = int -> bool option;

 (* ------------------------------------------------------------------------- *)

 (* a proof of unsatisfiability, to be interpreted as follows: each integer   *)

 (*      is a clause ID, each list 'xs' stored under the key 'x' in the table *)

 (*      contains the IDs of clauses that must be resolved (in the given      *)

 (*      order) to obtain the new clause 'x'.  Each list 'xs' must be         *)

 (*      non-empty, and the literal to be resolved upon must always be unique *)

 (*      (e.g. "A | ~B" must not be resolved with "~A | B").  Circular        *)

 (*      dependencies of clauses are not allowed.  (At least) one of the      *)

 (*      clauses in the table must be the empty clause (i.e. contain no       *)

 (*      literals); its ID is given by the second component of the proof.     *)

 (*      The clauses of the original problem passed to the SAT solver have    *)

 (*      consecutive IDs starting with 0.  Clause IDs must be non-negative,   *)

 (*      but do not need to be consecutive.                                   *)

 (* ------------------------------------------------------------------------- *)

   type proof = int list Inttab.table * int;

 (* ------------------------------------------------------------------------- *)

 (* return type of SAT solvers: if the result is 'SATISFIABLE', a satisfying  *)

 (*      assignment must be returned as well; if the result is                *)

 (*      'UNSATISFIABLE', a proof of unsatisfiability may be returned         *)

 (* ------------------------------------------------------------------------- *)

   datatype result = SATISFIABLE of assignment

                   | UNSATISFIABLE of proof option

                   | UNKNOWN;

 (* ------------------------------------------------------------------------- *)

 (* type of SAT solvers: given a propositional formula, a satisfying          *)

 (*      assignment may be returned                                           *)

 (* ------------------------------------------------------------------------- *)

   type solver = prop_formula -> result;

 (* ------------------------------------------------------------------------- *)

 (* write_dimacs_cnf_file: serializes a formula 'fm' of propositional logic   *)

 (*      to a file in DIMACS CNF format (see "Satisfiability Suggested        *)

 (*      Format", May 8 1993, Section 2.1)                                    *)

 (* Note: 'fm' must not contain a variable index less than 1.                 *)

 (* Note: 'fm' must be given in CNF.                                          *)

 (* ------------------------------------------------------------------------- *)

   (* Path.T -> prop_formula -> unit *)

   fun write_dimacs_cnf_file path fm =

   let

     (* prop_formula -> prop_formula *)

     fun cnf_True_False_elim True =

       Or (BoolVar 1, Not (BoolVar 1))

       | cnf_True_False_elim False =

       And (BoolVar 1, Not (BoolVar 1))

       | cnf_True_False_elim fm =

       fm  (* since 'fm' is in CNF, either 'fm'='True'/'False',

              or 'fm' does not contain 'True'/'False' at all *)

     (* prop_formula -> int *)

     fun cnf_number_of_clauses (And (fm1, fm2)) =

       (cnf_number_of_clauses fm1) + (cnf_number_of_clauses fm2)

       | cnf_number_of_clauses _ =

       1

     (* TextIO.outstream -> unit *)

     fun write_cnf_file out =

     let

       (* prop_formula -> unit *)

       fun write_formula True =

           error "formula is not in CNF"

         | write_formula False =

           error "formula is not in CNF"

         | write_formula (BoolVar i) =

           (i>=1 orelse error "formula contains a variable index less than 1";

            TextIO.output (out, string_of_int i))

         | write_formula (Not (BoolVar i)) =

           (TextIO.output (out, "-");

            write_formula (BoolVar i))

         | write_formula (Not _) =

           error "formula is not in CNF"

         | write_formula (Or (fm1, fm2)) =

           (write_formula fm1;

            TextIO.output (out, " ");

            write_formula fm2)

         | write_formula (And (fm1, fm2)) =

           (write_formula fm1;

            TextIO.output (out, " 0\n");

            write_formula fm2)

       val fm'               = cnf_True_False_elim fm

       val number_of_vars    = maxidx fm'

       val number_of_clauses = cnf_number_of_clauses fm'

     in

       TextIO.output (out, "c This file was generated by SatSolver.write_dimacs_cnf_file\n");

       TextIO.output (out, "p cnf " ^ string_of_int number_of_vars ^ " " ^

                             string_of_int number_of_clauses ^ "\n");

       write_formula fm';

       TextIO.output (out, " 0\n")

     end

   in

     File.open_output write_cnf_file path

   end;

 (* ------------------------------------------------------------------------- *)

 (* write_dimacs_sat_file: serializes a formula 'fm' of propositional logic   *)

 (*      to a file in DIMACS SAT format (see "Satisfiability Suggested        *)

 (*      Format", May 8 1993, Section 2.2)                                    *)

 (* Note: 'fm' must not contain a variable index less than 1.                 *)

 (* ------------------------------------------------------------------------- *)

   (* Path.T -> prop_formula -> unit *)

   fun write_dimacs_sat_file path fm =

   let

     (* TextIO.outstream -> unit *)

     fun write_sat_file out =

     let

       (* prop_formula -> unit *)

       fun write_formula True =

           TextIO.output (out, "*()")

         | write_formula False =

           TextIO.output (out, "+()")

         | write_formula (BoolVar i) =

           (i>=1 orelse error "formula contains a variable index less than 1";

