src/HOL/Tools/Meson/meson_clausify.ML
author blanchet
Tue Oct 05 10:28:11 2010 +0200 (2010-10-05)
changeset 39948 317010af8972
parent 39941 02fcd9cd1eac
child 39949 186a3b447e0b
permissions -rw-r--r--
factor out "Meson_Tactic" from "Meson_Clausify"
     1 (*  Title:      HOL/Tools/Meson/meson_clausify.ML
     2     Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA
     3     Author:     Jasmin Blanchette, TU Muenchen
     4 
     5 Transformation of HOL theorems into CNF forms.
     6 *)
     7 
     8 signature MESON_CLAUSIFY =
     9 sig
    10   val new_skolem_var_prefix : string
    11   val extensionalize_theorem : thm -> thm
    12   val introduce_combinators_in_cterm : cterm -> thm
    13   val introduce_combinators_in_theorem : thm -> thm
    14   val to_definitional_cnf_with_quantifiers : theory -> thm -> thm
    15   val cluster_of_zapped_var_name : string -> (int * (int * int)) * bool
    16   val cnf_axiom :
    17     Proof.context -> bool -> int -> thm -> (thm * term) option * thm list
    18 end;
    19 
    20 structure Meson_Clausify : MESON_CLAUSIFY =
    21 struct
    22 
    23 (* the extra "?" helps prevent clashes *)
    24 val new_skolem_var_prefix = "?SK"
    25 val new_nonskolem_var_prefix = "?V"
    26 
    27 (**** Transformation of Elimination Rules into First-Order Formulas****)
    28 
    29 val cfalse = cterm_of @{theory HOL} HOLogic.false_const;
    30 val ctp_false = cterm_of @{theory HOL} (HOLogic.mk_Trueprop HOLogic.false_const);
    31 
    32 (* Converts an elim-rule into an equivalent theorem that does not have the
    33    predicate variable. Leaves other theorems unchanged. We simply instantiate
    34    the conclusion variable to False. (Cf. "transform_elim_term" in
    35    "Sledgehammer_Util".) *)
    36 fun transform_elim_theorem th =
    37   case concl_of th of    (*conclusion variable*)
    38        @{const Trueprop} $ (v as Var (_, @{typ bool})) =>
    39            Thm.instantiate ([], [(cterm_of @{theory HOL} v, cfalse)]) th
    40     | v as Var(_, @{typ prop}) =>
    41            Thm.instantiate ([], [(cterm_of @{theory HOL} v, ctp_false)]) th
    42     | _ => th
    43 
    44 
    45 (**** SKOLEMIZATION BY INFERENCE (lcp) ****)
    46 
    47 fun mk_old_skolem_term_wrapper t =
    48   let val T = fastype_of t in
    49     Const (@{const_name skolem}, T --> T) $ t
    50   end
    51 
    52 fun beta_eta_in_abs_body (Abs (s, T, t')) = Abs (s, T, beta_eta_in_abs_body t')
    53   | beta_eta_in_abs_body t = Envir.beta_eta_contract t
    54 
    55 (*Traverse a theorem, accumulating Skolem function definitions.*)
    56 fun old_skolem_defs th =
    57   let
    58     fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (_, T, p))) rhss =
    59         (*Existential: declare a Skolem function, then insert into body and continue*)
    60         let
    61           val args = OldTerm.term_frees body
    62           (* Forms a lambda-abstraction over the formal parameters *)
    63           val rhs =
    64             list_abs_free (map dest_Free args,
    65                            HOLogic.choice_const T $ beta_eta_in_abs_body body)
    66             |> mk_old_skolem_term_wrapper
    67           val comb = list_comb (rhs, args)
    68         in dec_sko (subst_bound (comb, p)) (rhs :: rhss) end
    69       | dec_sko (Const (@{const_name All},_) $ Abs (a, T, p)) rhss =
    70         (*Universal quant: insert a free variable into body and continue*)
    71         let val fname = Name.variant (OldTerm.add_term_names (p,[])) a
    72         in dec_sko (subst_bound (Free(fname,T), p)) rhss end
    73       | dec_sko (@{const conj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
    74       | dec_sko (@{const disj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
    75       | dec_sko (@{const Trueprop} $ p) rhss = dec_sko p rhss
    76       | dec_sko _ rhss = rhss
    77   in  dec_sko (prop_of th) []  end;
    78 
    79 
    80 (**** REPLACING ABSTRACTIONS BY COMBINATORS ****)
    81 
    82 val fun_cong_all = @{thm fun_eq_iff [THEN iffD1]}
    83 
    84 (* Removes the lambdas from an equation of the form "t = (%x. u)".
