src/HOL/Tools/Sledgehammer/sledgehammer_translate.ML

+(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_translate.ML
+    Author:     Fabian Immler, TU Muenchen
+    Author:     Makarius
+    Author:     Jasmin Blanchette, TU Muenchen
+
+Translation of HOL to FOL.
+*)
+
+signature SLEDGEHAMMER_TRANSLATE =
+sig
+  type 'a problem = 'a ATP_Problem.problem
+
+  val axiom_prefix : string
+  val conjecture_prefix : string
+  val helper_prefix : string
+  val class_rel_clause_prefix : string
+  val arity_clause_prefix : string
+  val tfrees_name : string
+  val prepare_problem :
+    Proof.context -> bool -> bool -> bool -> bool -> term list -> term
+    -> (string * thm) list
+    -> string problem * string Symtab.table * int * string Vector.vector
+end;
+
+structure Sledgehammer_Translate : SLEDGEHAMMER_TRANSLATE =
+struct
+
+open ATP_Problem
+open Metis_Clauses
+open Sledgehammer_Util
+
+val axiom_prefix = "ax_"
+val conjecture_prefix = "conj_"
+val helper_prefix = "help_"
+val class_rel_clause_prefix = "clrel_";
+val arity_clause_prefix = "arity_"
+val tfrees_name = "tfrees"
+
+(* Freshness almost guaranteed! *)
+val sledgehammer_weak_prefix = "Sledgehammer:"
+
+datatype fol_formula =
+  FOLFormula of {name: string,
+                 kind: kind,
+                 combformula: (name, combterm) formula,
+                 ctypes_sorts: typ list}
+
+fun mk_anot phi = AConn (ANot, [phi])
+fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])
+fun mk_ahorn [] phi = phi
+  | mk_ahorn (phi :: phis) psi =
+    AConn (AImplies, [fold (mk_aconn AAnd) phis phi, psi])
+
+fun combformula_for_prop thy =
+  let
+    val do_term = combterm_from_term thy
+    fun do_quant bs q s T t' =
+      do_formula ((s, T) :: bs) t'
+      #>> (fn phi => AQuant (q, [`make_bound_var s], phi))
+    and do_conn bs c t1 t2 =
+      do_formula bs t1 ##>> do_formula bs t2
+      #>> (fn (phi1, phi2) => AConn (c, [phi1, phi2]))
+    and do_formula bs t =
+      case t of
+        @{const Not} $ t1 =>
+        do_formula bs t1 #>> (fn phi => AConn (ANot, [phi]))
+      | Const (@{const_name All}, _) $ Abs (s, T, t') =>
+        do_quant bs AForall s T t'
+      | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
+        do_quant bs AExists s T t'
+      | @{const "op &"} $ t1 $ t2 => do_conn bs AAnd t1 t2
+      | @{const "op |"} $ t1 $ t2 => do_conn bs AOr t1 t2
+      | @{const "op -->"} $ t1 $ t2 => do_conn bs AImplies t1 t2
+      | Const (@{const_name "op ="}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
+        do_conn bs AIff t1 t2
+      | _ => (fn ts => do_term bs (Envir.eta_contract t)
+                       |>> AAtom ||> union (op =) ts)
+  in do_formula [] end
+
+(* 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. (Cf. "transform_elim_term" in
+   "ATP_Systems".) *)
+fun transform_elim_term t =
+  case Logic.strip_imp_concl t of
+    @{const Trueprop} $ Var (z, @{typ bool}) =>
+    subst_Vars [(z, @{const False})] t
+  | Var (z, @{typ prop}) => subst_Vars [(z, @{prop False})] t
+  | _ => t
+
+fun presimplify_term thy =
+  Skip_Proof.make_thm thy
+  #> Meson.presimplify
+  #> prop_of
+
+fun concealed_bound_name j = sledgehammer_weak_prefix ^ Int.toString j
+fun conceal_bounds Ts t =
+  subst_bounds (map (Free o apfst concealed_bound_name)
+                    (0 upto length Ts - 1 ~~ Ts), t)
+fun reveal_bounds Ts =
+  subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
+                    (0 upto length Ts - 1 ~~ Ts))
+
+fun introduce_combinators_in_term ctxt kind t =
+  let
+    val thy = ProofContext.theory_of ctxt
+    fun aux Ts t =
+      case t of
+        @{const Not} $ t1 => @{const Not} $ aux Ts t1
+      | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
+        t0 $ Abs (s, T, aux (T :: Ts) t')
+      | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
+        t0 $ Abs (s, T, aux (T :: Ts) t')
+      | (t0 as @{const "op &"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
+      | (t0 as @{const "op |"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
+      | (t0 as @{const "op -->"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