            TextIO.output (out, string_of_int i))

         | write_formula (Not (BoolVar i)) =

           (TextIO.output (out, "-");

            write_formula (BoolVar i))

         | write_formula (Not fm) =

           (TextIO.output (out, "-(");

            write_formula fm;

            TextIO.output (out, ")"))

         | write_formula (Or (fm1, fm2)) =

           (TextIO.output (out, "+(");

            write_formula_or fm1;

            TextIO.output (out, " ");

            write_formula_or fm2;

            TextIO.output (out, ")"))

         | write_formula (And (fm1, fm2)) =

           (TextIO.output (out, "*(");

            write_formula_and fm1;

            TextIO.output (out, " ");

            write_formula_and fm2;

            TextIO.output (out, ")"))

       (* optimization to make use of n-ary disjunction/conjunction *)

       and write_formula_or (Or (fm1, fm2)) =

           (write_formula_or fm1;

            TextIO.output (out, " ");

            write_formula_or fm2)

         | write_formula_or fm =

           write_formula fm

       and write_formula_and (And (fm1, fm2)) =

           (write_formula_and fm1;

            TextIO.output (out, " ");

            write_formula_and fm2)

         | write_formula_and fm =

           write_formula fm

       val number_of_vars = Int.max (maxidx fm, 1)

     in

       TextIO.output (out, "c This file was generated by SatSolver.write_dimacs_sat_file\n");

       TextIO.output (out, "p sat " ^ string_of_int number_of_vars ^ "\n");

       TextIO.output (out, "(");

       write_formula fm;

       TextIO.output (out, ")\n")

     end

   in

     File.open_output write_sat_file path

   end;

 (* ------------------------------------------------------------------------- *)

 (* read_std_result_file: scans a SAT solver's output file for a satisfying   *)

 (*      variable assignment.  Returns the assignment, or 'UNSATISFIABLE' if  *)

 (*      the file contains 'unsatisfiable', or 'UNKNOWN' if the file contains *)

 (*      neither 'satisfiable' nor 'unsatisfiable'.  Empty lines are ignored. *)

 (*      The assignment must be given in one or more lines immediately after  *)

 (*      the line that contains 'satisfiable'.  These lines must begin with   *)

 (*      'assignment_prefix'.  Variables must be separated by " ".  Non-      *)

 (*      integer strings are ignored.  If variable i is contained in the      *)

 (*      assignment, then i is interpreted as 'true'.  If ~i is contained in  *)

 (*      the assignment, then i is interpreted as 'false'.  Otherwise the     *)

 (*      value of i is taken to be unspecified.                               *)

 (* ------------------------------------------------------------------------- *)

   (* Path.T -> string * string * string -> result *)

   fun read_std_result_file path (satisfiable, assignment_prefix, unsatisfiable) =

   let

     (* string -> int list *)

     fun int_list_from_string s =

       List.mapPartial Int.fromString (space_explode " " s)

     (* int list -> assignment *)

     fun assignment_from_list [] i =

       NONE  (* the SAT solver didn't provide a value for this variable *)

       | assignment_from_list (x::xs) i =

       if x=i then (SOME true)

       else if x=(~i) then (SOME false)

       else assignment_from_list xs i

     (* int list -> string list -> assignment *)

     fun parse_assignment xs [] =

       assignment_from_list xs

       | parse_assignment xs (line::lines) =

       if String.isPrefix assignment_prefix line then

         parse_assignment (xs @ int_list_from_string line) lines

       else

         assignment_from_list xs

     (* string -> string -> bool *)

     fun is_substring needle haystack =

     let

       val length1 = String.size needle

       val length2 = String.size haystack

     in

       if length2 < length1 then

         false

       else if needle = String.substring (haystack, 0, length1) then

         true

       else is_substring needle (String.substring (haystack, 1, length2-1))

     end

     (* string list -> result *)

     fun parse_lines [] =

       UNKNOWN

       | parse_lines (line::lines) =

       if is_substring unsatisfiable line then

         UNSATISFIABLE NONE

       else if is_substring satisfiable line then

         SATISFIABLE (parse_assignment [] lines)

       else

         parse_lines lines

   in

     (parse_lines o (List.filter (fn l => l <> "")) o split_lines o File.read) path

   end;

 (* ------------------------------------------------------------------------- *)

 (* make_external_solver: call 'writefn', execute 'cmd', call 'readfn'        *)

 (* ------------------------------------------------------------------------- *)

   (* string -> (PropLogic.prop_formula -> unit) -> (unit -> result) -> solver *)

   fun make_external_solver cmd writefn readfn fm =

     (writefn fm; system cmd; readfn ());

 (* ------------------------------------------------------------------------- *)

 (* read_dimacs_cnf_file: returns a propositional formula that corresponds to *)

 (*      a SAT problem given in DIMACS CNF format                             *)

 (* ------------------------------------------------------------------------- *)

   (* Path.T -> PropLogic.prop_formula *)

   fun read_dimacs_cnf_file path =

   let

     (* string list -> string list *)

     fun filter_preamble [] =

       error "problem line not found in DIMACS CNF file"

       | filter_preamble (line::lines) =

       if String.isPrefix "c " line orelse line = "c" then

         (* ignore comments *)

         filter_preamble lines

       else if String.isPrefix "p " line then

         (* ignore the problem line (which must be the last line of the preamble) *)