    85    (Cf. "extensionalize_term" in "Sledgehammer_Translate".) *)
    86 fun extensionalize_theorem th =
    87   case prop_of th of
    88     _ $ (Const (@{const_name HOL.eq}, Type (_, [Type (@{type_name fun}, _), _]))
    89          $ _ $ Abs _) => extensionalize_theorem (th RS fun_cong_all)
    90   | _ => th
    91 
    92 fun is_quasi_lambda_free (Const (@{const_name skolem}, _) $ _) = true
    93   | is_quasi_lambda_free (t1 $ t2) =
    94     is_quasi_lambda_free t1 andalso is_quasi_lambda_free t2
    95   | is_quasi_lambda_free (Abs _) = false
    96   | is_quasi_lambda_free _ = true
    97 
    98 val [f_B,g_B] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_B}));
    99 val [g_C,f_C] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_C}));
   100 val [f_S,g_S] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_S}));
   101 
   102 (* FIXME: Requires more use of cterm constructors. *)
   103 fun abstract ct =
   104   let
   105       val thy = theory_of_cterm ct
   106       val Abs(x,_,body) = term_of ct
   107       val Type(@{type_name fun}, [xT,bodyT]) = typ_of (ctyp_of_term ct)
   108       val cxT = ctyp_of thy xT
   109       val cbodyT = ctyp_of thy bodyT
   110       fun makeK () =
   111         instantiate' [SOME cxT, SOME cbodyT] [SOME (cterm_of thy body)]
   112                      @{thm abs_K}
   113   in
   114       case body of
   115           Const _ => makeK()
   116         | Free _ => makeK()
   117         | Var _ => makeK()  (*though Var isn't expected*)
   118         | Bound 0 => instantiate' [SOME cxT] [] @{thm abs_I} (*identity: I*)
   119         | rator$rand =>
   120             if loose_bvar1 (rator,0) then (*C or S*)
   121                if loose_bvar1 (rand,0) then (*S*)
   122                  let val crator = cterm_of thy (Abs(x,xT,rator))
   123                      val crand = cterm_of thy (Abs(x,xT,rand))
   124                      val abs_S' = cterm_instantiate [(f_S,crator),(g_S,crand)] @{thm abs_S}
   125                      val (_,rhs) = Thm.dest_equals (cprop_of abs_S')
   126                  in
   127                    Thm.transitive abs_S' (Conv.binop_conv abstract rhs)
   128                  end
   129                else (*C*)
   130                  let val crator = cterm_of thy (Abs(x,xT,rator))
   131                      val abs_C' = cterm_instantiate [(f_C,crator),(g_C,cterm_of thy rand)] @{thm abs_C}
   132                      val (_,rhs) = Thm.dest_equals (cprop_of abs_C')
   133                  in
   134                    Thm.transitive abs_C' (Conv.fun_conv (Conv.arg_conv abstract) rhs)
   135                  end
   136             else if loose_bvar1 (rand,0) then (*B or eta*)
   137                if rand = Bound 0 then Thm.eta_conversion ct
   138                else (*B*)
   139                  let val crand = cterm_of thy (Abs(x,xT,rand))
   140                      val crator = cterm_of thy rator
   141                      val abs_B' = cterm_instantiate [(f_B,crator),(g_B,crand)] @{thm abs_B}
   142                      val (_,rhs) = Thm.dest_equals (cprop_of abs_B')
   143                  in Thm.transitive abs_B' (Conv.arg_conv abstract rhs) end
   144             else makeK()
   145         | _ => raise Fail "abstract: Bad term"
   146   end;
   147 
   148 (* Traverse a theorem, remplacing lambda-abstractions with combinators. *)
   149 fun introduce_combinators_in_cterm ct =
   150   if is_quasi_lambda_free (term_of ct) then
   151     Thm.reflexive ct
   152   else case term_of ct of
   153     Abs _ =>
   154     let
   155       val (cv, cta) = Thm.dest_abs NONE ct
   156       val (v, _) = dest_Free (term_of cv)
   157       val u_th = introduce_combinators_in_cterm cta
   158       val cu = Thm.rhs_of u_th
   159       val comb_eq = abstract (Thm.cabs cv cu)
   160     in Thm.transitive (Thm.abstract_rule v cv u_th) comb_eq end
   161   | _ $ _ =>
   162     let val (ct1, ct2) = Thm.dest_comb ct in
   163         Thm.combination (introduce_combinators_in_cterm ct1)
   164                         (introduce_combinators_in_cterm ct2)
   165     end
   166 
   167 fun introduce_combinators_in_theorem th =
   168   if is_quasi_lambda_free (prop_of th) then
   169     th
   170   else
   171     let
   172       val th = Drule.eta_contraction_rule th
   173       val eqth = introduce_combinators_in_cterm (cprop_of th)
   174     in Thm.equal_elim eqth th end
   175     handle THM (msg, _, _) =>
   176            (warning ("Error in the combinator translation of " ^
   177                      Display.string_of_thm_without_context th ^
   178                      "\nException message: " ^ msg ^ ".");
   179             (* A type variable of sort "{}" will make abstraction fail. *)
   180             TrueI)
   181 
   182 (*cterms are used throughout for efficiency*)
   183 val cTrueprop = cterm_of @{theory HOL} HOLogic.Trueprop;
   184 
   185 (*Given an abstraction over n variables, replace the bound variables by free
   186   ones. Return the body, along with the list of free variables.*)
   187 fun c_variant_abs_multi (ct0, vars) =
   188       let val (cv,ct) = Thm.dest_abs NONE ct0
   189       in  c_variant_abs_multi (ct, cv::vars)  end
   190       handle CTERM _ => (ct0, rev vars);
   191 
   192 val skolem_def_raw = @{thms skolem_def_raw}
   193 
   194 (* Given the definition of a Skolem function, return a theorem to replace
   195    an existential formula by a use of that function.
