src/HOL/Tools/Sledgehammer/meson_clausifier.ML
author blanchet
Mon Sep 27 10:44:08 2010 +0200 (2010-09-27)
changeset 39720 0b93a954da4f
parent 39561 src/HOL/Tools/Sledgehammer/clausifier.ML@3857a4a81fa7
child 39721 76a61ca09d5d
permissions -rw-r--r--
rename "Clausifier" to "Meson_Clausifier" and merge with "Meson_Tactic"
     1 (*  Title:      HOL/Tools/Sledgehammer/clausifier.ML
     2     Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA
     3     Author:     Jasmin Blanchette, TU Muenchen
     4 
     5 Transformation of axiom rules (elim/intro/etc) into CNF forms.
     6 *)
     7 
     8 signature MESON_CLAUSIFIER =
     9 sig
    10   val extensionalize_theorem : thm -> thm
    11   val introduce_combinators_in_cterm : cterm -> thm
    12   val introduce_combinators_in_theorem : thm -> thm
    13   val to_definitional_cnf_with_quantifiers : theory -> thm -> thm
    14   val cnf_axiom : theory -> thm -> thm list
    15   val meson_general_tac : Proof.context -> thm list -> int -> tactic
    16   val setup: theory -> theory
    17 end;
    18 
    19 structure Meson_Clausifier : MESON_CLAUSIFIER =
    20 struct
    21 
    22 (**** Transformation of Elimination Rules into First-Order Formulas****)
    23 
    24 val cfalse = cterm_of @{theory HOL} HOLogic.false_const;
    25 val ctp_false = cterm_of @{theory HOL} (HOLogic.mk_Trueprop HOLogic.false_const);
    26 
    27 (* Converts an elim-rule into an equivalent theorem that does not have the
    28    predicate variable. Leaves other theorems unchanged. We simply instantiate
    29    the conclusion variable to False. (Cf. "transform_elim_term" in
    30    "Sledgehammer_Util".) *)
    31 fun transform_elim_theorem th =
    32   case concl_of th of    (*conclusion variable*)
    33        @{const Trueprop} $ (v as Var (_, @{typ bool})) =>
    34            Thm.instantiate ([], [(cterm_of @{theory HOL} v, cfalse)]) th
    35     | v as Var(_, @{typ prop}) =>
    36            Thm.instantiate ([], [(cterm_of @{theory HOL} v, ctp_false)]) th
    37     | _ => th
    38 
    39 
    40 (**** SKOLEMIZATION BY INFERENCE (lcp) ****)
    41 
    42 fun mk_skolem t =
    43   let val T = fastype_of t in
    44     Const (@{const_name skolem}, T --> T) $ t
    45   end
    46 
    47 fun beta_eta_under_lambdas (Abs (s, T, t')) =
    48     Abs (s, T, beta_eta_under_lambdas t')
    49   | beta_eta_under_lambdas t = Envir.beta_eta_contract t
    50 
    51 (*Traverse a theorem, accumulating Skolem function definitions.*)
    52 fun assume_skolem_funs th =
    53   let
    54     fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (_, T, p))) rhss =
    55         (*Existential: declare a Skolem function, then insert into body and continue*)
    56         let
    57           val args = OldTerm.term_frees body
    58           (* Forms a lambda-abstraction over the formal parameters *)
    59           val rhs =
    60             list_abs_free (map dest_Free args,
    61                            HOLogic.choice_const T $ beta_eta_under_lambdas body)
    62             |> mk_skolem
    63           val comb = list_comb (rhs, args)
    64         in dec_sko (subst_bound (comb, p)) (rhs :: rhss) end
    65       | dec_sko (Const (@{const_name All},_) $ Abs (a, T, p)) rhss =
    66         (*Universal quant: insert a free variable into body and continue*)
    67         let val fname = Name.variant (OldTerm.add_term_names (p,[])) a
    68         in dec_sko (subst_bound (Free(fname,T), p)) rhss end
    69       | dec_sko (@{const HOL.conj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
    70       | dec_sko (@{const HOL.disj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
    71       | dec_sko (@{const Trueprop} $ p) rhss = dec_sko p rhss
    72       | dec_sko _ rhss = rhss
    73   in  dec_sko (prop_of th) []  end;
    74 
    75 
    76 (**** REPLACING ABSTRACTIONS BY COMBINATORS ****)
    77 
    78 val fun_cong_all = @{thm fun_eq_iff [THEN iffD1]}
    79 
    80 (* Removes the lambdas from an equation of the form "t = (%x. u)".
