(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_fact_preprocessor.ML

    Author:     Jia Meng, Cambridge University Computer Laboratory

Transformation of axiom rules (elim/intro/etc) into CNF forms.

*)

signature SLEDGEHAMMER_FACT_PREPROCESSOR =

sig

  val trace: bool Unsynchronized.ref

  val trace_msg: (unit -> string) -> unit

  val skolem_prefix: string

  val cnf_axiom: theory -> thm -> thm list

  val pairname: thm -> string * thm

  val multi_base_blacklist: string list

  val bad_for_atp: thm -> bool

  val type_has_topsort: typ -> bool

  val cnf_rules_pairs: theory -> (string * thm) list -> (thm * (string * int)) list

  val neg_clausify: thm list -> thm list

  val combinators: thm -> thm

  val neg_conjecture_clauses: Proof.context -> thm -> int -> thm list * (string * typ) list

  val suppress_endtheory: bool Unsynchronized.ref

    (*for emergency use where endtheory causes problems*)

  val setup: theory -> theory

end;

structure Sledgehammer_Fact_Preprocessor : SLEDGEHAMMER_FACT_PREPROCESSOR =

struct

open Sledgehammer_FOL_Clause

val trace = Unsynchronized.ref false;

fun trace_msg msg = if !trace then tracing (msg ()) else ();

val skolem_prefix = "sko_"

fun freeze_thm th = #1 (Drule.legacy_freeze_thaw th);

val type_has_topsort = Term.exists_subtype

  (fn TFree (_, []) => true

    | TVar (_, []) => true

    | _ => false);

(**** Transformation of Elimination Rules into First-Order Formulas****)

val cfalse = cterm_of @{theory HOL} HOLogic.false_const;

val ctp_false = cterm_of @{theory HOL} (HOLogic.mk_Trueprop HOLogic.false_const);

(*Converts an elim-rule into an equivalent theorem that does not have the

  predicate variable.  Leaves other theorems unchanged.  We simply instantiate the

  conclusion variable to False.*)

fun transform_elim th =

  case concl_of th of    (*conclusion variable*)

       Const("Trueprop",_) $ (v as Var(_,Type("bool",[]))) =>

           Thm.instantiate ([], [(cterm_of @{theory HOL} v, cfalse)]) th

    | v as Var(_, Type("prop",[])) =>

           Thm.instantiate ([], [(cterm_of @{theory HOL} v, ctp_false)]) th

    | _ => th;

(*To enforce single-threading*)

exception Clausify_failure of theory;

(**** SKOLEMIZATION BY INFERENCE (lcp) ****)

fun rhs_extra_types lhsT rhs =

  let val lhs_vars = Term.add_tfreesT lhsT []

      fun add_new_TFrees (TFree v) =

            if member (op =) lhs_vars v then I else insert (op =) (TFree v)

        | add_new_TFrees _ = I

      val rhs_consts = fold_aterms (fn Const c => insert (op =) c | _ => I) rhs []

  in fold (#2 #> Term.fold_atyps add_new_TFrees) rhs_consts [] end;

(*Traverse a theorem, declaring Skolem function definitions. String s is the suggested

  prefix for the Skolem constant.*)

fun declare_skofuns s th =

  let

    val nref = Unsynchronized.ref 0    (* FIXME ??? *)

    fun dec_sko (Const ("Ex",_) $ (xtp as Abs (_, T, p))) (axs, thy) =

          (*Existential: declare a Skolem function, then insert into body and continue*)

          let

            val cname = skolem_prefix ^ s ^ "_" ^ Int.toString (Unsynchronized.inc nref)

            val args0 = OldTerm.term_frees xtp  (*get the formal parameter list*)

            val Ts = map type_of args0

            val extraTs = rhs_extra_types (Ts ---> T) xtp

            val argsx = map (fn T => Free (gensym "vsk", T)) extraTs

            val args = argsx @ args0

            val cT = extraTs ---> Ts ---> T

            val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp)

