(*  Title:      HOL/Sledgehammer/sledgehammer_hol_clause.ML

    Author:     Jia Meng, NICTA

FOL clauses translated from HOL formulae.

*)

signature SLEDGEHAMMER_HOL_CLAUSE =

sig

  type kind = Sledgehammer_FOL_Clause.kind

  type fol_type = Sledgehammer_FOL_Clause.fol_type

  type classrel_clause = Sledgehammer_FOL_Clause.classrel_clause

  type arity_clause = Sledgehammer_FOL_Clause.arity_clause

  type axiom_name = string

  type polarity = bool

  type hol_clause_id = int

  datatype combterm =

    CombConst of string * fol_type * fol_type list (* Const and Free *) |

    CombVar of string * fol_type |

    CombApp of combterm * combterm

  datatype literal = Literal of polarity * combterm

  datatype hol_clause =

    HOLClause of {clause_id: hol_clause_id, axiom_name: axiom_name, th: thm,

                  kind: kind, literals: literal list, ctypes_sorts: typ list}

  val type_of_combterm : combterm -> fol_type

  val strip_combterm_comb : combterm -> combterm * combterm list

  val literals_of_term : theory -> term -> literal list * typ list

  exception TRIVIAL

  val make_conjecture_clauses : bool -> theory -> thm list -> hol_clause list

  val make_axiom_clauses : bool -> theory ->

       (thm * (axiom_name * hol_clause_id)) list -> (axiom_name * hol_clause) list

  val get_helper_clauses : bool -> theory -> bool ->

       hol_clause list * (thm * (axiom_name * hol_clause_id)) list * string list ->

       hol_clause list

  val write_tptp_file : bool -> Path.T ->

    hol_clause list * hol_clause list * hol_clause list * hol_clause list *

    classrel_clause list * arity_clause list ->

    int * int

  val write_dfg_file : bool -> Path.T ->

    hol_clause list * hol_clause list * hol_clause list * hol_clause list *

    classrel_clause list * arity_clause list -> int * int

end

structure Sledgehammer_HOL_Clause : SLEDGEHAMMER_HOL_CLAUSE =

struct

open Sledgehammer_FOL_Clause

open Sledgehammer_Fact_Preprocessor

(* Parameter t_full below indicates that full type information is to be

exported *)

(* If true, each function will be directly applied to as many arguments as

   possible, avoiding use of the "apply" operator. Use of hBOOL is also

   minimized. *)

val minimize_applies = true;

fun min_arity_of const_min_arity c = the_default 0 (Symtab.lookup const_min_arity c);

(*True if the constant ever appears outside of the top-level position in literals.

  If false, the constant always receives all of its arguments and is used as a predicate.*)

fun needs_hBOOL const_needs_hBOOL c =

  not minimize_applies orelse

    the_default false (Symtab.lookup const_needs_hBOOL c);

(******************************************************)

(* data types for typed combinator expressions        *)

(******************************************************)

type axiom_name = string;

type polarity = bool;

type hol_clause_id = int;

datatype combterm =

  CombConst of string * fol_type * fol_type list (* Const and Free *) |

  CombVar of string * fol_type |

  CombApp of combterm * combterm

datatype literal = Literal of polarity * combterm;

datatype hol_clause =

  HOLClause of {clause_id: hol_clause_id, axiom_name: axiom_name, th: thm,

                kind: kind, literals: literal list, ctypes_sorts: typ list};

(*********************************************************************)

(* convert a clause with type Term.term to a clause with type clause *)