         (* Ignoring the problem line implies that if the file contains more clauses *)

         (* or variables than specified in its preamble, we will accept it anyway.   *)

         lines

       else

         error "preamble in DIMACS CNF file contains a line that does not begin with \"c \" or \"p \""

     (* string -> int *)

     fun int_from_string s =

       case Int.fromString s of

         SOME i => i

       | NONE   => error ("token " ^ quote s ^ "in DIMACS CNF file is not a number")

     (* int list -> int list list *)

     fun clauses xs =

       let

         val (xs1, xs2) = take_prefix (fn i => i <> 0) xs

       in

         case xs2 of

           []      => [xs1]

         | (0::[]) => [xs1]

         | (0::tl) => xs1 :: clauses tl

         | _       => sys_error "this cannot happen"

       end

     (* int -> PropLogic.prop_formula *)

     fun literal_from_int i =

       (i<>0 orelse error "variable index in DIMACS CNF file is 0";

       if i>0 then

         PropLogic.BoolVar i

       else

         PropLogic.Not (PropLogic.BoolVar (~i)))

     (* PropLogic.prop_formula list -> PropLogic.prop_formula *)

     fun disjunction [] =

       error "empty clause in DIMACS CNF file"

       | disjunction (x::xs) =

       (case xs of

         [] => x

       | _  => PropLogic.Or (x, disjunction xs))

     (* PropLogic.prop_formula list -> PropLogic.prop_formula *)

     fun conjunction [] =

       error "no clause in DIMACS CNF file"

       | conjunction (x::xs) =

       (case xs of

         [] => x

       | _  => PropLogic.And (x, conjunction xs))

   in

     (conjunction

     o (map disjunction)

     o (map (map literal_from_int))

     o clauses

     o (map int_from_string)

     o List.concat

     o (map (String.fields (fn c => c mem [#" ", #"\t", #"\n"])))

     o filter_preamble

     o (List.filter (fn l => l <> ""))

     o split_lines

     o File.read)

       path

   end;

 (* ------------------------------------------------------------------------- *)

 (* solvers: a (reference to a) table of all registered SAT solvers           *)

 (* ------------------------------------------------------------------------- *)

   val solvers = ref ([] : (string * solver) list);

 (* ------------------------------------------------------------------------- *)

 (* add_solver: updates 'solvers' by adding a new solver                      *)

 (* ------------------------------------------------------------------------- *)

   (* string * solver -> unit *)

     fun add_solver (name, new_solver) =

       let

         val the_solvers = !solvers;

         val _ = if AList.defined (op =) the_solvers name

           then warning ("SAT solver " ^ quote name ^ " was defined before")

           else ();

       in solvers := AList.update (op =) (name, new_solver) the_solvers end;

 (* ------------------------------------------------------------------------- *)

 (* invoke_solver: returns the solver associated with the given 'name'        *)

 (* Note: If no solver is associated with 'name', exception 'Option' will be  *)

 (*       raised.                                                             *)

 (* ------------------------------------------------------------------------- *)

   (* string -> solver *)

   fun invoke_solver name =

     (the o AList.lookup (op =) (!solvers)) name;

 end;  (* SatSolver *)

 (* ------------------------------------------------------------------------- *)

 (* Predefined SAT solvers                                                    *)

 (* ------------------------------------------------------------------------- *)

 (* ------------------------------------------------------------------------- *)

 (* Internal SAT solver, available as 'SatSolver.invoke_solver "enumerate"'   *)

 (* -- simply enumerates assignments until a satisfying assignment is found   *)

 (* ------------------------------------------------------------------------- *)

 let

   fun enum_solver fm =

   let

     (* int list *)

     val indices = PropLogic.indices fm

     (* int list -> int list -> int list option *)

     (* binary increment: list 'xs' of current bits, list 'ys' of all bits (lower bits first) *)

     fun next_list _ ([]:int list) =

       NONE

       | next_list [] (y::ys) =

       SOME [y]

       | next_list (x::xs) (y::ys) =

       if x=y then

         (* reset the bit, continue *)

         next_list xs ys

       else

         (* set the lowest bit that wasn't set before, keep the higher bits *)

         SOME (y::x::xs)

     (* int list -> int -> bool *)

     fun assignment_from_list xs i =

       i mem xs

     (* int list -> SatSolver.result *)

     fun solver_loop xs =

       if PropLogic.eval (assignment_from_list xs) fm then

         SatSolver.SATISFIABLE (SOME o (assignment_from_list xs))

       else

         (case next_list xs indices of

           SOME xs' => solver_loop xs'

         | NONE     => SatSolver.UNSATISFIABLE NONE)

   in

     (* start with the "first" assignment (all variables are interpreted as 'false') *)

     solver_loop []

   end

 in

   SatSolver.add_solver ("enumerate", enum_solver)

 end;

 (* ------------------------------------------------------------------------- *)

 (* Internal SAT solver, available as 'SatSolver.invoke_solver "dpll"' -- a   *)

 (* simple implementation of the DPLL algorithm (cf. L. Zhang, S. Malik: "The *)