   196    Example: "EX x. x : A & x ~: B ==> sko A B : A & sko A B ~: B"  [.] *)
   197 fun old_skolem_theorem_from_def thy rhs0 =
   198   let
   199     val rhs = rhs0 |> Type.legacy_freeze_thaw |> #1 |> cterm_of thy
   200     val rhs' = rhs |> Thm.dest_comb |> snd
   201     val (ch, frees) = c_variant_abs_multi (rhs', [])
   202     val (hilbert, cabs) = ch |> Thm.dest_comb |>> term_of
   203     val T =
   204       case hilbert of
   205         Const (_, Type (@{type_name fun}, [_, T])) => T
   206       | _ => raise TERM ("old_skolem_theorem_from_def: expected \"Eps\"",
   207                          [hilbert])
   208     val cex = cterm_of thy (HOLogic.exists_const T)
   209     val ex_tm = Thm.capply cTrueprop (Thm.capply cex cabs)
   210     val conc =
   211       Drule.list_comb (rhs, frees)
   212       |> Drule.beta_conv cabs |> Thm.capply cTrueprop
   213     fun tacf [prem] =
   214       rewrite_goals_tac skolem_def_raw
   215       THEN rtac ((prem |> rewrite_rule skolem_def_raw)
   216                  RS Global_Theory.get_thm thy "someI_ex") 1
   217   in
   218     Goal.prove_internal [ex_tm] conc tacf
   219     |> forall_intr_list frees
   220     |> Thm.forall_elim_vars 0  (*Introduce Vars, but don't discharge defs.*)
   221     |> Thm.varifyT_global
   222   end
   223 
   224 fun to_definitional_cnf_with_quantifiers thy th =
   225   let
   226     val eqth = cnf.make_cnfx_thm thy (HOLogic.dest_Trueprop (prop_of th))
   227     val eqth = eqth RS @{thm eq_reflection}
   228     val eqth = eqth RS @{thm TruepropI}
   229   in Thm.equal_elim eqth th end
   230 
   231 fun zapped_var_name ((ax_no, cluster_no), skolem) index_no s =
   232   (if skolem then new_skolem_var_prefix else new_nonskolem_var_prefix) ^
   233   "_" ^ string_of_int ax_no ^ "_" ^ string_of_int cluster_no ^ "_" ^
   234   string_of_int index_no ^ "_" ^ s
   235 
   236 fun cluster_of_zapped_var_name s =
   237   let val get_int = the o Int.fromString o nth (space_explode "_" s) in
   238     ((get_int 1, (get_int 2, get_int 3)),
   239      String.isPrefix new_skolem_var_prefix s)
   240   end
   241 
   242 fun zap (cluster as (cluster_no, cluster_skolem)) index_no pos ct =
   243   ct
   244   |> (case term_of ct of
   245         Const (s, _) $ Abs (s', _, _) =>
   246         if s = @{const_name all} orelse s = @{const_name All} orelse
   247            s = @{const_name Ex} then
   248           let
   249             val skolem = (pos = (s = @{const_name Ex}))
   250             val (cluster, index_no) =
   251               if skolem = cluster_skolem then (cluster, index_no)
   252               else ((cluster_no ||> cluster_skolem ? Integer.add 1, skolem), 0)
   253           in
   254             Thm.dest_comb #> snd
   255             #> Thm.dest_abs (SOME (zapped_var_name cluster index_no s'))
   256             #> snd #> zap cluster (index_no + 1) pos
   257           end
   258         else
   259           Conv.all_conv
   260       | Const (s, _) $ _ $ _ =>
   261         if s = @{const_name "==>"} orelse s = @{const_name implies} then
   262           Conv.combination_conv (Conv.arg_conv (zap cluster index_no (not pos)))
   263                                 (zap cluster index_no pos)
   264         else if s = @{const_name conj} orelse s = @{const_name disj} then
   265           Conv.combination_conv (Conv.arg_conv (zap cluster index_no pos))
   266                                 (zap cluster index_no pos)
   267         else
   268           Conv.all_conv
   269       | Const (s, _) $ _ =>
   270         if s = @{const_name Trueprop} then
   271           Conv.arg_conv (zap cluster index_no pos)
   272         else if s = @{const_name Not} then
   273           Conv.arg_conv (zap cluster index_no (not pos))
   274         else
   275           Conv.all_conv
   276       | _ => Conv.all_conv)
   277 
   278 fun ss_only ths = MetaSimplifier.clear_ss HOL_basic_ss addsimps ths