+      | (t0 as Const (@{const_name "op ="}, Type (_, [@{typ bool}, _])))
+          $ t1 $ t2 =>
+        t0 $ aux Ts t1 $ aux Ts t2
+      | _ => if not (exists_subterm (fn Abs _ => true | _ => false) t) then
+               t
+             else
+               let
+                 val t = t |> conceal_bounds Ts
+                           |> Envir.eta_contract
+                 val ([t], ctxt') = Variable.import_terms true [t] ctxt
+               in
+                 t |> cterm_of thy
+                   |> Clausifier.introduce_combinators_in_cterm
+                   |> singleton (Variable.export ctxt' ctxt)
+                   |> prop_of |> Logic.dest_equals |> snd
+                   |> reveal_bounds Ts
+               end
+  in t |> not (Meson.is_fol_term thy t) ? aux [] end
+  handle THM _ =>
+         (* A type variable of sort "{}" will make abstraction fail. *)
+         case kind of
+           Axiom => HOLogic.true_const
+         | Conjecture => HOLogic.false_const
+
+(* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
+   same in Sledgehammer to prevent the discovery of unreplable proofs. *)
+fun freeze_term t =
+  let
+    fun aux (t $ u) = aux t $ aux u
+      | aux (Abs (s, T, t)) = Abs (s, T, aux t)
+      | aux (Var ((s, i), T)) =
+        Free (sledgehammer_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
+      | aux t = t
+  in t |> exists_subterm is_Var t ? aux end
+
+(* making axiom and conjecture formulas *)
+fun make_formula ctxt presimp (name, kind, t) =
+  let
+    val thy = ProofContext.theory_of ctxt
+    val t = t |> transform_elim_term
+              |> Object_Logic.atomize_term thy
+    val t = t |> fastype_of t = HOLogic.boolT ? HOLogic.mk_Trueprop
+              |> extensionalize_term
+              |> presimp ? presimplify_term thy
+              |> perhaps (try (HOLogic.dest_Trueprop))
+              |> introduce_combinators_in_term ctxt kind
+              |> kind = Conjecture ? freeze_term
+    val (combformula, ctypes_sorts) = combformula_for_prop thy t []
+  in
+    FOLFormula {name = name, combformula = combformula, kind = kind,
+                ctypes_sorts = ctypes_sorts}
+  end
+
+fun make_axiom ctxt presimp (name, th) =
+  (name, make_formula ctxt presimp (name, Axiom, prop_of th))
+fun make_conjectures ctxt ts =
+  map2 (fn j => fn t => make_formula ctxt true (Int.toString j, Conjecture, t))
+       (0 upto length ts - 1) ts
+
+(** Helper facts **)
+
+fun count_combterm (CombConst ((s, _), _, _)) =
+    Symtab.map_entry s (Integer.add 1)
+  | count_combterm (CombVar _) = I
+  | count_combterm (CombApp (t1, t2)) = fold count_combterm [t1, t2]
+fun count_combformula (AQuant (_, _, phi)) = count_combformula phi
+  | count_combformula (AConn (_, phis)) = fold count_combformula phis
+  | count_combformula (AAtom tm) = count_combterm tm
+fun count_fol_formula (FOLFormula {combformula, ...}) =
+  count_combformula combformula
+
+val optional_helpers =
+  [(["c_COMBI", "c_COMBK"], @{thms COMBI_def COMBK_def}),
+   (["c_COMBB", "c_COMBC"], @{thms COMBB_def COMBC_def}),
+   (["c_COMBS"], @{thms COMBS_def})]
+val optional_typed_helpers =
+  [(["c_True", "c_False"], @{thms True_or_False}),
+   (["c_If"], @{thms if_True if_False True_or_False})]
+val mandatory_helpers = @{thms fequal_imp_equal equal_imp_fequal}
+
+val init_counters =
+  Symtab.make (maps (maps (map (rpair 0) o fst))
+                    [optional_helpers, optional_typed_helpers])
+
+fun get_helper_facts ctxt is_FO full_types conjectures axioms =
+  let
+    val ct = fold (fold count_fol_formula) [conjectures, axioms] init_counters
+    fun is_needed c = the (Symtab.lookup ct c) > 0
+  in
+    (optional_helpers
+     |> full_types ? append optional_typed_helpers
+     |> maps (fn (ss, ths) =>
+                 if exists is_needed ss then map (`Thm.get_name_hint) ths
+                 else [])) @
+    (if is_FO then [] else map (`Thm.get_name_hint) mandatory_helpers)
+    |> map (snd o make_axiom ctxt false)
+  end
+
+fun meta_not t = @{const "==>"} $ t $ @{prop False}
+