(*********************************************************************)

fun isFalse (Literal(pol, CombConst(c,_,_))) =

      (pol andalso c = "c_False") orelse (not pol andalso c = "c_True")

  | isFalse _ = false;

fun isTrue (Literal (pol, CombConst(c,_,_))) =

      (pol andalso c = "c_True") orelse

      (not pol andalso c = "c_False")

  | isTrue _ = false;

fun isTaut (HOLClause {literals,...}) = exists isTrue literals;

fun type_of dfg (Type (a, Ts)) =

      let val (folTypes,ts) = types_of dfg Ts

      in  (Comp(make_fixed_type_const dfg a, folTypes), ts)  end

  | type_of _ (tp as TFree (a, _)) = (AtomF (make_fixed_type_var a), [tp])

  | type_of _ (tp as TVar (v, _)) = (AtomV (make_schematic_type_var v), [tp])

and types_of dfg Ts =

      let val (folTyps,ts) = ListPair.unzip (map (type_of dfg) Ts)

      in  (folTyps, union_all ts)  end;

(* same as above, but no gathering of sort information *)

fun simp_type_of dfg (Type (a, Ts)) =

      Comp(make_fixed_type_const dfg a, map (simp_type_of dfg) Ts)

  | simp_type_of _ (TFree (a, _)) = AtomF (make_fixed_type_var a)

  | simp_type_of _ (TVar (v, _)) = AtomV (make_schematic_type_var v);

fun const_type_of dfg thy (c,t) =

      let val (tp,ts) = type_of dfg t

      in  (tp, ts, map (simp_type_of dfg) (Sign.const_typargs thy (c,t))) end;

(* convert a Term.term (with combinators) into a combterm, also accummulate sort info *)

fun combterm_of dfg thy (Const(c,t)) =

      let val (tp,ts,tvar_list) = const_type_of dfg thy (c,t)

          val c' = CombConst(make_fixed_const dfg c, tp, tvar_list)

      in  (c',ts)  end

  | combterm_of dfg _ (Free(v,t)) =

      let val (tp,ts) = type_of dfg t

          val v' = CombConst(make_fixed_var v, tp, [])

      in  (v',ts)  end

  | combterm_of dfg _ (Var(v,t)) =

      let val (tp,ts) = type_of dfg t

          val v' = CombVar(make_schematic_var v,tp)

      in  (v',ts)  end

  | combterm_of dfg thy (P $ Q) =

      let val (P',tsP) = combterm_of dfg thy P

          val (Q',tsQ) = combterm_of dfg thy Q

      in  (CombApp(P',Q'), union (op =) tsP tsQ)  end

  | combterm_of _ _ (t as Abs _) = raise CLAUSE ("HOL CLAUSE", t);

fun predicate_of dfg thy ((@{const Not} $ P), polarity) = predicate_of dfg thy (P, not polarity)

  | predicate_of dfg thy (t,polarity) = (combterm_of dfg thy (Envir.eta_contract t), polarity);

fun literals_of_term1 dfg thy args (@{const Trueprop} $ P) = literals_of_term1 dfg thy args P

  | literals_of_term1 dfg thy args (@{const "op |"} $ P $ Q) =

      literals_of_term1 dfg thy (literals_of_term1 dfg thy args P) Q

  | literals_of_term1 dfg thy (lits,ts) P =

      let val ((pred,ts'),pol) = predicate_of dfg thy (P,true)

      in

          (Literal(pol,pred)::lits, union (op =) ts ts')

      end;

fun literals_of_term_dfg dfg thy P = literals_of_term1 dfg thy ([],[]) P;

val literals_of_term = literals_of_term_dfg false;

(* Trivial problem, which resolution cannot handle (empty clause) *)

exception TRIVIAL;

(* making axiom and conjecture clauses *)

fun make_clause dfg thy (clause_id, axiom_name, kind, th) =

    let val (lits,ctypes_sorts) = literals_of_term_dfg dfg thy (prop_of th)

    in

        if forall isFalse lits then

            raise TRIVIAL

        else

            HOLClause {clause_id = clause_id, axiom_name = axiom_name, th = th,

                       kind = kind, literals = lits, ctypes_sorts = ctypes_sorts}

    end;

fun add_axiom_clause dfg thy ((th,(name,id)), pairs) =

  let val cls = make_clause dfg thy (id, name, Axiom, th)

  in

      if isTaut cls then pairs else (name,cls)::pairs

  end;

fun make_axiom_clauses dfg thy = List.foldl (add_axiom_clause dfg thy) [];

fun make_conjecture_clauses_aux _ _ _ [] = []

  | make_conjecture_clauses_aux dfg thy n (th::ths) =

      make_clause dfg thy (n,"conjecture", Conjecture, th) ::

      make_conjecture_clauses_aux dfg thy (n+1) ths;

fun make_conjecture_clauses dfg thy = make_conjecture_clauses_aux dfg thy 0;