    81    (Cf. "extensionalize_term" in "Sledgehammer_Translate".) *)
    82 fun extensionalize_theorem th =
    83   case prop_of th of
    84     _ $ (Const (@{const_name HOL.eq}, Type (_, [Type (@{type_name fun}, _), _]))
    85          $ _ $ Abs _) => extensionalize_theorem (th RS fun_cong_all)
    86   | _ => th
    87 
    88 fun is_quasi_lambda_free (Const (@{const_name skolem}, _) $ _) = true
    89   | is_quasi_lambda_free (t1 $ t2) =
    90     is_quasi_lambda_free t1 andalso is_quasi_lambda_free t2
    91   | is_quasi_lambda_free (Abs _) = false
    92   | is_quasi_lambda_free _ = true
    93 
    94 val [f_B,g_B] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_B}));
    95 val [g_C,f_C] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_C}));
    96 val [f_S,g_S] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_S}));
    97 
    98 (* FIXME: Requires more use of cterm constructors. *)
    99 fun abstract ct =
   100   let
   101       val thy = theory_of_cterm ct
   102       val Abs(x,_,body) = term_of ct
   103       val Type(@{type_name fun}, [xT,bodyT]) = typ_of (ctyp_of_term ct)
   104       val cxT = ctyp_of thy xT
   105       val cbodyT = ctyp_of thy bodyT
   106       fun makeK () =
   107         instantiate' [SOME cxT, SOME cbodyT] [SOME (cterm_of thy body)]
   108                      @{thm abs_K}
   109   in
   110       case body of
   111           Const _ => makeK()
   112         | Free _ => makeK()
   113         | Var _ => makeK()  (*though Var isn't expected*)
   114         | Bound 0 => instantiate' [SOME cxT] [] @{thm abs_I} (*identity: I*)
   115         | rator$rand =>
   116             if loose_bvar1 (rator,0) then (*C or S*)
   117                if loose_bvar1 (rand,0) then (*S*)
   118                  let val crator = cterm_of thy (Abs(x,xT,rator))
   119                      val crand = cterm_of thy (Abs(x,xT,rand))
   120                      val abs_S' = cterm_instantiate [(f_S,crator),(g_S,crand)] @{thm abs_S}
   121                      val (_,rhs) = Thm.dest_equals (cprop_of abs_S')
   122                  in
   123                    Thm.transitive abs_S' (Conv.binop_conv abstract rhs)
   124                  end
   125                else (*C*)
   126                  let val crator = cterm_of thy (Abs(x,xT,rator))
   127                      val abs_C' = cterm_instantiate [(f_C,crator),(g_C,cterm_of thy rand)] @{thm abs_C}
   128                      val (_,rhs) = Thm.dest_equals (cprop_of abs_C')
   129                  in
   130                    Thm.transitive abs_C' (Conv.fun_conv (Conv.arg_conv abstract) rhs)
   131                  end
   132             else if loose_bvar1 (rand,0) then (*B or eta*)
   133                if rand = Bound 0 then Thm.eta_conversion ct
   134                else (*B*)
   135                  let val crand = cterm_of thy (Abs(x,xT,rand))
   136                      val crator = cterm_of thy rator
   137                      val abs_B' = cterm_instantiate [(f_B,crator),(g_B,crand)] @{thm abs_B}
   138                      val (_,rhs) = Thm.dest_equals (cprop_of abs_B')
   139                  in Thm.transitive abs_B' (Conv.arg_conv abstract rhs) end
   140             else makeK()
   141         | _ => raise Fail "abstract: Bad term"
   142   end;
   143 
   144 (* Traverse a theorem, remplacing lambda-abstractions with combinators. *)
   145 fun introduce_combinators_in_cterm ct =
   146   if is_quasi_lambda_free (term_of ct) then
   147     Thm.reflexive ct
   148   else case term_of ct of
   149     Abs _ =>
   150     let
   151       val (cv, cta) = Thm.dest_abs NONE ct
   152       val (v, _) = dest_Free (term_of cv)
   153       val u_th = introduce_combinators_in_cterm cta
   154       val cu = Thm.rhs_of u_th
   155       val comb_eq = abstract (Thm.cabs cv cu)
   156     in Thm.transitive (Thm.abstract_rule v cv u_th) comb_eq end
   157   | _ $ _ =>
   158     let val (ct1, ct2) = Thm.dest_comb ct in
   159         Thm.combination (introduce_combinators_in_cterm ct1)
   160                         (introduce_combinators_in_cterm ct2)
   161     end
   162 
   163 fun introduce_combinators_in_theorem th =
   164   if is_quasi_lambda_free (prop_of th) then
   165     th
   166   else
   167     let
   168       val th = Drule.eta_contraction_rule th
   169       val eqth = introduce_combinators_in_cterm (cprop_of th)
   170     in Thm.equal_elim eqth th end
   171     handle THM (msg, _, _) =>
   172            (warning ("Error in the combinator translation of " ^
   173                      Display.string_of_thm_without_context th ^
   174                      "\nException message: " ^ msg ^ ".");
   175             (* A type variable of sort "{}" will make abstraction fail. *)
   176             TrueI)
   177 
   178 (*cterms are used throughout for efficiency*)
   179 val cTrueprop = cterm_of @{theory HOL} HOLogic.Trueprop;
   180 
   181 (*Given an abstraction over n variables, replace the bound variables by free
   182   ones. Return the body, along with the list of free variables.*)
   183 fun c_variant_abs_multi (ct0, vars) =
   184       let val (cv,ct) = Thm.dest_abs NONE ct0
   185       in  c_variant_abs_multi (ct, cv::vars)  end
   186       handle CTERM _ => (ct0, rev vars);
   187 
   188 val skolem_def_raw = @{thms skolem_def_raw}
   189 
   190 (* Given the definition of a Skolem function, return a theorem to replace
   191    an existential formula by a use of that function.
