(*  Title:      HOL/Tools/Meson/meson_clausify.ML

     Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA

     Author:     Jasmin Blanchette, TU Muenchen

 Transformation of HOL theorems into CNF forms.

 *)

 signature MESON_CLAUSIFY =

 sig

   val new_skolem_var_prefix : string

   val new_nonskolem_var_prefix : string

   val is_zapped_var_name : string -> bool

   val introduce_combinators_in_cterm : cterm -> thm

   val introduce_combinators_in_theorem : thm -> thm

   val cluster_of_zapped_var_name : string -> (int * (int * int)) * bool

   val ss_only : thm list -> simpset

   val cnf_axiom :

     Proof.context -> bool -> int -> thm -> (thm * term) option * thm list

 end;

 structure Meson_Clausify : MESON_CLAUSIFY =

 struct

 open Meson

 (* the extra "Meson" helps prevent clashes (FIXME) *)

 val new_skolem_var_prefix = "MesonSK"

 val new_nonskolem_var_prefix = "MesonV"

 fun is_zapped_var_name s =

   exists (fn prefix => String.isPrefix prefix s)

          [new_skolem_var_prefix, new_nonskolem_var_prefix]

 (**** 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. (Cf. "transform_elim_prop" in

    "Sledgehammer_Util".) *)

 fun transform_elim_theorem th =

   case concl_of th of    (*conclusion variable*)

        @{const Trueprop} $ (v as Var (_, @{typ bool})) =>

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

     | v as Var(_, @{typ prop}) =>

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

     | _ => th

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

 fun mk_old_skolem_term_wrapper t =

   let val T = fastype_of t in

     Const (@{const_name Meson.skolem}, T --> T) $ t

   end

 fun beta_eta_in_abs_body (Abs (s, T, t')) = Abs (s, T, beta_eta_in_abs_body t')

   | beta_eta_in_abs_body t = Envir.beta_eta_contract t

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

 fun old_skolem_defs th =

   let

     fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (_, T, p))) rhss =

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

         let

           val args = OldTerm.term_frees body

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

           val rhs =

             list_abs_free (map dest_Free args,

                            HOLogic.choice_const T $ beta_eta_in_abs_body body)

             |> mk_old_skolem_term_wrapper

           val comb = list_comb (rhs, args)

         in dec_sko (subst_bound (comb, p)) (rhs :: rhss) end

       | dec_sko (Const (@{const_name All},_) $ Abs (a, T, p)) rhss =

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

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

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

       | dec_sko (@{const conj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q

       | dec_sko (@{const disj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q

       | dec_sko (@{const Trueprop} $ p) rhss = dec_sko p rhss

       | dec_sko _ rhss = rhss

   in  dec_sko (prop_of th) []  end;

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

 fun is_quasi_lambda_free (Const (@{const_name Meson.skolem}, _) $ _) = true

   | is_quasi_lambda_free (t1 $ t2) =

     is_quasi_lambda_free t1 andalso is_quasi_lambda_free t2

   | is_quasi_lambda_free (Abs _) = false

   | is_quasi_lambda_free _ = true

 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(@{type_name fun}, [xT,bodyT]) = typ_of (ctyp_of_term ct)

       val cxT = ctyp_of thy xT

       val 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 Term.is_dependent rator then (*C or S*)

                if Term.is_dependent rand 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 Term.is_dependent rand then (*B or eta*)

                if rand = Bound 0 then Thm.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()

         | _ => raise Fail "abstract: Bad term"

   end;

 (* Traverse a theorem, remplacing lambda-abstractions with combinators. *)

 fun introduce_combinators_in_cterm ct =

   if is_quasi_lambda_free (term_of ct) then

     Thm.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 = introduce_combinators_in_cterm cta

       val cu = Thm.rhs_of u_th

       val comb_eq = abstract (Thm.cabs cv cu)

     in Thm.transitive (Thm.abstract_rule v cv u_th) comb_eq end

   | _ $ _ =>

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

         Thm.combination (introduce_combinators_in_cterm ct1)

                         (introduce_combinators_in_cterm ct2)

     end

 fun introduce_combinators_in_theorem th =

   if is_quasi_lambda_free (prop_of th) then

     th

   else

     let

       val th = Drule.eta_contraction_rule th

       val eqth = introduce_combinators_in_cterm (cprop_of th)

     in Thm.equal_elim eqth th end

     handle THM (msg, _, _) =>

            (warning ("Error in the combinator translation of " ^

                      Display.string_of_thm_without_context th ^

                      "\nException message: " ^ msg ^ ".");

             (* A type variable of sort "{}" will make abstraction fail. *)

