src/HOL/Tools/Meson/meson_clausify.ML

eliminated old List.nth;
tuned;

(*  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 extensionalize_theorem : thm -> thm
  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_term" 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 = Name.variant (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 ****)

val fun_cong_all = @{thm fun_eq_iff [THEN iffD1]}

(* Removes the lambdas from an equation of the form "t = (%x. u)".
   (Cf. "extensionalize_term" in "Sledgehammer_Translate".) *)
fun extensionalize_theorem th =
  case prop_of th of
    _ $ (Const (@{const_name HOL.eq}, Type (_, [Type (@{type_name fun}, _), _]))
         $ _ $ Abs _) => extensionalize_theorem (th RS fun_cong_all)
  | _ => th

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
                |> extensionalize_theorem
                |> 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;