                    (*Forms a lambda-abstraction over the formal parameters*)

            val (c, thy') =

              Sign.declare_const ((Binding.conceal (Binding.name cname), cT), NoSyn) thy

            val cdef = cname ^ "_def"

            val thy'' =

              Theory.add_defs_i true false [(Binding.name cdef, Logic.mk_equals (c, rhs))] thy'

            val ax = Thm.axiom thy'' (Sign.full_bname thy'' cdef)

          in dec_sko (subst_bound (list_comb (c, args), p)) (ax :: axs, thy'') end

      | dec_sko (Const ("All", _) $ (Abs (a, T, p))) thx =

          (*Universal quant: insert a free variable into body and continue*)

          let val fname = Name.variant (OldTerm.add_term_names (p, [])) a

          in dec_sko (subst_bound (Free (fname, T), p)) thx end

      | dec_sko (Const ("op &", _) $ p $ q) thx = dec_sko q (dec_sko p thx)

      | dec_sko (Const ("op |", _) $ p $ q) thx = dec_sko q (dec_sko p thx)

      | dec_sko (Const ("Trueprop", _) $ p) thx = dec_sko p thx

      | dec_sko t thx = thx (*Do nothing otherwise*)

  in fn thy => dec_sko (Thm.prop_of th) ([], thy) end;

(*Traverse a theorem, accumulating Skolem function definitions.*)

fun assume_skofuns s th =

  let val sko_count = Unsynchronized.ref 0   (* FIXME ??? *)

      fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) defs =

            (*Existential: declare a Skolem function, then insert into body and continue*)

            let val skos = map (#1 o Logic.dest_equals) defs  (*existing sko fns*)

                val args = subtract (op =) skos (OldTerm.term_frees xtp) (*the formal parameters*)

                val Ts = map type_of args

                val cT = Ts ---> T

                val id = skolem_prefix ^ s ^ "_" ^ Int.toString (Unsynchronized.inc sko_count)

                val c = Free (id, cT)

                val rhs = list_abs_free (map dest_Free args,

                                         HOLogic.choice_const T $ xtp)

                      (*Forms a lambda-abstraction over the formal parameters*)

                val def = Logic.mk_equals (c, rhs)

            in dec_sko (subst_bound (list_comb(c,args), p))

                       (def :: defs)

            end

        | dec_sko (Const ("All",_) $ Abs (a, T, p)) defs =

            (*Universal quant: insert a free variable into body and continue*)

            let val fname = Name.variant (OldTerm.add_term_names (p,[])) a

            in dec_sko (subst_bound (Free(fname,T), p)) defs end

        | dec_sko (Const ("op &", _) $ p $ q) defs = dec_sko q (dec_sko p defs)

        | dec_sko (Const ("op |", _) $ p $ q) defs = dec_sko q (dec_sko p defs)

        | dec_sko (Const ("Trueprop", _) $ p) defs = dec_sko p defs

        | dec_sko t defs = defs (*Do nothing otherwise*)

  in  dec_sko (prop_of th) []  end;

(**** REPLACING ABSTRACTIONS BY COMBINATORS ****)

(*Returns the vars of a theorem*)

fun vars_of_thm th =

  map (Thm.cterm_of (theory_of_thm th) o Var) (Thm.fold_terms Term.add_vars th []);

(*Make a version of fun_cong with a given variable name*)

local

    val fun_cong' = fun_cong RS asm_rl; (*renumber f, g to prevent clashes with (a,0)*)

    val cx = hd (vars_of_thm fun_cong');

    val ty = typ_of (ctyp_of_term cx);

    val thy = theory_of_thm fun_cong;

    fun mkvar a = cterm_of thy (Var((a,0),ty));

in

fun xfun_cong x = Thm.instantiate ([], [(cx, mkvar x)]) fun_cong'

end;