             TrueI)

 (*cterms are used throughout for efficiency*)

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

 (*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);

 val skolem_def_raw = @{thms skolem_def_raw}

 (* 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 old_skolem_theorem_from_def thy rhs0 =

   let

     val rhs = rhs0 |> Type.legacy_freeze_thaw |> #1 |> cterm_of thy

     val rhs' = rhs |> Thm.dest_comb |> snd

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

     val (hilbert, cabs) = ch |> Thm.dest_comb |>> term_of

     val T =

       case hilbert of

         Const (_, Type (@{type_name fun}, [_, T])) => T

       | _ => raise TERM ("old_skolem_theorem_from_def: expected \"Eps\"",

                          [hilbert])

     val cex = cterm_of thy (HOLogic.exists_const T)

     val ex_tm = Thm.capply cTrueprop (Thm.capply cex cabs)

     val conc =

       Drule.list_comb (rhs, frees)

       |> Drule.beta_conv cabs |> Thm.capply cTrueprop

     fun tacf [prem] =

       rewrite_goals_tac skolem_def_raw

       THEN rtac ((prem |> rewrite_rule skolem_def_raw)

                  RS Global_Theory.get_thm thy "Hilbert_Choice.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

 fun to_definitional_cnf_with_quantifiers ctxt th =

   let

     val eqth = cnf.make_cnfx_thm ctxt (HOLogic.dest_Trueprop (prop_of th))

     val eqth = eqth RS @{thm eq_reflection}

     val eqth = eqth RS @{thm TruepropI}

   in Thm.equal_elim eqth th end

 fun zapped_var_name ((ax_no, cluster_no), skolem) index_no s =

   (if skolem then new_skolem_var_prefix else new_nonskolem_var_prefix) ^

   "_" ^ string_of_int ax_no ^ "_" ^ string_of_int cluster_no ^ "_" ^

   string_of_int index_no ^ "_" ^ Name.desymbolize false s

 fun cluster_of_zapped_var_name s =

   let val get_int = the o Int.fromString o nth (space_explode "_" s) in

     ((get_int 1, (get_int 2, get_int 3)),

      String.isPrefix new_skolem_var_prefix s)

   end

 fun rename_bound_vars_to_be_zapped ax_no =

   let

     fun aux (cluster as (cluster_no, cluster_skolem)) index_no pos t =

       case t of

         (t1 as Const (s, _)) $ Abs (s', T, t') =>

         if s = @{const_name all} orelse s = @{const_name All} orelse

            s = @{const_name Ex} then

           let

             val skolem = (pos = (s = @{const_name Ex}))

             val (cluster, index_no) =

               if skolem = cluster_skolem then (cluster, index_no)

               else ((cluster_no ||> cluster_skolem ? Integer.add 1, skolem), 0)

             val s' = zapped_var_name cluster index_no s'

           in t1 $ Abs (s', T, aux cluster (index_no + 1) pos t') end

         else

           t

       | (t1 as Const (s, _)) $ t2 $ t3 =>

         if s = @{const_name "==>"} orelse s = @{const_name HOL.implies} then

           t1 $ aux cluster index_no (not pos) t2 $ aux cluster index_no pos t3

         else if s = @{const_name HOL.conj} orelse

                 s = @{const_name HOL.disj} then

           t1 $ aux cluster index_no pos t2 $ aux cluster index_no pos t3

         else

           t

       | (t1 as Const (s, _)) $ t2 =>

         if s = @{const_name Trueprop} then

           t1 $ aux cluster index_no pos t2

         else if s = @{const_name Not} then

           t1 $ aux cluster index_no (not pos) t2

         else

           t

       | _ => t

   in aux ((ax_no, 0), true) 0 true end

 fun zap pos ct =

   ct

   |> (case term_of ct of

         Const (s, _) $ Abs (s', _, _) =>

         if s = @{const_name all} orelse s = @{const_name All} orelse

            s = @{const_name Ex} then

           Thm.dest_comb #> snd #> Thm.dest_abs (SOME s') #> snd #> zap pos

         else

           Conv.all_conv

       | Const (s, _) $ _ $ _ =>

         if s = @{const_name "==>"} orelse s = @{const_name implies} then

           Conv.combination_conv (Conv.arg_conv (zap (not pos))) (zap pos)

         else if s = @{const_name conj} orelse s = @{const_name disj} then

           Conv.combination_conv (Conv.arg_conv (zap pos)) (zap pos)

         else

           Conv.all_conv

       | Const (s, _) $ _ =>

         if s = @{const_name Trueprop} then Conv.arg_conv (zap pos)