+fun prepare_formulas ctxt full_types hyp_ts concl_t axioms =
+  let
+    val thy = ProofContext.theory_of ctxt
+    val axiom_ts = map (prop_of o snd) axioms
+    val hyp_ts =
+      if null hyp_ts then
+        []
+      else
+        (* Remove existing axioms from the conjecture, as this can dramatically
+           boost an ATP's performance (for some reason). *)
+        let
+          val axiom_table = fold (Termtab.update o rpair ()) axiom_ts
+                                 Termtab.empty
+        in hyp_ts |> filter_out (Termtab.defined axiom_table) end
+    val goal_t = Logic.list_implies (hyp_ts, concl_t)
+    val is_FO = Meson.is_fol_term thy goal_t
+    val subs = tfree_classes_of_terms [goal_t]
+    val supers = tvar_classes_of_terms axiom_ts
+    val tycons = type_consts_of_terms thy (goal_t :: axiom_ts)
+    (* TFrees in the conjecture; TVars in the axioms *)
+    val conjectures = map meta_not hyp_ts @ [concl_t] |> make_conjectures ctxt
+    val (axiom_names, axioms) =
+      ListPair.unzip (map (make_axiom ctxt true) axioms)
+    val helper_facts = get_helper_facts ctxt is_FO full_types conjectures axioms
+    val (supers', arity_clauses) = make_arity_clauses thy tycons supers
+    val class_rel_clauses = make_class_rel_clauses thy subs supers'
+  in
+    (Vector.fromList axiom_names,
+     (conjectures, axioms, helper_facts, class_rel_clauses, arity_clauses))
+  end
+
+fun wrap_type ty t = ATerm ((type_wrapper_name, type_wrapper_name), [ty, t])
+
+fun fo_term_for_combtyp (CombTVar name) = ATerm (name, [])
+  | fo_term_for_combtyp (CombTFree name) = ATerm (name, [])
+  | fo_term_for_combtyp (CombType (name, tys)) =
+    ATerm (name, map fo_term_for_combtyp tys)
+
+fun fo_literal_for_type_literal (TyLitVar (class, name)) =
+    (true, ATerm (class, [ATerm (name, [])]))
+  | fo_literal_for_type_literal (TyLitFree (class, name)) =
+    (true, ATerm (class, [ATerm (name, [])]))
+
+fun formula_for_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
+
+fun fo_term_for_combterm full_types =
+  let
+    fun aux top_level u =
+      let
+        val (head, args) = strip_combterm_comb u
+        val (x, ty_args) =
+          case head of
+            CombConst (name as (s, s'), _, ty_args) =>
+            if s = "equal" then
+              (if top_level andalso length args = 2 then name
+               else ("c_fequal", @{const_name fequal}), [])
+            else if top_level then
+              case s of
+                "c_False" => (("$false", s'), [])
+              | "c_True" => (("$true", s'), [])
+              | _ => (name, if full_types then [] else ty_args)
+            else
+              (name, if full_types then [] else ty_args)
+          | CombVar (name, _) => (name, [])
+          | CombApp _ => raise Fail "impossible \"CombApp\""
+        val t = ATerm (x, map fo_term_for_combtyp ty_args @
+                          map (aux false) args)
+    in
+      if full_types then wrap_type (fo_term_for_combtyp (combtyp_of u)) t else t
+    end
+  in aux true end
+
+fun formula_for_combformula full_types =
+  let
+    fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)
+      | aux (AConn (c, phis)) = AConn (c, map aux phis)
+      | aux (AAtom tm) = AAtom (fo_term_for_combterm full_types tm)
+  in aux end
+
+fun formula_for_axiom full_types (FOLFormula {combformula, ctypes_sorts, ...}) =
+  mk_ahorn (map (formula_for_fo_literal o fo_literal_for_type_literal)
+                (type_literals_for_types ctypes_sorts))
+           (formula_for_combformula full_types combformula)
+
+fun problem_line_for_fact prefix full_types
+                          (formula as FOLFormula {name, kind, ...}) =
+  Fof (prefix ^ ascii_of name, kind, formula_for_axiom full_types formula)
+
+fun problem_line_for_class_rel_clause (ClassRelClause {name, subclass,
+                                                       superclass, ...}) =
+  let val ty_arg = ATerm (("T", "T"), []) in
+    Fof (class_rel_clause_prefix ^ ascii_of name, Axiom,
+         AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
+                           AAtom (ATerm (superclass, [ty_arg]))]))