 (* Quest for Efficient Boolean Satisfiability Solvers", July 2002, Fig. 1).  *)

 (* ------------------------------------------------------------------------- *)

 let

   local

     open PropLogic

   in

     fun dpll_solver fm =

     let

       (* We could use 'PropLogic.defcnf fm' instead of 'PropLogic.nnf fm' *)

       (* but that sometimes leads to worse performance due to the         *)

       (* introduction of additional variables.                            *)

       val fm' = PropLogic.nnf fm

       (* int list *)

       val indices = PropLogic.indices fm'

       (* int list -> int -> prop_formula *)

       fun partial_var_eval []      i = BoolVar i

         | partial_var_eval (x::xs) i = if x=i then True else if x=(~i) then False else partial_var_eval xs i

       (* int list -> prop_formula -> prop_formula *)

       fun partial_eval xs True             = True

         | partial_eval xs False            = False

         | partial_eval xs (BoolVar i)      = partial_var_eval xs i

         | partial_eval xs (Not fm)         = SNot (partial_eval xs fm)

         | partial_eval xs (Or (fm1, fm2))  = SOr (partial_eval xs fm1, partial_eval xs fm2)

         | partial_eval xs (And (fm1, fm2)) = SAnd (partial_eval xs fm1, partial_eval xs fm2)

       (* prop_formula -> int list *)

       fun forced_vars True              = []

         | forced_vars False             = []

         | forced_vars (BoolVar i)       = [i]

         | forced_vars (Not (BoolVar i)) = [~i]

         | forced_vars (Or (fm1, fm2))   = (forced_vars fm1) inter_int (forced_vars fm2)

         | forced_vars (And (fm1, fm2))  = (forced_vars fm1) union_int (forced_vars fm2)

         (* Above, i *and* ~i may be forced.  In this case the first occurrence takes   *)

         (* precedence, and the next partial evaluation of the formula returns 'False'. *)

         | forced_vars _                 = error "formula is not in negation normal form"

       (* int list -> prop_formula -> (int list * prop_formula) *)

       fun eval_and_force xs fm =

       let

         val fm' = partial_eval xs fm

         val xs' = forced_vars fm'

       in

         if null xs' then

           (xs, fm')

         else

           eval_and_force (xs@xs') fm'  (* xs and xs' should be distinct, so '@' here should have *)

                                        (* the same effect as 'union_int'                         *)

       end

       (* int list -> int option *)

       fun fresh_var xs =

         Library.find_first (fn i => not (i mem_int xs) andalso not ((~i) mem_int xs)) indices

       (* int list -> prop_formula -> int list option *)

       (* partial assignment 'xs' *)

       fun dpll xs fm =

       let

         val (xs', fm') = eval_and_force xs fm

       in

         case fm' of

           True  => SOME xs'

         | False => NONE

         | _     =>

           let

             val x = valOf (fresh_var xs')  (* a fresh variable must exist since 'fm' did not evaluate to 'True'/'False' *)

           in

             case dpll (x::xs') fm' of  (* passing fm' rather than fm should save some simplification work *)

               SOME xs'' => SOME xs''

             | NONE      => dpll ((~x)::xs') fm'  (* now try interpreting 'x' as 'False' *)

           end

       end

       (* int list -> assignment *)

       fun assignment_from_list [] i =

         NONE  (* the DPLL procedure didn't provide a value for this variable *)

         | assignment_from_list (x::xs) i =

         if x=i then (SOME true)

         else if x=(~i) then (SOME false)

         else assignment_from_list xs i

     in

       (* initially, no variable is interpreted yet *)

       case dpll [] fm' of

         SOME assignment => SatSolver.SATISFIABLE (assignment_from_list assignment)

       | NONE            => SatSolver.UNSATISFIABLE NONE

     end

   end  (* local *)

 in

   SatSolver.add_solver ("dpll", dpll_solver)

 end;

 (* ------------------------------------------------------------------------- *)

 (* Internal SAT solver, available as 'SatSolver.invoke_solver "auto"': uses  *)

 (* the last installed solver (other than "auto" itself) that does not raise  *)

 (* 'NOT_CONFIGURED'.  (However, the solver may return 'UNKNOWN'.)            *)

 (* ------------------------------------------------------------------------- *)

 let

   fun auto_solver fm =

   let

     fun loop [] =

       SatSolver.UNKNOWN

       | loop ((name, solver)::solvers) =

       if name="auto" then

         (* do not call solver "auto" from within "auto" *)

         loop solvers

       else (

         (if name="dpll" orelse name="enumerate" then

           warning ("Using SAT solver " ^ quote name ^ "; for better performance, consider using an external solver.")

         else

           tracing ("Using SAT solver " ^ quote name ^ "."));

         (* apply 'solver' to 'fm' *)

         solver fm

           handle SatSolver.NOT_CONFIGURED => loop solvers

       )

   in

     loop (!SatSolver.solvers)

   end

 in

   SatSolver.add_solver ("auto", auto_solver)

 end;

 (* ------------------------------------------------------------------------- *)

 (* MiniSat 1.14                                                              *)

 (* (http://www.cs.chalmers.se/Cs/Research/FormalMethods/MiniSat/)            *)