(*Removes the lambdas from an equation of the form t = (%x. u).  A non-negative n,

  serves as an upper bound on how many to remove.*)

fun strip_lambdas 0 th = th

  | strip_lambdas n th =

      case prop_of th of

          _ $ (Const ("op =", _) $ _ $ Abs (x,_,_)) =>

              strip_lambdas (n-1) (freeze_thm (th RS xfun_cong x))

        | _ => th;

val lambda_free = not o Term.has_abs;

val [f_B,g_B] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_B}));

val [g_C,f_C] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_C}));

val [f_S,g_S] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_S}));

(*FIXME: requires more use of cterm constructors*)

fun abstract ct =

  let

      val thy = theory_of_cterm ct

      val Abs(x,_,body) = term_of ct

      val Type("fun",[xT,bodyT]) = typ_of (ctyp_of_term ct)

      val cxT = ctyp_of thy xT and cbodyT = ctyp_of thy bodyT

      fun makeK() = instantiate' [SOME cxT, SOME cbodyT] [SOME (cterm_of thy body)] @{thm abs_K}

  in

      case body of

          Const _ => makeK()

        | Free _ => makeK()

        | Var _ => makeK()  (*though Var isn't expected*)

        | Bound 0 => instantiate' [SOME cxT] [] @{thm abs_I} (*identity: I*)

        | rator$rand =>

            if loose_bvar1 (rator,0) then (*C or S*)

               if loose_bvar1 (rand,0) then (*S*)

                 let val crator = cterm_of thy (Abs(x,xT,rator))

                     val crand = cterm_of thy (Abs(x,xT,rand))

                     val abs_S' = cterm_instantiate [(f_S,crator),(g_S,crand)] @{thm abs_S}

                     val (_,rhs) = Thm.dest_equals (cprop_of abs_S')

                 in

                   Thm.transitive abs_S' (Conv.binop_conv abstract rhs)

                 end

               else (*C*)

                 let val crator = cterm_of thy (Abs(x,xT,rator))

                     val abs_C' = cterm_instantiate [(f_C,crator),(g_C,cterm_of thy rand)] @{thm abs_C}

                     val (_,rhs) = Thm.dest_equals (cprop_of abs_C')

                 in

                   Thm.transitive abs_C' (Conv.fun_conv (Conv.arg_conv abstract) rhs)

                 end

            else if loose_bvar1 (rand,0) then (*B or eta*)

               if rand = Bound 0 then eta_conversion ct

               else (*B*)

                 let val crand = cterm_of thy (Abs(x,xT,rand))

                     val crator = cterm_of thy rator

                     val abs_B' = cterm_instantiate [(f_B,crator),(g_B,crand)] @{thm abs_B}

                     val (_,rhs) = Thm.dest_equals (cprop_of abs_B')

                 in

                   Thm.transitive abs_B' (Conv.arg_conv abstract rhs)

                 end

            else makeK()

        | _ => error "abstract: Bad term"

  end;

(*Traverse a theorem, declaring abstraction function definitions. String s is the suggested

  prefix for the constants.*)

fun combinators_aux ct =

  if lambda_free (term_of ct) then reflexive ct

  else

  case term_of ct of

      Abs _ =>

        let val (cv, cta) = Thm.dest_abs NONE ct

            val (v, _) = dest_Free (term_of cv)

            val u_th = combinators_aux cta

            val cu = Thm.rhs_of u_th

            val comb_eq = abstract (Thm.cabs cv cu)

        in transitive (abstract_rule v cv u_th) comb_eq end

    | _ $ _ =>

        let val (ct1, ct2) = Thm.dest_comb ct

        in  combination (combinators_aux ct1) (combinators_aux ct2)  end;

fun combinators th =

  if lambda_free (prop_of th) then th

  else

    let val th = Drule.eta_contraction_rule th

        val eqth = combinators_aux (cprop_of th)

    in  equal_elim eqth th   end

    handle THM (msg,_,_) =>

      (warning (cat_lines

        ["Error in the combinator translation of " ^ Display.string_of_thm_without_context th,

          "  Exception message: " ^ msg]);