+  end
+
+fun fo_literal_for_arity_literal (TConsLit (c, t, args)) =
+    (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
+  | fo_literal_for_arity_literal (TVarLit (c, sort)) =
+    (false, ATerm (c, [ATerm (sort, [])]))
+
+fun problem_line_for_arity_clause (ArityClause {name, conclLit, premLits,
+                                                ...}) =
+  Fof (arity_clause_prefix ^ ascii_of name, Axiom,
+       mk_ahorn (map (formula_for_fo_literal o apfst not
+                      o fo_literal_for_arity_literal) premLits)
+                (formula_for_fo_literal
+                     (fo_literal_for_arity_literal conclLit)))
+
+fun problem_line_for_conjecture full_types
+                                (FOLFormula {name, kind, combformula, ...}) =
+  Fof (conjecture_prefix ^ name, kind,
+       formula_for_combformula full_types combformula)
+
+fun free_type_literals_for_conjecture (FOLFormula {ctypes_sorts, ...}) =
+  map fo_literal_for_type_literal (type_literals_for_types ctypes_sorts)
+
+fun problem_line_for_free_type lit =
+  Fof (tfrees_name, Conjecture, mk_anot (formula_for_fo_literal lit))
+fun problem_lines_for_free_types conjectures =
+  let
+    val litss = map free_type_literals_for_conjecture conjectures
+    val lits = fold (union (op =)) litss []
+  in map problem_line_for_free_type lits end
+
+(** "hBOOL" and "hAPP" **)
+
+type const_info = {min_arity: int, max_arity: int, sub_level: bool}
+
+fun consider_term top_level (ATerm ((s, _), ts)) =
+  (if is_tptp_variable s then
+     I
+   else
+     let val n = length ts in
+       Symtab.map_default
+           (s, {min_arity = n, max_arity = 0, sub_level = false})
+           (fn {min_arity, max_arity, sub_level} =>
+               {min_arity = Int.min (n, min_arity),
+                max_arity = Int.max (n, max_arity),
+                sub_level = sub_level orelse not top_level})
+     end)
+  #> fold (consider_term (top_level andalso s = type_wrapper_name)) ts
+fun consider_formula (AQuant (_, _, phi)) = consider_formula phi
+  | consider_formula (AConn (_, phis)) = fold consider_formula phis
+  | consider_formula (AAtom tm) = consider_term true tm
+
+fun consider_problem_line (Fof (_, _, phi)) = consider_formula phi
+fun consider_problem problem = fold (fold consider_problem_line o snd) problem
+
+fun const_table_for_problem explicit_apply problem =
+  if explicit_apply then NONE
+  else SOME (Symtab.empty |> consider_problem problem)
+
+fun min_arity_of thy full_types NONE s =
+    (if s = "equal" orelse s = type_wrapper_name orelse
+        String.isPrefix type_const_prefix s orelse
+        String.isPrefix class_prefix s then
+       16383 (* large number *)
+     else if full_types then
+       0
+     else case strip_prefix_and_undo_ascii const_prefix s of
+       SOME s' => num_type_args thy (invert_const s')
+     | NONE => 0)
+  | min_arity_of _ _ (SOME the_const_tab) s =
+    case Symtab.lookup the_const_tab s of
+      SOME ({min_arity, ...} : const_info) => min_arity
+    | NONE => 0
+
+fun full_type_of (ATerm ((s, _), [ty, _])) =
+    if s = type_wrapper_name then ty else raise Fail "expected type wrapper"
+  | full_type_of _ = raise Fail "expected type wrapper"
+
+fun list_hAPP_rev _ t1 [] = t1
+  | list_hAPP_rev NONE t1 (t2 :: ts2) =
+    ATerm (`I "hAPP", [list_hAPP_rev NONE t1 ts2, t2])
+  | list_hAPP_rev (SOME ty) t1 (t2 :: ts2) =
+    let val ty' = ATerm (`make_fixed_type_const @{type_name fun},
+                         [full_type_of t2, ty]) in
+      ATerm (`I "hAPP", [wrap_type ty' (list_hAPP_rev (SOME ty') t1 ts2), t2])
+    end
+
+fun repair_applications_in_term thy full_types const_tab =
+  let
+    fun aux opt_ty (ATerm (name as (s, _), ts)) =
+      if s = type_wrapper_name then
+        case ts of
+          [t1, t2] => ATerm (name, [aux NONE t1, aux (SOME t1) t2])
+        | _ => raise Fail "malformed type wrapper"
+      else
+        let
+          val ts = map (aux NONE) ts
+          val (ts1, ts2) = chop (min_arity_of thy full_types const_tab s) ts
+        in list_hAPP_rev opt_ty (ATerm (name, ts1)) (rev ts2) end
+  in aux NONE end