         else if s = @{const_name Not} then Conv.arg_conv (zap (not pos))

         else Conv.all_conv

       | _ => Conv.all_conv)

 fun ss_only ths = Simplifier.clear_ss HOL_basic_ss addsimps ths

 val cheat_choice =

   @{prop "ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)"}

   |> Logic.varify_global

   |> Skip_Proof.make_thm @{theory}

 (* Converts an Isabelle theorem into NNF. *)

 fun nnf_axiom choice_ths new_skolemizer ax_no th ctxt =

   let

     val thy = Proof_Context.theory_of ctxt

     val th =

       th |> transform_elim_theorem

          |> zero_var_indexes

          |> new_skolemizer ? forall_intr_vars

     val (th, ctxt) = Variable.import true [th] ctxt |>> snd |>> the_single

     val th = th |> Conv.fconv_rule Object_Logic.atomize

                 |> Raw_Simplifier.rewrite_rule (unfold_set_const_simps ctxt)

                 |> extensionalize_theorem ctxt

                 |> make_nnf ctxt

   in

     if new_skolemizer then

       let

         fun skolemize choice_ths =

           skolemize_with_choice_theorems ctxt choice_ths

           #> simplify (ss_only @{thms all_simps[symmetric]})

         val no_choice = null choice_ths

         val pull_out =

           if no_choice then

             simplify (ss_only @{thms all_simps[symmetric] ex_simps[symmetric]})

           else

             skolemize choice_ths

         val discharger_th = th |> pull_out

         val discharger_th =

           discharger_th |> has_too_many_clauses ctxt (concl_of discharger_th)

                            ? (to_definitional_cnf_with_quantifiers ctxt

                               #> pull_out)

         val zapped_th =

           discharger_th |> prop_of |> rename_bound_vars_to_be_zapped ax_no

           |> (if no_choice then

                 Skip_Proof.make_thm thy #> skolemize [cheat_choice] #> cprop_of

               else

                 cterm_of thy)

           |> zap true

         val fixes =

           [] |> Term.add_free_names (prop_of zapped_th)

              |> filter is_zapped_var_name

         val ctxt' = ctxt |> Variable.add_fixes_direct fixes

         val fully_skolemized_t =

           zapped_th |> singleton (Variable.export ctxt' ctxt)

                     |> cprop_of |> Thm.dest_equals |> snd |> term_of

       in

         if exists_subterm (fn Var ((s, _), _) =>

                               String.isPrefix new_skolem_var_prefix s

                             | _ => false) fully_skolemized_t then

           let

             val (fully_skolemized_ct, ctxt) =

               Variable.import_terms true [fully_skolemized_t] ctxt

               |>> the_single |>> cterm_of thy

           in

             (SOME (discharger_th, fully_skolemized_ct),

              (Thm.assume fully_skolemized_ct, ctxt))

           end

        else

          (NONE, (th, ctxt))

       end

     else

       (NONE, (th |> has_too_many_clauses ctxt (concl_of th)

                     ? to_definitional_cnf_with_quantifiers ctxt, ctxt))

   end

 (* Convert a theorem to CNF, with additional premises due to skolemization. *)

 fun cnf_axiom ctxt0 new_skolemizer ax_no th =

   let

     val thy = Proof_Context.theory_of ctxt0

     val choice_ths = choice_theorems thy

     val (opt, (nnf_th, ctxt)) =

       nnf_axiom choice_ths new_skolemizer ax_no th ctxt0

     fun clausify th =

       make_cnf (if new_skolemizer orelse null choice_ths then []

                 else map (old_skolem_theorem_from_def thy) (old_skolem_defs th))

                th ctxt

     val (cnf_ths, ctxt) = clausify nnf_th

     fun intr_imp ct th =

       Thm.instantiate ([], map (pairself (cterm_of thy))

                                [(Var (("i", 0), @{typ nat}),

                                  HOLogic.mk_nat ax_no)])

                       (zero_var_indexes @{thm skolem_COMBK_D})

       RS Thm.implies_intr ct th

   in

     (opt |> Option.map (I #>> singleton (Variable.export ctxt ctxt0)

                         ##> (term_of #> HOLogic.dest_Trueprop

                              #> singleton (Variable.export_terms ctxt ctxt0))),

      cnf_ths |> map (introduce_combinators_in_theorem

                      #> (case opt of SOME (_, ct) => intr_imp ct | NONE => I))

              |> Variable.export ctxt ctxt0

              |> finish_cnf

              |> map Thm.close_derivation)

   end

   handle THM _ => (NONE, [])

 end;