   279 
   280 val no_choice =
   281   @{prop "ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)"}
   282   |> Logic.varify_global
   283   |> Skip_Proof.make_thm @{theory}
   284 
   285 (* Converts an Isabelle theorem into NNF. *)
   286 fun nnf_axiom choice_ths new_skolemizer ax_no th ctxt =
   287   let
   288     val thy = ProofContext.theory_of ctxt
   289     val th =
   290       th |> transform_elim_theorem
   291          |> zero_var_indexes
   292          |> new_skolemizer ? forall_intr_vars
   293     val (th, ctxt) = Variable.import true [th] ctxt |>> snd |>> the_single
   294     val th = th |> Conv.fconv_rule Object_Logic.atomize
   295                 |> extensionalize_theorem
   296                 |> Meson.make_nnf ctxt
   297   in
   298     if new_skolemizer then
   299       let
   300         fun skolemize choice_ths =
   301           Meson.skolemize_with_choice_thms ctxt choice_ths
   302           #> simplify (ss_only @{thms all_simps[symmetric]})
   303         val pull_out =
   304           simplify (ss_only @{thms all_simps[symmetric] ex_simps[symmetric]})
   305         val (discharger_th, fully_skolemized_th) =
   306           if null choice_ths then
   307             th |> `I |>> pull_out ||> skolemize [no_choice]
   308           else
   309             th |> skolemize choice_ths |> `I
   310         val t =
   311           fully_skolemized_th |> cprop_of
   312           |> zap ((ax_no, 0), true) 0 true |> Drule.export_without_context
   313           |> cprop_of |> Thm.dest_equals |> snd |> term_of
   314       in
   315         if exists_subterm (fn Var ((s, _), _) =>
   316                               String.isPrefix new_skolem_var_prefix s
   317                             | _ => false) t then
   318           let
   319             val (ct, ctxt) =
   320               Variable.import_terms true [t] ctxt
   321               |>> the_single |>> cterm_of thy
   322           in (SOME (discharger_th, ct), Thm.assume ct, ctxt) end
   323        else
   324          (NONE, th, ctxt)
   325       end
   326     else
   327       (NONE, th, ctxt)
   328   end
   329 
   330 (* Convert a theorem to CNF, with additional premises due to skolemization. *)
   331 fun cnf_axiom ctxt0 new_skolemizer ax_no th =
   332   let
   333     val thy = ProofContext.theory_of ctxt0
   334     val choice_ths = Meson_Choices.get ctxt0
   335     val (opt, nnf_th, ctxt) = nnf_axiom choice_ths new_skolemizer ax_no th ctxt0
   336     fun clausify th =
   337       Meson.make_cnf (if new_skolemizer then
   338                         []
   339                       else
   340                         map (old_skolem_theorem_from_def thy)
   341                             (old_skolem_defs th)) th ctxt
   342     val (cnf_ths, ctxt) =
   343       clausify nnf_th
   344       |> (fn ([], _) =>
   345              clausify (to_definitional_cnf_with_quantifiers thy nnf_th)
   346            | p => p)
   347     fun intr_imp ct th =
   348       Thm.instantiate ([], map (pairself (cterm_of @{theory}))
   349                                [(Var (("i", 1), @{typ nat}),
   350                                  HOLogic.mk_nat ax_no)])
   351                       @{thm skolem_COMBK_D}
   352       RS Thm.implies_intr ct th
   353   in
   354     (opt |> Option.map (I #>> singleton (Variable.export ctxt ctxt0)
   355                         ##> (term_of #> HOLogic.dest_Trueprop
   356                              #> singleton (Variable.export_terms ctxt ctxt0))),
   357      cnf_ths |> map (introduce_combinators_in_theorem
   358                      #> (case opt of SOME (_, ct) => intr_imp ct | NONE => I))
   359              |> Variable.export ctxt ctxt0
   360              |> Meson.finish_cnf
   361              |> map Thm.close_derivation)
   362   end
   363   handle THM _ => (NONE, [])
   364 
   365 end;