 (* ------------------------------------------------------------------------- *)

 (* ------------------------------------------------------------------------- *)

 (* "minisat_with_proofs" requires a modified version of MiniSat 1.14 by John *)

 (* Matthews, which can output ASCII proof traces.  Replaying binary proof    *)

 (* traces generated by MiniSat-p_v1.14 has _not_ been implemented.           *)

 (* ------------------------------------------------------------------------- *)

 (* add "minisat_with_proofs" _before_ "minisat" to the available solvers, so *)

 (* that the latter is preferred by the "auto" solver                         *)

 (* There is a complication that is dealt with in the code below: MiniSat     *)

 (* introduces IDs for original clauses in the proof trace.  It does not (in  *)

 (* general) follow the convention that the original clauses are numbered     *)

 (* from 0 to n-1 (where n is the number of clauses in the formula).          *)

 let

   exception INVALID_PROOF of string

   fun minisat_with_proofs fm =

   let

     val _          = if (getenv "MINISAT_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

     val serial_str = serial_string ()

     val inpath     = File.tmp_path (Path.explode ("isabelle" ^ serial_str ^ ".cnf"))

     val outpath    = File.tmp_path (Path.explode ("result" ^ serial_str))

     val proofpath  = File.tmp_path (Path.explode ("result" ^ serial_str ^ ".prf"))

     val cmd        = (getenv "MINISAT_HOME") ^ "/minisat " ^ (Path.implode inpath) ^ " -r " ^ (Path.implode outpath) ^ " -t " ^ (Path.implode proofpath) ^ "> /dev/null"

     fun writefn fm = SatSolver.write_dimacs_cnf_file inpath fm

     fun readfn ()  = SatSolver.read_std_result_file outpath ("SAT", "", "UNSAT")

     val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

     val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

     val cnf        = PropLogic.defcnf fm

     val result     = SatSolver.make_external_solver cmd writefn readfn cnf

     val _          = try File.rm inpath

     val _          = try File.rm outpath

   in  case result of

     SatSolver.UNSATISFIABLE NONE =>

     (let

       (* string list *)

       val proof_lines = (split_lines o File.read) proofpath

         handle IO.Io _ => raise INVALID_PROOF "Could not read file \"result.prf\""

       (* representation of clauses as ordered lists of literals (with duplicates removed) *)

       (* prop_formula -> int list *)

       fun clause_to_lit_list (PropLogic.Or (fm1, fm2)) =

         OrdList.union int_ord (clause_to_lit_list fm1) (clause_to_lit_list fm2)

         | clause_to_lit_list (PropLogic.BoolVar i) =

         [i]

         | clause_to_lit_list (PropLogic.Not (PropLogic.BoolVar i)) =

         [~i]

         | clause_to_lit_list _ =

         raise INVALID_PROOF "Error: invalid clause in CNF formula."

       (* prop_formula -> int *)

       fun cnf_number_of_clauses (PropLogic.And (fm1, fm2)) =

         cnf_number_of_clauses fm1 + cnf_number_of_clauses fm2

         | cnf_number_of_clauses _ =

         1

       val number_of_clauses = cnf_number_of_clauses cnf

       (* int list array *)

       val clauses = Array.array (number_of_clauses, [])

       (* initialize the 'clauses' array *)

       (* prop_formula * int -> int *)

       fun init_array (PropLogic.And (fm1, fm2), n) =

         init_array (fm2, init_array (fm1, n))

         | init_array (fm, n) =

         (Array.update (clauses, n, clause_to_lit_list fm); n+1)

       val _ = init_array (cnf, 0)

       (* optimization for the common case where MiniSat "R"s clauses in their *)

       (* original order:                                                      *)

       val last_ref_clause = ref (number_of_clauses - 1)

       (* search the 'clauses' array for the given list of literals 'lits', *)

       (* starting at index '!last_ref_clause + 1'                          *)

       (* int list -> int option *)

       fun original_clause_id lits =

       let

         fun original_clause_id_from index =

           if index = number_of_clauses then

             (* search from beginning again *)

             original_clause_id_from 0

           (* both 'lits' and the list of literals used in 'clauses' are sorted, so *)

           (* testing for equality should suffice -- barring duplicate literals     *)

           else if Array.sub (clauses, index) = lits then (

             (* success *)

             last_ref_clause := index;

             SOME index

           ) else if index = !last_ref_clause then

             (* failure *)

             NONE

           else

             (* continue search *)

             original_clause_id_from (index + 1)

       in

         original_clause_id_from (!last_ref_clause + 1)

       end

       (* string -> int *)

       fun int_from_string s = (

         case Int.fromString s of

           SOME i => i

         | NONE   => raise INVALID_PROOF ("File format error: number expected (" ^ quote s ^ " encountered).")

       )

       (* parse the proof file *)

       val clause_table  = ref (Inttab.empty : int list Inttab.table)

       val empty_id      = ref ~1

       (* contains a mapping from clause IDs as used by MiniSat to clause IDs in *)

       (* our proof format, where original clauses are numbered starting from 0  *)

       val clause_id_map = ref (Inttab.empty : int Inttab.table)

       fun sat_to_proof id = (

         case Inttab.lookup (!clause_id_map) id of

           SOME id' => id'

         | NONE     => raise INVALID_PROOF ("Clause ID " ^ Int.toString id ^ " used, but not defined.")

