(*  Title:      HOL/Tools/Metis/metis_tactic.ML

     Author:     Kong W. Susanto, Cambridge University Computer Laboratory

     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   Cambridge University 2007

 HOL setup for the Metis prover.

 *)

 signature METIS_TACTIC =

 sig

   val trace : bool Config.T

   val verbose : bool Config.T

   val new_skolemizer : bool Config.T

   val type_has_top_sort : typ -> bool

   val metis_tac :

     string list -> string -> Proof.context -> thm list -> int -> tactic

   val metis_lam_transs : string list

   val parse_metis_options : (string list option * string option) parser

   val setup : theory -> theory

 end

 structure Metis_Tactic : METIS_TACTIC =

 struct

 open ATP_Translate

 open ATP_Reconstruct

 open Metis_Translate

 open Metis_Reconstruct

 val new_skolemizer =

   Attrib.setup_config_bool @{binding metis_new_skolemizer} (K false)

 (* Designed to work also with monomorphic instances of polymorphic theorems. *)

 fun have_common_thm ths1 ths2 =

   exists (member (Term.aconv_untyped o pairself prop_of) ths1)

          (map Meson.make_meta_clause ths2)

 (*Determining which axiom clauses are actually used*)

 fun used_axioms axioms (th, Metis_Proof.Axiom _) = SOME (lookth axioms th)

   | used_axioms _ _ = NONE

 (* Lightweight predicate type information comes in two flavors, "t = t'" and

    "t => t'", where "t" and "t'" are the same term modulo type tags.

    In Isabelle, type tags are stripped away, so we are left with "t = t" or

    "t => t". Type tag idempotence is also handled this way. *)

 fun reflexive_or_trivial_from_metis ctxt type_enc sym_tab concealed mth =

   let val thy = Proof_Context.theory_of ctxt in

     case hol_clause_from_metis ctxt type_enc sym_tab concealed mth of

       Const (@{const_name HOL.eq}, _) $ _ $ t =>

       let

         val ct = cterm_of thy t

         val cT = ctyp_of_term ct

       in refl |> Drule.instantiate' [SOME cT] [SOME ct] end

     | Const (@{const_name disj}, _) $ t1 $ t2 =>

       (if can HOLogic.dest_not t1 then t2 else t1)

       |> HOLogic.mk_Trueprop |> cterm_of thy |> Thm.trivial

     | _ => raise Fail "expected reflexive or trivial clause"

   end

   |> Meson.make_meta_clause

 fun lam_lifted_from_metis ctxt type_enc sym_tab concealed mth =

   let

     val thy = Proof_Context.theory_of ctxt

     val tac = rewrite_goals_tac @{thms lambda_def_raw} THEN rtac refl 1

     val t = hol_clause_from_metis ctxt type_enc sym_tab concealed mth

     val ct = cterm_of thy (HOLogic.mk_Trueprop t)

   in Goal.prove_internal [] ct (K tac) |> Meson.make_meta_clause end

 fun add_vars_and_frees (t $ u) = fold (add_vars_and_frees) [t, u]

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

   | add_vars_and_frees (t as Var _) = insert (op =) t

   | add_vars_and_frees (t as Free _) = insert (op =) t

   | add_vars_and_frees _ = I

 fun introduce_lam_wrappers ctxt th =

   if Meson_Clausify.is_quasi_lambda_free (prop_of th) then

     th

   else

     let

       val thy = Proof_Context.theory_of ctxt

       fun conv first ctxt ct =

         if Meson_Clausify.is_quasi_lambda_free (term_of ct) then

           Thm.reflexive ct

         else case term_of ct of

           Abs (_, _, u) =>

           if first then

             case add_vars_and_frees u [] of

               [] =>

               Conv.abs_conv (conv false o snd) ctxt ct

               |> (fn th => Meson.first_order_resolve th @{thm Metis.eq_lambdaI})

             | v :: _ =>

               Abs (Name.uu, fastype_of v, abstract_over (v, term_of ct)) $ v

               |> cterm_of thy

               |> Conv.comb_conv (conv true ctxt)

           else

             Conv.abs_conv (conv false o snd) ctxt ct

         | Const (@{const_name Meson.skolem}, _) $ _ => Thm.reflexive ct

         | _ => Conv.comb_conv (conv true ctxt) ct

       val eq_th = conv true ctxt (cprop_of th)