       TrueI);  (*A type variable of sort {} will cause make abstraction fail.*)

(*cterms are used throughout for efficiency*)

val cTrueprop = Thm.cterm_of @{theory HOL} HOLogic.Trueprop;

(*cterm version of mk_cTrueprop*)

fun c_mkTrueprop A = Thm.capply cTrueprop A;

(*Given an abstraction over n variables, replace the bound variables by free

  ones. Return the body, along with the list of free variables.*)

fun c_variant_abs_multi (ct0, vars) =

      let val (cv,ct) = Thm.dest_abs NONE ct0

      in  c_variant_abs_multi (ct, cv::vars)  end

      handle CTERM _ => (ct0, rev vars);

(*Given the definition of a Skolem function, return a theorem to replace

  an existential formula by a use of that function.

   Example: "EX x. x : A & x ~: B ==> sko A B : A & sko A B ~: B"  [.] *)

fun skolem_of_def def =

  let val (c,rhs) = Thm.dest_equals (cprop_of (freeze_thm def))

      val (ch, frees) = c_variant_abs_multi (rhs, [])

      val (chilbert,cabs) = Thm.dest_comb ch

      val thy = Thm.theory_of_cterm chilbert

      val t = Thm.term_of chilbert

      val T = case t of Const ("Hilbert_Choice.Eps", Type("fun",[_,T])) => T

                      | _ => raise THM ("skolem_of_def: expected Eps", 0, [def])

      val cex = Thm.cterm_of thy (HOLogic.exists_const T)

      val ex_tm = c_mkTrueprop (Thm.capply cex cabs)

      and conc =  c_mkTrueprop (Drule.beta_conv cabs (Drule.list_comb(c,frees)));

      fun tacf [prem] = rewrite_goals_tac [def] THEN rtac (prem RS @{thm someI_ex}) 1

  in  Goal.prove_internal [ex_tm] conc tacf

       |> forall_intr_list frees

       |> Thm.forall_elim_vars 0  (*Introduce Vars, but don't discharge defs.*)

       |> Thm.varifyT_global

  end;

(*Converts an Isabelle theorem (intro, elim or simp format, even higher-order) into NNF.*)

fun to_nnf th ctxt0 =

  let val th1 = th |> transform_elim |> zero_var_indexes

      val ((_, [th2]), ctxt) = Variable.import true [th1] ctxt0

      val th3 = th2

        |> Conv.fconv_rule Object_Logic.atomize

        |> Meson.make_nnf ctxt |> strip_lambdas ~1

  in  (th3, ctxt)  end;

(*Generate Skolem functions for a theorem supplied in nnf*)

fun assume_skolem_of_def s th =

  map (skolem_of_def o assume o (cterm_of (theory_of_thm th))) (assume_skofuns s th);

(*** Blacklisting (duplicated in "Sledgehammer_Fact_Filter"?) ***)

val max_lambda_nesting = 3;

fun excessive_lambdas (f$t, k) = excessive_lambdas (f,k) orelse excessive_lambdas (t,k)

  | excessive_lambdas (Abs(_,_,t), k) = k=0 orelse excessive_lambdas (t,k-1)

  | excessive_lambdas _ = false;

fun is_formula_type T = (T = HOLogic.boolT orelse T = propT);

(*Don't count nested lambdas at the level of formulas, as they are quantifiers*)

fun excessive_lambdas_fm Ts (Abs(_,T,t)) = excessive_lambdas_fm (T::Ts) t

  | excessive_lambdas_fm Ts t =

      if is_formula_type (fastype_of1 (Ts, t))

      then exists (excessive_lambdas_fm Ts) (#2 (strip_comb t))

      else excessive_lambdas (t, max_lambda_nesting);

