(* 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.
The "meson" proof method for HOL.
*)
signature MESON_CLAUSIFY =
sig
val new_skolem_var_prefix : string
val extensionalize_theorem : thm -> thm
val introduce_combinators_in_cterm : cterm -> thm
val introduce_combinators_in_theorem : thm -> thm
val to_definitional_cnf_with_quantifiers : theory -> thm -> thm
val cluster_of_zapped_var_name : string -> (int * (int * int)) * bool
val cnf_axiom :
Proof.context -> bool -> int -> thm -> (thm * term) option * thm list
val meson_general_tac : Proof.context -> thm list -> int -> tactic
val setup: theory -> theory
end;
structure Meson_Clausify : MESON_CLAUSIFY =
struct
(* the extra "?" helps prevent clashes *)
val new_skolem_var_prefix = "?SK"
val new_nonskolem_var_prefix = "?V"
(**** 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 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 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 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_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 "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 thy th =
let
val eqth = cnf.make_cnfx_thm thy (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 ^ "_" ^ 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 zap (cluster as (cluster_no, cluster_skolem)) index_no 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
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)
in
Thm.dest_comb #> snd
#> Thm.dest_abs (SOME (zapped_var_name cluster index_no s'))
#> snd #> zap cluster (index_no + 1) pos
end
else
Conv.all_conv
| Const (s, _) $ _ $ _ =>
if s = @{const_name "==>"} orelse s = @{const_name implies} then
Conv.combination_conv (Conv.arg_conv (zap cluster index_no (not pos)))
(zap cluster index_no pos)
else if s = @{const_name conj} orelse s = @{const_name disj} then
Conv.combination_conv (Conv.arg_conv (zap cluster index_no pos))
(zap cluster index_no pos)
else
Conv.all_conv
| Const (s, _) $ _ =>
if s = @{const_name Trueprop} then
Conv.arg_conv (zap cluster index_no pos)
else if s = @{const_name Not} then
Conv.arg_conv (zap cluster index_no (not pos))
else
Conv.all_conv
| _ => Conv.all_conv)
fun ss_only ths = MetaSimplifier.clear_ss HOL_basic_ss addsimps ths
val no_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 = ProofContext.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
|> Meson.make_nnf ctxt
in
if new_skolemizer then
let
fun skolemize choice_ths =
Meson.skolemize_with_choice_thms ctxt choice_ths
#> simplify (ss_only @{thms all_simps[symmetric]})
val pull_out =
simplify (ss_only @{thms all_simps[symmetric] ex_simps[symmetric]})
val (discharger_th, fully_skolemized_th) =
if null choice_ths then
th |> `I |>> pull_out ||> skolemize [no_choice]
else
th |> skolemize choice_ths |> `I
val t =
fully_skolemized_th |> cprop_of
|> zap ((ax_no, 0), true) 0 true |> Drule.export_without_context
|> cprop_of |> Thm.dest_equals |> snd |> term_of
in
if exists_subterm (fn Var ((s, _), _) =>
String.isPrefix new_skolem_var_prefix s
| _ => false) t then
let
val (ct, ctxt) =
Variable.import_terms true [t] ctxt
|>> the_single |>> cterm_of thy
in (SOME (discharger_th, ct), Thm.assume ct, ctxt) end
else
(NONE, th, ctxt)
end
else
(NONE, th, 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 = ProofContext.theory_of ctxt0
val choice_ths = Meson_Choices.get ctxt0
val (opt, nnf_th, ctxt) = nnf_axiom choice_ths new_skolemizer ax_no th ctxt0
fun clausify th =
Meson.make_cnf (if new_skolemizer then
[]
else
map (old_skolem_theorem_from_def thy)
(old_skolem_defs th)) th ctxt
val (cnf_ths, ctxt) =
clausify nnf_th
|> (fn ([], _) =>
clausify (to_definitional_cnf_with_quantifiers thy nnf_th)
| p => p)
fun intr_imp ct th =
Thm.instantiate ([], map (pairself (cterm_of @{theory}))
[(Var (("i", 1), @{typ nat}),
HOLogic.mk_nat ax_no)])
@{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
|> Meson.finish_cnf
|> map Thm.close_derivation)
end
handle THM _ => (NONE, [])
fun meson_general_tac ctxt ths =
let val ctxt = Classical.put_claset HOL_cs ctxt in
Meson.meson_tac ctxt (maps (snd o cnf_axiom ctxt false 0) ths)
end
val setup =
Method.setup @{binding meson} (Attrib.thms >> (fn ths => fn ctxt =>
SIMPLE_METHOD' (CHANGED_PROP o meson_general_tac ctxt ths)))
"MESON resolution proof procedure"
end;