(**********************************************************************)

(* convert clause into ATP specific formats:                          *)

(* TPTP used by Vampire and E                                         *)

(* DFG used by SPASS                                                  *)

(**********************************************************************)

(*Result of a function type; no need to check that the argument type matches.*)

fun result_type (Comp ("tc_fun", [_, tp2])) = tp2

  | result_type _ = error "result_type"

fun type_of_combterm (CombConst (_, tp, _)) = tp

  | type_of_combterm (CombVar (_, tp)) = tp

  | type_of_combterm (CombApp (t1, _)) = result_type (type_of_combterm t1);

(*gets the head of a combinator application, along with the list of arguments*)

fun strip_combterm_comb u =

    let fun stripc (CombApp(t,u), ts) = stripc (t, u::ts)

        |   stripc  x =  x

    in  stripc(u,[])  end;

val type_wrapper = "ti";

fun head_needs_hBOOL const_needs_hBOOL (CombConst(c,_,_)) = needs_hBOOL const_needs_hBOOL c

  | head_needs_hBOOL _ _ = true;

fun wrap_type t_full (s, tp) =

  if t_full then

      type_wrapper ^ paren_pack [s, string_of_fol_type tp]

  else s;

fun apply ss = "hAPP" ^ paren_pack ss;

fun rev_apply (v, []) = v

  | rev_apply (v, arg::args) = apply [rev_apply (v, args), arg];

fun string_apply (v, args) = rev_apply (v, rev args);

(*Apply an operator to the argument strings, using either the "apply" operator or

  direct function application.*)

fun string_of_applic t_full cma (CombConst (c, _, tvars), args) =

      let val c = if c = "equal" then "c_fequal" else c

          val nargs = min_arity_of cma c

          val args1 = List.take(args, nargs)

            handle Subscript => error ("string_of_applic: " ^ c ^ " has arity " ^

                                         Int.toString nargs ^ " but is applied to " ^

                                         space_implode ", " args)

          val args2 = List.drop(args, nargs)

          val targs = if not t_full then map string_of_fol_type tvars else []

      in

          string_apply (c ^ paren_pack (args1@targs), args2)

      end

  | string_of_applic _ _ (CombVar (v, _), args) = string_apply (v, args)

  | string_of_applic _ _ _ = error "string_of_applic";

fun wrap_type_if t_full cnh (head, s, tp) =

  if head_needs_hBOOL cnh head then wrap_type t_full (s, tp) else s;

fun string_of_combterm (params as (t_full, cma, cnh)) t =

  let val (head, args) = strip_combterm_comb t

  in  wrap_type_if t_full cnh (head,

                    string_of_applic t_full cma (head, map (string_of_combterm (params)) args),

                    type_of_combterm t)

  end;

(*Boolean-valued terms are here converted to literals.*)

fun boolify params t =

  "hBOOL" ^ paren_pack [string_of_combterm params t];

fun string_of_predicate (params as (_,_,cnh)) t =

  case t of

      (CombApp(CombApp(CombConst("equal",_,_), t1), t2)) =>

          (*DFG only: new TPTP prefers infix equality*)

          ("equal" ^ paren_pack [string_of_combterm params t1, string_of_combterm params t2])

    | _ =>

          case #1 (strip_combterm_comb t) of

              CombConst(c,_,_) => if needs_hBOOL cnh c then boolify params t else string_of_combterm params t

            | _ => boolify params t;