       (* We replace the equation's left-hand side with a beta-equivalent term

          so that "Thm.equal_elim" works below. *)

       val t0 $ _ $ t2 = prop_of eq_th

       val eq_ct = t0 $ prop_of th $ t2 |> cterm_of thy

       val eq_th' = Goal.prove_internal [] eq_ct (K (Tactic.rtac eq_th 1))

     in Thm.equal_elim eq_th' th end

 val clause_params =

   {ordering = Metis_KnuthBendixOrder.default,

    orderLiterals = Metis_Clause.UnsignedLiteralOrder,

    orderTerms = true}

 val active_params =

   {clause = clause_params,

    prefactor = #prefactor Metis_Active.default,

    postfactor = #postfactor Metis_Active.default}

 val waiting_params =

   {symbolsWeight = 1.0,

    variablesWeight = 0.0,

    literalsWeight = 0.0,

    models = []}

 val resolution_params = {active = active_params, waiting = waiting_params}

 (* Main function to start Metis proof and reconstruction *)

 fun FOL_SOLVE (type_enc :: fallback_type_encs) lam_trans ctxt cls ths0 =

   let val thy = Proof_Context.theory_of ctxt

       val new_skolemizer =

         Config.get ctxt new_skolemizer orelse null (Meson.choice_theorems thy)

       val do_lams = lam_trans = lam_liftingN ? introduce_lam_wrappers ctxt

       val th_cls_pairs =

         map2 (fn j => fn th =>

                 (Thm.get_name_hint th,

                  th |> Drule.eta_contraction_rule

                     |> Meson_Clausify.cnf_axiom ctxt new_skolemizer

                                                 (lam_trans = combinatorsN) j

                     ||> map do_lams))

              (0 upto length ths0 - 1) ths0

       val ths = maps (snd o snd) th_cls_pairs

       val dischargers = map (fst o snd) th_cls_pairs

       val cls = cls |> map (Drule.eta_contraction_rule #> do_lams)

       val _ = trace_msg ctxt (fn () => "FOL_SOLVE: CONJECTURE CLAUSES")

       val _ = app (fn th => trace_msg ctxt (fn () => Display.string_of_thm ctxt th)) cls

       val _ = trace_msg ctxt (fn () => "type_enc = " ^ type_enc)

       val type_enc = type_enc_from_string Strict type_enc

       val (sym_tab, axioms, concealed) =

         prepare_metis_problem ctxt type_enc lam_trans cls ths

       fun get_isa_thm mth Isa_Reflexive_or_Trivial =

           reflexive_or_trivial_from_metis ctxt type_enc sym_tab concealed mth

         | get_isa_thm mth Isa_Lambda_Lifted =

           lam_lifted_from_metis ctxt type_enc sym_tab concealed mth

         | get_isa_thm _ (Isa_Raw ith) = ith

       val axioms = axioms |> map (fn (mth, ith) => (mth, get_isa_thm mth ith))

       val _ = trace_msg ctxt (fn () => "ISABELLE CLAUSES")

       val _ = app (fn (_, ith) => trace_msg ctxt (fn () => Display.string_of_thm ctxt ith)) axioms

       val _ = trace_msg ctxt (fn () => "METIS CLAUSES")

       val _ = app (fn (mth, _) => trace_msg ctxt (fn () => Metis_Thm.toString mth)) axioms

       val _ = trace_msg ctxt (fn () => "START METIS PROVE PROCESS")

   in

       case filter (fn t => prop_of t aconv @{prop False}) cls of

           false_th :: _ => [false_th RS @{thm FalseE}]

         | [] =>

       case Metis_Resolution.new resolution_params

                                 {axioms = axioms |> map fst, conjecture = []}

            |> Metis_Resolution.loop of

           Metis_Resolution.Contradiction mth =>

             let val _ = trace_msg ctxt (fn () => "METIS RECONSTRUCTION START: " ^

                           Metis_Thm.toString mth)

                 val ctxt' = fold Variable.declare_constraints (map prop_of cls) ctxt

                              (*add constraints arising from converting goal to clause form*)

                 val proof = Metis_Proof.proof mth

                 val result =

                   axioms

                   |> fold (replay_one_inference ctxt' type_enc concealed sym_tab) proof

                 val used = proof |> map_filter (used_axioms axioms)

                 val _ = trace_msg ctxt (fn () => "METIS COMPLETED...clauses actually used:")

                 val _ = app (fn th => trace_msg ctxt (fn () => Display.string_of_thm ctxt th)) used

                 val names = th_cls_pairs |> map fst

                 val used_names =

                   th_cls_pairs

                   |> map_filter (fn (name, (_, cls)) =>

                                     if have_common_thm used cls then SOME name

                                     else NONE)

                 val unused_names = names |> subtract (op =) used_names

             in

                 if not (null cls) andalso not (have_common_thm used cls) then

                   verbose_warning ctxt "The assumptions are inconsistent"

                 else

                   ();

                 if not (null unused_names) then

                   "Unused theorems: " ^ commas_quote unused_names

                   |> verbose_warning ctxt

                 else

                   ();

                 case result of

                     (_,ith)::_ =>

                         (trace_msg ctxt (fn () => "Success: " ^ Display.string_of_thm ctxt ith);