       )

       val next_id = ref (number_of_clauses - 1)

       (* string list -> unit *)

       fun process_tokens [] =

         ()

         | process_tokens (tok::toks) =

         if tok="R" then (

           case toks of

             id::sep::lits =>

             let

               val _        = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: \"R\" disallowed after \"X\"."

               val cid      = int_from_string id

               val _        = if sep = "<=" then () else raise INVALID_PROOF ("File format error: \"<=\" expected (" ^ quote sep ^ " encountered).")

               val ls       = sort int_ord (map int_from_string lits)

               val proof_id = case original_clause_id ls of

                                SOME orig_id => orig_id

                              | NONE         => raise INVALID_PROOF ("Original clause (new ID is " ^ id ^ ") not found.")

             in

               (* extend the mapping of clause IDs with this newly defined ID *)

               clause_id_map := Inttab.update_new (cid, proof_id) (!clause_id_map)

                 handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ id ^ " defined more than once (in \"R\").")

               (* the proof itself doesn't change *)

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"R\" followed by an insufficient number of tokens."

         ) else if tok="C" then (

           case toks of

             id::sep::ids =>

             let

               val _        = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: \"C\" disallowed after \"X\"."

               val cid      = int_from_string id

               val _        = if sep = "<=" then () else raise INVALID_PROOF ("File format error: \"<=\" expected (" ^ quote sep ^ " encountered).")

               (* ignore the pivot literals in MiniSat's trace *)

               fun unevens []             = raise INVALID_PROOF "File format error: \"C\" followed by an even number of IDs."

                 | unevens (x :: [])      = x :: []

                 | unevens (x :: _ :: xs) = x :: unevens xs

               val rs       = (map sat_to_proof o unevens o map int_from_string) ids

               (* extend the mapping of clause IDs with this newly defined ID *)

               val proof_id = inc next_id

               val _        = clause_id_map := Inttab.update_new (cid, proof_id) (!clause_id_map)

                                handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ id ^ " defined more than once (in \"C\").")

             in

               (* update clause table *)

               clause_table := Inttab.update_new (proof_id, rs) (!clause_table)

                 handle Inttab.DUP _ => raise INVALID_PROOF ("Error: internal ID for clause " ^ id ^ " already used.")

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"C\" followed by an insufficient number of tokens."

         ) else if tok="D" then (

           case toks of

             [id] =>

             let

               val _ = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: \"D\" disallowed after \"X\"."

               val _ = sat_to_proof (int_from_string id)

             in

               (* simply ignore "D" *)

               ()

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"D\" followed by an illegal number of tokens."

         ) else if tok="X" then (

           case toks of

             [id1, id2] =>

             let

               val _            = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: more than one end-of-proof statement."

               val _            = sat_to_proof (int_from_string id1)

               val new_empty_id = sat_to_proof (int_from_string id2)

             in

               (* update conflict id *)

               empty_id := new_empty_id

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"X\" followed by an illegal number of tokens."

         ) else

           raise INVALID_PROOF ("File format error: unknown token " ^ quote tok ^ " encountered.")

       (* string list -> unit *)

       fun process_lines [] =

         ()

         | process_lines (l::ls) = (

           process_tokens (String.tokens (fn c => c = #" " orelse c = #"\t") l);

           process_lines ls

         )

       (* proof *)

       val _ = process_lines proof_lines

       val _ = if !empty_id <> ~1 then () else raise INVALID_PROOF "File format error: no conflicting clause specified."

     in

       SatSolver.UNSATISFIABLE (SOME (!clause_table, !empty_id))

     end handle INVALID_PROOF reason => (warning reason; SatSolver.UNSATISFIABLE NONE))

   | result =>

     result

   end

 in

   SatSolver.add_solver ("minisat_with_proofs", minisat_with_proofs)

 end;

 let

   fun minisat fm =

   let

     val _          = if (getenv "MINISAT_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

     val serial_str = serial_string ()

     val inpath     = File.tmp_path (Path.explode ("isabelle" ^ serial_str ^ ".cnf"))

     val outpath    = File.tmp_path (Path.explode ("result" ^ serial_str))

     val cmd        = (getenv "MINISAT_HOME") ^ "/minisat " ^ (Path.implode inpath) ^ " -r " ^ (Path.implode outpath) ^ " > /dev/null"

     fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

     fun readfn ()  = SatSolver.read_std_result_file outpath ("SAT", "", "UNSAT")

     val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

     val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

     val result     = SatSolver.make_external_solver cmd writefn readfn fm

     val _          = try File.rm inpath

     val _          = try File.rm outpath

   in

     result

   end

 in

   SatSolver.add_solver ("minisat", minisat)

 end;

 (* ------------------------------------------------------------------------- *)

 (* zChaff (http://www.princeton.edu/~chaff/zchaff.html)                      *)

 (* ------------------------------------------------------------------------- *)

 (* ------------------------------------------------------------------------- *)

 (* 'zchaff_with_proofs' applies the "zchaff" prover to a formula, and if     *)

 (* zChaff finds that the formula is unsatisfiable, a proof of this is read   *)

 (* from a file "resolve_trace" that was generated by zChaff.  See the code   *)