(*** tptp format ***)

fun tptp_of_equality params pol (t1,t2) =

  let val eqop = if pol then " = " else " != "

  in  string_of_combterm params t1 ^ eqop ^ string_of_combterm params t2  end;

fun tptp_literal params (Literal(pol, CombApp(CombApp(CombConst("equal", _, _), t1), t2))) =

      tptp_of_equality params pol (t1,t2)

  | tptp_literal params (Literal(pol,pred)) =

      tptp_sign pol (string_of_predicate params pred);

(*Given a clause, returns its literals paired with a list of literals concerning TFrees;

  the latter should only occur in conjecture clauses.*)

fun tptp_type_lits params pos (HOLClause {literals, ctypes_sorts, ...}) =

      (map (tptp_literal params) literals, 

       map (tptp_of_typeLit pos) (add_typs ctypes_sorts));

fun clause2tptp params (cls as HOLClause {axiom_name, clause_id, kind, ...}) =

  let val (lits,tylits) = tptp_type_lits params (kind = Conjecture) cls

  in

      (gen_tptp_cls (clause_id, axiom_name, kind, lits, tylits), tylits)

  end;

(*** dfg format ***)

fun dfg_literal params (Literal(pol,pred)) = dfg_sign pol (string_of_predicate params pred);

fun dfg_type_lits params pos (HOLClause {literals, ctypes_sorts, ...}) =

      (map (dfg_literal params) literals, 

       map (dfg_of_typeLit pos) (add_typs ctypes_sorts));

fun get_uvars (CombConst _) vars = vars

  | get_uvars (CombVar(v,_)) vars = (v::vars)

  | get_uvars (CombApp(P,Q)) vars = get_uvars P (get_uvars Q vars);

fun get_uvars_l (Literal(_,c)) = get_uvars c [];

fun dfg_vars (HOLClause {literals,...}) = union_all (map get_uvars_l literals);

fun clause2dfg params (cls as HOLClause {axiom_name, clause_id, kind,

                                         ctypes_sorts, ...}) =

  let val (lits,tylits) = dfg_type_lits params (kind = Conjecture) cls

      val vars = dfg_vars cls

      val tvars = get_tvar_strs ctypes_sorts

  in

      (gen_dfg_cls (clause_id, axiom_name, kind, lits, tylits, tvars@vars), tylits)

  end;

(** For DFG format: accumulate function and predicate declarations **)

fun addtypes tvars tab = List.foldl add_foltype_funcs tab tvars;

fun add_decls (t_full, cma, cnh) (CombConst (c, _, tvars), (funcs, preds)) =

      if c = "equal" then (addtypes tvars funcs, preds)

      else

        let val arity = min_arity_of cma c

            val ntys = if not t_full then length tvars else 0

            val addit = Symtab.update(c, arity+ntys)

        in

            if needs_hBOOL cnh c then (addtypes tvars (addit funcs), preds)

            else (addtypes tvars funcs, addit preds)

        end

  | add_decls _ (CombVar(_,ctp), (funcs,preds)) =

      (add_foltype_funcs (ctp,funcs), preds)

  | add_decls params (CombApp(P,Q),decls) = add_decls params (P,add_decls params (Q,decls));

fun add_literal_decls params (Literal (_,c), decls) = add_decls params (c,decls);

fun add_clause_decls params (HOLClause {literals, ...}, decls) =

    List.foldl (add_literal_decls params) decls literals

    handle Symtab.DUP a => error ("function " ^ a ^ " has multiple arities")

fun decls_of_clauses params clauses arity_clauses =

  let val init_functab = Symtab.update (type_wrapper,2) (Symtab.update ("hAPP",2) init_functab)

      val init_predtab = Symtab.update ("hBOOL",1) Symtab.empty

      val (functab,predtab) = (List.foldl (add_clause_decls params) (init_functab, init_predtab) clauses)

  in

      (Symtab.dest (List.foldl add_arity_clause_funcs functab arity_clauses),

       Symtab.dest predtab)

  end;

fun add_clause_preds (HOLClause {ctypes_sorts, ...}, preds) =

  List.foldl add_type_sort_preds preds ctypes_sorts

  handle Symtab.DUP a => error ("predicate " ^ a ^ " has multiple arities")

