(* Title: HOL/Tools/Sledgehammer/clausifier.ML
Author: Jia Meng, Cambridge University Computer Laboratory and NICTA
Author: Jasmin Blanchette, TU Muenchen
Transformation of axiom rules (elim/intro/etc) into CNF forms.
*)
signature CLAUSIFIER =
sig
val transform_elim_theorem : thm -> thm
val extensionalize_theorem : thm -> thm
val introduce_combinators_in_cterm : cterm -> thm
val introduce_combinators_in_theorem : thm -> thm
val cnf_axiom: theory -> thm -> thm list
end;
structure Clausifier : CLAUSIFIER =
struct
(**** 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
"ATP_Systems".) *)
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
(*To enforce single-threading*)
exception Clausify_failure of theory;
(**** SKOLEMIZATION BY INFERENCE (lcp) ****)
fun mk_skolem_id t =
let val T = fastype_of t in
Const (@{const_name skolem_id}, T --> T) $ t
end
fun beta_eta_under_lambdas (Abs (s, T, t')) =
Abs (s, T, beta_eta_under_lambdas t')
| beta_eta_under_lambdas t = Envir.beta_eta_contract t
(*Traverse a theorem, accumulating Skolem function definitions.*)
fun assume_skolem_funs th =
let
fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (s', T, p))) rhss =
(*Existential: declare a Skolem function, then insert into body and continue*)
let
val args = OldTerm.term_frees body
val Ts = map type_of args
val cT = Ts ---> T
(* Forms a lambda-abstraction over the formal parameters *)
val rhs =
list_abs_free (map dest_Free args,
HOLogic.choice_const T $ beta_eta_under_lambdas body)
|> mk_skolem_id
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 "op &"} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
| dec_sko (@{const "op |"} $ 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 ****)
(*Returns the vars of a theorem*)
fun vars_of_thm th =
map (cterm_of (theory_of_thm th) o Var) (Thm.fold_terms Term.add_vars th []);
val fun_cong_all = @{thm expand_fun_eq [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 "op ="}, Type (_, [Type (@{type_name fun}, _), _]))
$ _ $ Abs (s, _, _)) => extensionalize_theorem (th RS fun_cong_all)
| _ => th
fun is_quasi_lambda_free (Const (@{const_name skolem_id}, _) $ _) = 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 loose_bvar1 (rator,0) then (*C or S*)
if loose_bvar1 (rand,0) 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 loose_bvar1 (rand,0) 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_id_def_raw = @{thms skolem_id_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 skolem_theorem_of_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 (@{const_name Eps}, Type (@{type_name fun}, [_, T])) => T
| _ => raise TERM ("skolem_theorem_of_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_id_def_raw
THEN rtac ((prem |> rewrite_rule skolem_id_def_raw) RS @{thm 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
(* Converts an Isabelle theorem (intro, elim or simp format, even higher-order)
into NNF. *)
fun to_nnf th ctxt0 =
let
val th1 = th |> transform_elim_theorem |> zero_var_indexes
val ((_, [th2]), ctxt) = Variable.import true [th1] ctxt0
val th3 = th2 |> Conv.fconv_rule Object_Logic.atomize
|> extensionalize_theorem
|> Meson.make_nnf ctxt
in (th3, ctxt) end
(* Convert a theorem to CNF, with Skolem functions as additional premises. *)
fun cnf_axiom thy th =
let
val ctxt0 = Variable.global_thm_context th
val (nnfth, ctxt) = to_nnf th ctxt0
val sko_ths = map (skolem_theorem_of_def thy)
(assume_skolem_funs nnfth)
val (cnfs, ctxt) = Meson.make_cnf sko_ths nnfth ctxt
in
cnfs |> map introduce_combinators_in_theorem
|> Variable.export ctxt ctxt0
|> Meson.finish_cnf
|> map Thm.close_derivation
end
handle THM _ => []
end;