   192    Example: "EX x. x : A & x ~: B ==> sko A B : A & sko A B ~: B"  [.] *)
   193 fun skolem_theorem_of_def thy rhs0 =
   194   let
   195     val rhs = rhs0 |> Type.legacy_freeze_thaw |> #1 |> cterm_of thy
   196     val rhs' = rhs |> Thm.dest_comb |> snd
   197     val (ch, frees) = c_variant_abs_multi (rhs', [])
   198     val (hilbert, cabs) = ch |> Thm.dest_comb |>> term_of
   199     val T =
   200       case hilbert of
   201         Const (@{const_name Eps}, Type (@{type_name fun}, [_, T])) => T
   202       | _ => raise TERM ("skolem_theorem_of_def: expected \"Eps\"", [hilbert])
   203     val cex = cterm_of thy (HOLogic.exists_const T)
   204     val ex_tm = Thm.capply cTrueprop (Thm.capply cex cabs)
   205     val conc =
   206       Drule.list_comb (rhs, frees)
   207       |> Drule.beta_conv cabs |> Thm.capply cTrueprop
   208     fun tacf [prem] =
   209       rewrite_goals_tac skolem_def_raw
   210       THEN rtac ((prem |> rewrite_rule skolem_def_raw) RS @{thm someI_ex}) 1
   211   in
   212     Goal.prove_internal [ex_tm] conc tacf
   213     |> forall_intr_list frees
   214     |> Thm.forall_elim_vars 0  (*Introduce Vars, but don't discharge defs.*)
   215     |> Thm.varifyT_global
   216   end
   217 
   218 (* Converts an Isabelle theorem (intro, elim or simp format, even higher-order)
   219    into NNF. *)
   220 fun to_nnf th ctxt0 =
   221   let
   222     val th1 = th |> transform_elim_theorem |> zero_var_indexes
   223     val ((_, [th2]), ctxt) = Variable.import true [th1] ctxt0
   224     val th3 = th2 |> Conv.fconv_rule Object_Logic.atomize
   225                   |> extensionalize_theorem
   226                   |> Meson.make_nnf ctxt
   227   in (th3, ctxt) end
   228 
   229 fun to_definitional_cnf_with_quantifiers thy th =
   230   let
   231     val eqth = cnf.make_cnfx_thm thy (HOLogic.dest_Trueprop (prop_of th))
   232     val eqth = eqth RS @{thm eq_reflection}
   233     val eqth = eqth RS @{thm TruepropI}
   234   in Thm.equal_elim eqth th end
   235 
   236 (* Convert a theorem to CNF, with Skolem functions as additional premises. *)
   237 fun cnf_axiom thy th =
   238   let
   239     val ctxt0 = Variable.global_thm_context th
   240     val (nnf_th, ctxt) = to_nnf th ctxt0
   241     fun aux th =
   242       Meson.make_cnf (map (skolem_theorem_of_def thy) (assume_skolem_funs th))
   243                      th ctxt
   244     val (cnf_ths, ctxt) =
   245       aux nnf_th
   246       |> (fn ([], _) => aux (to_definitional_cnf_with_quantifiers thy nnf_th)
   247            | p => p)
   248   in
   249     cnf_ths |> map introduce_combinators_in_theorem
   250             |> Variable.export ctxt ctxt0
   251             |> Meson.finish_cnf
   252             |> map Thm.close_derivation
   253   end
   254   handle THM _ => []
   255 
   256 fun meson_general_tac ctxt ths =
   257   let
   258     val thy = ProofContext.theory_of ctxt
   259     val ctxt0 = Classical.put_claset HOL_cs ctxt
   260   in Meson.meson_tac ctxt0 (maps (cnf_axiom thy) ths) end
   261 
   262 val setup =
   263   Method.setup @{binding meson} (Attrib.thms >> (fn ths => fn ctxt =>
   264     SIMPLE_METHOD' (CHANGED_PROP o meson_general_tac ctxt ths)))
   265     "MESON resolution proof procedure";
   266 
   267 end;