 (* below for the expected format of the "resolve_trace" file.  Aside from    *)

 (* some basic syntactic checks, no verification of the proof is performed.   *)

 (* ------------------------------------------------------------------------- *)

 (* add "zchaff_with_proofs" _before_ "zchaff" to the available solvers, so   *)

 (* that the latter is preferred by the "auto" solver                         *)

 let

   exception INVALID_PROOF of string

   fun zchaff_with_proofs fm =

   case SatSolver.invoke_solver "zchaff" fm of

     SatSolver.UNSATISFIABLE NONE =>

     (let

       (* string list *)

       val proof_lines = ((split_lines o File.read) (Path.explode "resolve_trace"))

         handle IO.Io _ => raise INVALID_PROOF "Could not read file \"resolve_trace\""

       (* PropLogic.prop_formula -> int *)

       fun cnf_number_of_clauses (PropLogic.And (fm1, fm2)) = cnf_number_of_clauses fm1 + cnf_number_of_clauses fm2

         | cnf_number_of_clauses _                          = 1

       (* string -> int *)

       fun int_from_string s = (

         case Int.fromString s of

           SOME i => i

         | NONE   => raise INVALID_PROOF ("File format error: number expected (" ^ quote s ^ " encountered).")

       )

       (* parse the "resolve_trace" file *)

       val clause_offset = ref ~1

       val clause_table  = ref (Inttab.empty : int list Inttab.table)

       val empty_id      = ref ~1

       (* string list -> unit *)

       fun process_tokens [] =

         ()

         | process_tokens (tok::toks) =

         if tok="CL:" then (

           case toks of

             id::sep::ids =>

             let

               val _   = if !clause_offset = ~1 then () else raise INVALID_PROOF ("File format error: \"CL:\" disallowed after \"VAR:\".")

               val _   = if !empty_id = ~1 then () else raise INVALID_PROOF ("File format error: \"CL:\" disallowed after \"CONF:\".")

               val cid = int_from_string id

               val _   = if sep = "<=" then () else raise INVALID_PROOF ("File format error: \"<=\" expected (" ^ quote sep ^ " encountered).")

               val rs  = map int_from_string ids

             in

               (* update clause table *)

               clause_table := Inttab.update_new (cid, rs) (!clause_table)

                 handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ id ^ " defined more than once.")

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"CL:\" followed by an insufficient number of tokens."

         ) else if tok="VAR:" then (

           case toks of

             id::levsep::levid::valsep::valid::antesep::anteid::litsep::lits =>

             let

               val _   = if !empty_id = ~1 then () else raise INVALID_PROOF ("File format error: \"VAR:\" disallowed after \"CONF:\".")

               (* set 'clause_offset' to the largest used clause ID *)

               val _   = if !clause_offset = ~1 then clause_offset :=

                 (case Inttab.max_key (!clause_table) of

                   SOME id => id

                 | NONE    => cnf_number_of_clauses (PropLogic.defcnf fm) - 1  (* the first clause ID is 0, not 1 *))

                 else

                   ()

               val vid = int_from_string id

               val _   = if levsep = "L:" then () else raise INVALID_PROOF ("File format error: \"L:\" expected (" ^ quote levsep ^ " encountered).")

               val _   = int_from_string levid

               val _   = if valsep = "V:" then () else raise INVALID_PROOF ("File format error: \"V:\" expected (" ^ quote valsep ^ " encountered).")

               val _   = int_from_string valid

               val _   = if antesep = "A:" then () else raise INVALID_PROOF ("File format error: \"A:\" expected (" ^ quote antesep ^ " encountered).")

               val aid = int_from_string anteid

               val _   = if litsep = "Lits:" then () else raise INVALID_PROOF ("File format error: \"Lits:\" expected (" ^ quote litsep ^ " encountered).")

               val ls  = map int_from_string lits

               (* convert the data provided by zChaff to our resolution-style proof format *)

               (* each "VAR:" line defines a unit clause, the resolvents are implicitly    *)

               (* given by the literals in the antecedent clause                           *)

               (* we use the sum of '!clause_offset' and the variable ID as clause ID for the unit clause *)

               val cid = !clause_offset + vid

               (* the low bit of each literal gives its sign (positive/negative), therefore  *)

               (* we have to divide each literal by 2 to obtain the proper variable ID; then *)

               (* we add '!clause_offset' to obtain the ID of the corresponding unit clause  *)

               val vids = filter (not_equal vid) (map (fn l => l div 2) ls)

               val rs   = aid :: map (fn v => !clause_offset + v) vids

             in

               (* update clause table *)

               clause_table := Inttab.update_new (cid, rs) (!clause_table)

                 handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ string_of_int cid ^ " (derived from antecedent for variable " ^ id ^ ") already defined.")

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"VAR:\" followed by an insufficient number of tokens."