(*Higher-order clauses have only the predicates hBOOL and type classes.*)

fun preds_of_clauses clauses clsrel_clauses arity_clauses =

    Symtab.dest

        (List.foldl add_classrel_clause_preds

               (List.foldl add_arity_clause_preds

                      (List.foldl add_clause_preds Symtab.empty clauses)

                      arity_clauses)

               clsrel_clauses)

(**********************************************************************)

(* write clauses to files                                             *)

(**********************************************************************)

val init_counters =

  Symtab.make [("c_COMBI", 0), ("c_COMBK", 0), ("c_COMBB", 0), ("c_COMBC", 0),

               ("c_COMBS", 0)];

fun count_combterm (CombConst (c, _, _), ct) =

     (case Symtab.lookup ct c of NONE => ct  (*no counter*)

                               | SOME n => Symtab.update (c,n+1) ct)

  | count_combterm (CombVar _, ct) = ct

  | count_combterm (CombApp(t1,t2), ct) = count_combterm(t1, count_combterm(t2, ct));

fun count_literal (Literal(_,t), ct) = count_combterm(t,ct);

fun count_clause (HOLClause {literals, ...}, ct) =

  List.foldl count_literal ct literals;

fun count_user_clause user_lemmas (HOLClause {axiom_name, literals, ...}, ct) =

  if axiom_name mem_string user_lemmas then List.foldl count_literal ct literals

  else ct;

fun cnf_helper_thms thy = cnf_rules_pairs thy o map pairname

fun get_helper_clauses dfg thy isFO (conjectures, axcls, user_lemmas) =

  if isFO then

    []

  else

    let

        val axclauses = map #2 (make_axiom_clauses dfg thy axcls)

        val ct0 = List.foldl count_clause init_counters conjectures

        val ct = List.foldl (count_user_clause user_lemmas) ct0 axclauses

        fun needed c = the (Symtab.lookup ct c) > 0

        val IK = if needed "c_COMBI" orelse needed "c_COMBK"

                 then cnf_helper_thms thy [@{thm COMBI_def}, @{thm COMBK_def}]

                 else []

        val BC = if needed "c_COMBB" orelse needed "c_COMBC"

                 then cnf_helper_thms thy [@{thm COMBB_def}, @{thm COMBC_def}]

                 else []

        val S = if needed "c_COMBS" then cnf_helper_thms thy [@{thm COMBS_def}]

                else []

        val other = cnf_helper_thms thy [@{thm fequal_imp_equal},

                                         @{thm equal_imp_fequal}]

    in

        map #2 (make_axiom_clauses dfg thy (other @ IK @ BC @ S))

    end;

(*Find the minimal arity of each function mentioned in the term. Also, note which uses

  are not at top level, to see if hBOOL is needed.*)

fun count_constants_term toplev t (const_min_arity, const_needs_hBOOL) =

  let val (head, args) = strip_combterm_comb t

      val n = length args

      val (const_min_arity, const_needs_hBOOL) = fold (count_constants_term false) args (const_min_arity, const_needs_hBOOL)

  in

      case head of

          CombConst (a,_,_) => (*predicate or function version of "equal"?*)

            let val a = if a="equal" andalso not toplev then "c_fequal" else a

            val const_min_arity = Symtab.map_default (a, n) (Integer.min n) const_min_arity

            in

              if toplev then (const_min_arity, const_needs_hBOOL)

              else (const_min_arity, Symtab.update (a,true) (const_needs_hBOOL))

            end

        | _ => (const_min_arity, const_needs_hBOOL)

  end;