(*The max apply_depth of any metis call in Metis_Examples (on 31-10-2007) was 11.*)

val max_apply_depth = 15;

fun apply_depth (f$t) = Int.max (apply_depth f, apply_depth t + 1)

  | apply_depth (Abs(_,_,t)) = apply_depth t

  | apply_depth _ = 0;

fun too_complex t =

  apply_depth t > max_apply_depth orelse

  Meson.too_many_clauses NONE t orelse

  excessive_lambdas_fm [] t;

fun is_strange_thm th =

  case head_of (concl_of th) of

      Const (a, _) => (a <> "Trueprop" andalso a <> "==")

    | _ => false;

fun bad_for_atp th =

  too_complex (prop_of th)

  orelse exists_type type_has_topsort (prop_of th)

  orelse is_strange_thm th;

val multi_base_blacklist =

  ["defs","select_defs","update_defs","induct","inducts","split","splits","split_asm",

   "cases","ext_cases"];  (* FIXME put other record thms here, or declare as "no_atp" *)

(*Keep the full complexity of the original name*)

fun flatten_name s = space_implode "_X" (Long_Name.explode s);

fun fake_name th =

  if Thm.has_name_hint th then flatten_name (Thm.get_name_hint th)

  else gensym "unknown_thm_";

(*Skolemize a named theorem, with Skolem functions as additional premises.*)

fun skolem_thm (s, th) =

  if member (op =) multi_base_blacklist (Long_Name.base_name s) orelse bad_for_atp th then []

  else

    let

      val ctxt0 = Variable.thm_context th

      val (nnfth, ctxt1) = to_nnf th ctxt0

      val (cnfs, ctxt2) = Meson.make_cnf (assume_skolem_of_def s nnfth) nnfth ctxt1

    in  cnfs |> map combinators |> Variable.export ctxt2 ctxt0 |> Meson.finish_cnf  end

    handle THM _ => [];

(*The cache prevents repeated clausification of a theorem, and also repeated declaration of

  Skolem functions.*)

structure ThmCache = Theory_Data

(

  type T = thm list Thmtab.table * unit Symtab.table;

  val empty = (Thmtab.empty, Symtab.empty);

  val extend = I;

  fun merge ((cache1, seen1), (cache2, seen2)) : T =

    (Thmtab.merge (K true) (cache1, cache2), Symtab.merge (K true) (seen1, seen2));

);

val lookup_cache = Thmtab.lookup o #1 o ThmCache.get;

val already_seen = Symtab.defined o #2 o ThmCache.get;

val update_cache = ThmCache.map o apfst o Thmtab.update;

fun mark_seen name = ThmCache.map (apsnd (Symtab.update (name, ())));

(*Exported function to convert Isabelle theorems into axiom clauses*)

fun cnf_axiom thy th0 =

  let val th = Thm.transfer thy th0 in

    case lookup_cache thy th of

      NONE => map Thm.close_derivation (skolem_thm (fake_name th, th))

    | SOME cls => cls

  end;

(**** Rules from the context ****)

fun pairname th = (Thm.get_name_hint th, th);

(**** Translate a set of theorems into CNF ****)

fun pair_name_cls k (n, []) = []

  | pair_name_cls k (n, cls::clss) = (cls, (n,k)) :: pair_name_cls (k+1) (n, clss)

fun cnf_rules_pairs_aux _ pairs [] = pairs

  | cnf_rules_pairs_aux thy pairs ((name,th)::ths) =

      let val pairs' = (pair_name_cls 0 (name, cnf_axiom thy th)) @ pairs

                       handle THM _ => pairs |

                              CLAUSE _ => pairs

      in  cnf_rules_pairs_aux thy pairs' ths  end;

(*The combination of rev and tail recursion preserves the original order*)

fun cnf_rules_pairs thy l = cnf_rules_pairs_aux thy [] (rev l);