         ) else if tok="CONF:" then (

           case toks of

             id::sep::ids =>

             let

               val _   = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: more than one conflicting clause specified."

               val cid = int_from_string id

               val _   = if sep = "==" then () else raise INVALID_PROOF ("File format error: \"==\" expected (" ^ quote sep ^ " encountered).")

               val ls  = map int_from_string ids

               (* the conflict clause must be resolved with the unit clauses *)

               (* for its literals to obtain the empty clause                *)

               val vids         = map (fn l => l div 2) ls

               val rs           = cid :: map (fn v => !clause_offset + v) vids

               val new_empty_id = getOpt (Inttab.max_key (!clause_table), !clause_offset) + 1

             in

               (* update clause table and conflict id *)

               clause_table := Inttab.update_new (new_empty_id, rs) (!clause_table)

                 handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ string_of_int new_empty_id ^ " (empty clause derived from clause " ^ id ^ ") already defined.");

               empty_id     := new_empty_id

             end

           | _ =>

             raise INVALID_PROOF "File format error: \"CONF:\" followed by an insufficient number of tokens."

         ) else

           raise INVALID_PROOF ("File format error: unknown token " ^ quote tok ^ " encountered.")

       (* string list -> unit *)

       fun process_lines [] =

         ()

         | process_lines (l::ls) = (

           process_tokens (String.tokens (fn c => c = #" " orelse c = #"\t") l);

           process_lines ls

         )

       (* proof *)

       val _ = process_lines proof_lines

       val _ = if !empty_id <> ~1 then () else raise INVALID_PROOF "File format error: no conflicting clause specified."

     in

       SatSolver.UNSATISFIABLE (SOME (!clause_table, !empty_id))

     end handle INVALID_PROOF reason => (warning reason; SatSolver.UNSATISFIABLE NONE))

   | result =>

     result

 in

   SatSolver.add_solver ("zchaff_with_proofs", zchaff_with_proofs)

 end;

 let

   fun zchaff fm =

   let

     val _          = if (getenv "ZCHAFF_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

     val serial_str = serial_string ()

     val inpath     = File.tmp_path (Path.explode ("isabelle" ^ serial_str ^ ".cnf"))

     val outpath    = File.tmp_path (Path.explode ("result" ^ serial_str))

     val cmd        = (getenv "ZCHAFF_HOME") ^ "/zchaff " ^ (Path.implode inpath) ^ " > " ^ (Path.implode outpath)

     fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

     fun readfn ()  = SatSolver.read_std_result_file outpath ("Instance Satisfiable", "", "Instance Unsatisfiable")

     val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

     val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

     val result     = SatSolver.make_external_solver cmd writefn readfn fm

     val _          = try File.rm inpath

     val _          = try File.rm outpath

   in

     result

   end

 in

   SatSolver.add_solver ("zchaff", zchaff)

 end;

 (* ------------------------------------------------------------------------- *)

 (* BerkMin 561 (http://eigold.tripod.com/BerkMin.html)                       *)

 (* ------------------------------------------------------------------------- *)

 let

   fun berkmin fm =

   let

     val _          = if (getenv "BERKMIN_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

     val serial_str = serial_string ()

     val inpath     = File.tmp_path (Path.explode ("isabelle" ^ serial_str ^ ".cnf"))

     val outpath    = File.tmp_path (Path.explode ("result" ^ serial_str))

     val exec       = getenv "BERKMIN_EXE"

     val cmd        = (getenv "BERKMIN_HOME") ^ "/" ^ (if exec = "" then "BerkMin561" else exec) ^ " " ^ (Path.implode inpath) ^ " > " ^ (Path.implode outpath)

     fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

     fun readfn ()  = SatSolver.read_std_result_file outpath ("Satisfiable          !!", "solution =", "UNSATISFIABLE          !!")

     val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

     val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

     val result     = SatSolver.make_external_solver cmd writefn readfn fm

     val _          = try File.rm inpath

     val _          = try File.rm outpath

   in

     result

   end

 in

   SatSolver.add_solver ("berkmin", berkmin)

 end;

 (* ------------------------------------------------------------------------- *)

 (* Jerusat 1.3 (http://www.cs.tau.ac.il/~ale1/)                              *)

 (* ------------------------------------------------------------------------- *)

 let

   fun jerusat fm =

   let

     val _          = if (getenv "JERUSAT_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

     val serial_str = serial_string ()

     val inpath     = File.tmp_path (Path.explode ("isabelle" ^ serial_str ^ ".cnf"))

     val outpath    = File.tmp_path (Path.explode ("result" ^ serial_str))

     val cmd        = (getenv "JERUSAT_HOME") ^ "/Jerusat1.3 " ^ (Path.implode inpath) ^ " > " ^ (Path.implode outpath)

     fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

     fun readfn ()  = SatSolver.read_std_result_file outpath ("s SATISFIABLE", "v ", "s UNSATISFIABLE")

     val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

     val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

     val result     = SatSolver.make_external_solver cmd writefn readfn fm

     val _          = try File.rm inpath

     val _          = try File.rm outpath

   in

     result

   end

 in

   SatSolver.add_solver ("jerusat", jerusat)

 end;

 (* ------------------------------------------------------------------------- *)

 (* Add code for other SAT solvers below.                                     *)

 (* ------------------------------------------------------------------------- *)

 (*

 let

   fun mysolver fm = ...

 in

   SatSolver.add_solver ("myname", (fn fm => if mysolver_is_configured then mysolver fm else raise SatSolver.NOT_CONFIGURED));  -- register the solver

 end;

 -- the solver is now available as SatSolver.invoke_solver "myname"

 *)