                          [discharge_skolem_premises ctxt dischargers ith])

                   | _ => (trace_msg ctxt (fn () => "Metis: No result"); [])

             end

         | Metis_Resolution.Satisfiable _ =>

             (trace_msg ctxt (fn () => "Metis: No first-order proof with the lemmas supplied");

              if null fallback_type_encs then

                ()

              else

                raise METIS ("FOL_SOLVE",

                             "No first-order proof with the lemmas supplied");

              [])

   end

   handle METIS (loc, msg) =>

          case fallback_type_encs of

            [] => error ("Failed to replay Metis proof in Isabelle." ^

                         (if Config.get ctxt verbose then "\n" ^ loc ^ ": " ^ msg

                          else ""))

          | first_fallback :: _ =>

            (verbose_warning ctxt

                 ("Falling back on " ^

                  quote (metis_call first_fallback lam_trans) ^ "...");

             FOL_SOLVE fallback_type_encs lam_trans ctxt cls ths0)

 fun neg_clausify ctxt combinators =

   single

   #> Meson.make_clauses_unsorted ctxt

   #> combinators ? map Meson_Clausify.introduce_combinators_in_theorem

   #> Meson.finish_cnf

 fun preskolem_tac ctxt st0 =

   (if exists (Meson.has_too_many_clauses ctxt)

              (Logic.prems_of_goal (prop_of st0) 1) then

      Simplifier.full_simp_tac (Meson_Clausify.ss_only @{thms not_all not_ex}) 1

      THEN cnf.cnfx_rewrite_tac ctxt 1

    else

      all_tac) st0

 val type_has_top_sort =

   exists_subtype (fn TFree (_, []) => true | TVar (_, []) => true | _ => false)

 fun generic_metis_tac type_encs lam_trans ctxt ths i st0 =

   let

     val _ = trace_msg ctxt (fn () =>

         "Metis called with theorems\n" ^

         cat_lines (map (Display.string_of_thm ctxt) ths))

     val type_encs = type_encs |> maps unalias_type_enc

     fun tac clause =

       resolve_tac (FOL_SOLVE type_encs lam_trans ctxt clause ths) 1

   in

     if exists_type type_has_top_sort (prop_of st0) then

       verbose_warning ctxt "Proof state contains the universal sort {}"

     else

       ();

     Meson.MESON (preskolem_tac ctxt)

         (maps (neg_clausify ctxt (lam_trans = combinatorsN))) tac ctxt i st0

   end

 fun metis_tac [] = generic_metis_tac partial_type_encs

   | metis_tac type_encs = generic_metis_tac type_encs

 (* Whenever "X" has schematic type variables, we treat "using X by metis" as

    "by (metis X)" to prevent "Subgoal.FOCUS" from freezing the type variables.

    We don't do it for nonschematic facts "X" because this breaks a few proofs

    (in the rare and subtle case where a proof relied on extensionality not being

    applied) and brings few benefits. *)

 val has_tvar =

   exists_type (exists_subtype (fn TVar _ => true | _ => false)) o prop_of

 fun method default_type_encs ((override_type_encs, lam_trans), ths) ctxt facts =

   let

     val _ =

       if default_type_encs = full_type_encs then

         legacy_feature "Old \"metisFT\" method -- use \"metis (full_types)\" instead"

       else

         ()

     val (schem_facts, nonschem_facts) = List.partition has_tvar facts

     val type_encs = override_type_encs |> the_default default_type_encs

     val lam_trans = lam_trans |> the_default metis_default_lam_trans

   in

     HEADGOAL (Method.insert_tac nonschem_facts THEN'

               CHANGED_PROP o generic_metis_tac type_encs lam_trans ctxt

                                                (schem_facts @ ths))

   end

 val metis_lam_transs = [hide_lamsN, lam_liftingN, combinatorsN]

 fun set_opt _ x NONE = SOME x

   | set_opt get x (SOME x0) =

     error ("Cannot specify both " ^ quote (get x0) ^ " and " ^ quote (get x) ^

            ".")

 fun consider_opt s =

   if member (op =) metis_lam_transs s then apsnd (set_opt I s)

   else apfst (set_opt hd [s])

 val parse_metis_options =

   Scan.optional

       (Args.parens (Parse.short_ident

                     -- Scan.option (Parse.$$$ "," |-- Parse.short_ident))

        >> (fn (s, s') =>

               (NONE, NONE) |> consider_opt s

                            |> (case s' of SOME s' => consider_opt s' | _ => I)))

       (NONE, NONE)

 fun setup_method (binding, type_encs) =

   Scan.lift parse_metis_options -- Attrib.thms >> (METHOD oo method type_encs)

   |> Method.setup binding

 val setup =

   [((@{binding metis}, partial_type_encs),

     "Metis for FOL and HOL problems"),

    ((@{binding metisFT}, full_type_encs),

     "Metis for FOL/HOL problems with fully-typed translation")]

   |> fold (uncurry setup_method)

 end;