+
+fun boolify t = ATerm (`I "hBOOL", [t])
+
+(* True if the constant ever appears outside of the top-level position in
+   literals, or if it appears with different arities (e.g., because of different
+   type instantiations). If false, the constant always receives all of its
+   arguments and is used as a predicate. *)
+fun is_predicate NONE s =
+    s = "equal" orelse String.isPrefix type_const_prefix s orelse
+    String.isPrefix class_prefix s
+  | is_predicate (SOME the_const_tab) s =
+    case Symtab.lookup the_const_tab s of
+      SOME {min_arity, max_arity, sub_level} =>
+      not sub_level andalso min_arity = max_arity
+    | NONE => false
+
+fun repair_predicates_in_term const_tab (t as ATerm ((s, _), ts)) =
+  if s = type_wrapper_name then
+    case ts of
+      [_, t' as ATerm ((s', _), _)] =>
+      if is_predicate const_tab s' then t' else boolify t
+    | _ => raise Fail "malformed type wrapper"
+  else
+    t |> not (is_predicate const_tab s) ? boolify
+
+fun close_universally phi =
+  let
+    fun term_vars bounds (ATerm (name as (s, _), tms)) =
+        (is_tptp_variable s andalso not (member (op =) bounds name))
+          ? insert (op =) name
+        #> fold (term_vars bounds) tms
+    fun formula_vars bounds (AQuant (q, xs, phi)) =
+        formula_vars (xs @ bounds) phi
+      | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
+      | formula_vars bounds (AAtom tm) = term_vars bounds tm
+  in
+    case formula_vars [] phi [] of [] => phi | xs => AQuant (AForall, xs, phi)
+  end
+
+fun repair_formula thy explicit_forall full_types const_tab =
+  let
+    fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)
+      | aux (AConn (c, phis)) = AConn (c, map aux phis)
+      | aux (AAtom tm) =
+        AAtom (tm |> repair_applications_in_term thy full_types const_tab
+                  |> repair_predicates_in_term const_tab)
+  in aux #> explicit_forall ? close_universally end
+
+fun repair_problem_line thy explicit_forall full_types const_tab
+                        (Fof (ident, kind, phi)) =
+  Fof (ident, kind, repair_formula thy explicit_forall full_types const_tab phi)
+fun repair_problem_with_const_table thy =
+  map o apsnd o map ooo repair_problem_line thy
+
+fun repair_problem thy explicit_forall full_types explicit_apply problem =
+  repair_problem_with_const_table thy explicit_forall full_types
+      (const_table_for_problem explicit_apply problem) problem
+
+fun prepare_problem ctxt readable_names explicit_forall full_types
+                    explicit_apply hyp_ts concl_t axiom_ts =
+  let
+    val thy = ProofContext.theory_of ctxt
+    val (axiom_names, (conjectures, axioms, helper_facts, class_rel_clauses,
+                       arity_clauses)) =
+      prepare_formulas ctxt full_types hyp_ts concl_t axiom_ts
+    val axiom_lines = map (problem_line_for_fact axiom_prefix full_types) axioms
+    val helper_lines =
+      map (problem_line_for_fact helper_prefix full_types) helper_facts
+    val conjecture_lines =
+      map (problem_line_for_conjecture full_types) conjectures
+    val tfree_lines = problem_lines_for_free_types conjectures
+    val class_rel_lines =
+      map problem_line_for_class_rel_clause class_rel_clauses
+    val arity_lines = map problem_line_for_arity_clause arity_clauses
+    (* Reordering these might or might not confuse the proof reconstruction
+       code or the SPASS Flotter hack. *)
+    val problem =
+      [("Relevant facts", axiom_lines),
+       ("Class relationships", class_rel_lines),
+       ("Arity declarations", arity_lines),
+       ("Helper facts", helper_lines),
+       ("Conjectures", conjecture_lines),
+       ("Type variables", tfree_lines)]
+      |> repair_problem thy explicit_forall full_types explicit_apply
+    val (problem, pool) = nice_tptp_problem readable_names problem
+    val conjecture_offset =
+      length axiom_lines + length class_rel_lines + length arity_lines
+      + length helper_lines
+  in
+    (problem,
+     case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
+     conjecture_offset, axiom_names)
+  end
+
+end;