(**** Convert all facts of the theory into FOL or HOL clauses ****)

local

fun skolem_def (name, th) thy =

  let val ctxt0 = Variable.thm_context th in

    (case try (to_nnf th) ctxt0 of

      NONE => (NONE, thy)

    | SOME (nnfth, ctxt1) =>

        let val (defs, thy') = declare_skofuns (flatten_name name) nnfth thy

        in (SOME (th, ctxt0, ctxt1, nnfth, defs), thy') end)

  end;

fun skolem_cnfs (th, ctxt0, ctxt1, nnfth, defs) =

  let

    val (cnfs, ctxt2) = Meson.make_cnf (map skolem_of_def defs) nnfth ctxt1;

    val cnfs' = cnfs

      |> map combinators

      |> Variable.export ctxt2 ctxt0

      |> Meson.finish_cnf

      |> map Thm.close_derivation;

    in (th, cnfs') end;

in

fun saturate_skolem_cache thy =

  let

    val facts = PureThy.facts_of thy;

    val new_facts = (facts, []) |-> Facts.fold_static (fn (name, ths) =>

      if Facts.is_concealed facts name orelse already_seen thy name then I

      else cons (name, ths));

    val new_thms = (new_facts, []) |-> fold (fn (name, ths) =>

      if member (op =) multi_base_blacklist (Long_Name.base_name name) then I

      else fold_index (fn (i, th) =>

        if bad_for_atp th orelse is_some (lookup_cache thy th) then I

        else cons (name ^ "_" ^ string_of_int (i + 1), Thm.transfer thy th)) ths);

  in

    if null new_facts then NONE

    else

      let

        val (defs, thy') = thy

          |> fold (mark_seen o #1) new_facts

          |> fold_map skolem_def (sort_distinct (Thm.thm_ord o pairself snd) new_thms)

          |>> map_filter I;

        val cache_entries = Par_List.map skolem_cnfs defs;

      in SOME (fold update_cache cache_entries thy') end

  end;

end;

val suppress_endtheory = Unsynchronized.ref false;

fun clause_cache_endtheory thy =

  if ! suppress_endtheory then NONE

  else saturate_skolem_cache thy;

(*The cache can be kept smaller by inspecting the prop of each thm. Can ignore all that are

  lambda_free, but then the individual theory caches become much bigger.*)

(*** Converting a subgoal into negated conjecture clauses. ***)

fun neg_skolemize_tac ctxt =

  EVERY' [rtac ccontr, Object_Logic.atomize_prems_tac, Meson.skolemize_tac ctxt];

val neg_clausify =

  Meson.make_clauses_unsorted #> map combinators #> Meson.finish_cnf;

fun neg_conjecture_clauses ctxt st0 n =

  let

    val st = Seq.hd (neg_skolemize_tac ctxt n st0)

    val ({params, prems, ...}, _) = Subgoal.focus (Variable.set_body false ctxt) n st

  in (neg_clausify prems, map (Term.dest_Free o Thm.term_of o #2) params) end;

(*Conversion of a subgoal to conjecture clauses. Each clause has

  leading !!-bound universal variables, to express generality. *)

fun neg_clausify_tac ctxt =

  neg_skolemize_tac ctxt THEN'

  SUBGOAL (fn (prop, i) =>

    let val ts = Logic.strip_assums_hyp prop in

      EVERY'

       [Subgoal.FOCUS

         (fn {prems, ...} =>

           (Method.insert_tac

             (map forall_intr_vars (neg_clausify prems)) i)) ctxt,

        REPEAT_DETERM_N (length ts) o etac thin_rl] i

     end);

val neg_clausify_setup =

  Method.setup @{binding neg_clausify} (Scan.succeed (SIMPLE_METHOD' o neg_clausify_tac))

  "conversion of goal to conjecture clauses";

(** Attribute for converting a theorem into clauses **)

val clausify_setup =

  Attrib.setup @{binding clausify}

    (Scan.lift OuterParse.nat >>

      (fn i => Thm.rule_attribute (fn context => fn th =>

          Meson.make_meta_clause (nth (cnf_axiom (Context.theory_of context) th) i))))

  "conversion of theorem to clauses";

(** setup **)

val setup =

  neg_clausify_setup #>

  clausify_setup #>

  perhaps saturate_skolem_cache #>

  Theory.at_end clause_cache_endtheory;

end;