(*A literal is a top-level term*)

fun count_constants_lit (Literal (_,t)) (const_min_arity, const_needs_hBOOL) =

  count_constants_term true t (const_min_arity, const_needs_hBOOL);

fun count_constants_clause (HOLClause {literals, ...})

                           (const_min_arity, const_needs_hBOOL) =

  fold count_constants_lit literals (const_min_arity, const_needs_hBOOL);

fun display_arity const_needs_hBOOL (c,n) =

  trace_msg (fn () => "Constant: " ^ c ^

                " arity:\t" ^ Int.toString n ^

                (if needs_hBOOL const_needs_hBOOL c then " needs hBOOL" else ""));

fun count_constants (conjectures, _, extra_clauses, helper_clauses, _, _) =

  if minimize_applies then

     let val (const_min_arity, const_needs_hBOOL) =

          fold count_constants_clause conjectures (Symtab.empty, Symtab.empty)

       |> fold count_constants_clause extra_clauses

       |> fold count_constants_clause helper_clauses

     val _ = List.app (display_arity const_needs_hBOOL) (Symtab.dest (const_min_arity))

     in (const_min_arity, const_needs_hBOOL) end

  else (Symtab.empty, Symtab.empty);

(* TPTP format *)

fun write_tptp_file t_full file clauses =

  let

    val (conjectures, axclauses, _, helper_clauses,

      classrel_clauses, arity_clauses) = clauses

    val (cma, cnh) = count_constants clauses

    val params = (t_full, cma, cnh)

    val (tptp_clss,tfree_litss) = ListPair.unzip (map (clause2tptp params) conjectures)

    val tfree_clss = map tptp_tfree_clause (List.foldl (uncurry (union (op =))) [] tfree_litss)

    val _ =

      File.write_list file (

        map (#1 o (clause2tptp params)) axclauses @

        tfree_clss @

        tptp_clss @

        map tptp_classrel_clause classrel_clauses @

        map tptp_arity_clause arity_clauses @

        map (#1 o (clause2tptp params)) helper_clauses)

    in (length axclauses + 1, length tfree_clss + length tptp_clss)

  end;

(* DFG format *)

fun write_dfg_file t_full file clauses =

  let

    val (conjectures, axclauses, _, helper_clauses,

      classrel_clauses, arity_clauses) = clauses

    val (cma, cnh) = count_constants clauses

    val params = (t_full, cma, cnh)

    val (dfg_clss, tfree_litss) = ListPair.unzip (map (clause2dfg params) conjectures)

    and probname = Path.implode (Path.base file)

    val axstrs = map (#1 o (clause2dfg params)) axclauses

    val tfree_clss = map dfg_tfree_clause (union_all tfree_litss)

    val helper_clauses_strs = map (#1 o (clause2dfg params)) helper_clauses

    val (funcs,cl_preds) = decls_of_clauses params (helper_clauses @ conjectures @ axclauses) arity_clauses

    and ty_preds = preds_of_clauses axclauses classrel_clauses arity_clauses

    val _ =

      File.write_list file (

        string_of_start probname ::

        string_of_descrip probname ::

        string_of_symbols (string_of_funcs funcs)

          (string_of_preds (cl_preds @ ty_preds)) ::

        "list_of_clauses(axioms,cnf).\n" ::

        axstrs @

        map dfg_classrel_clause classrel_clauses @

        map dfg_arity_clause arity_clauses @

        helper_clauses_strs @

        ["end_of_list.\n\nlist_of_clauses(conjectures,cnf).\n"] @

        tfree_clss @

        dfg_clss @

        ["end_of_list.\n\n",

        (*VarWeight=3 helps the HO problems, probably by counteracting the presence of hAPP*)

         "list_of_settings(SPASS).\n{*\nset_flag(VarWeight,3).\n*}\nend_of_list.\n\n",

         "end_problem.\n"])

    in (length axclauses + length classrel_clauses + length arity_clauses +

      length helper_clauses + 1, length tfree_clss + length dfg_clss)

  end;

end;