(* Title: HOL/Tools/Sledgehammer/clausifier.ML
Author: Jia Meng, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Transformation of axiom rules (elim/intro/etc) into CNF forms.
*)
signature CLAUSIFIER =
sig
type cnf_thm = thm * ((string * int) * thm)
val trace: bool Unsynchronized.ref
val skolem_theory_name: string
val skolem_prefix: string
val skolem_infix: string
val is_skolem_const_name: string -> bool
val num_type_args: theory -> string -> int
val cnf_axiom: theory -> thm -> thm list
val multi_base_blacklist: string list
val is_theorem_bad_for_atps: thm -> bool
val type_has_topsort: typ -> bool
val cnf_rules_pairs : theory -> (string * thm) list -> cnf_thm list
val saturate_cache: theory -> theory option
val auto_saturate_cache: bool Unsynchronized.ref
val neg_clausify: thm -> thm list
val neg_conjecture_clauses:
Proof.context -> thm -> int -> thm list list * (string * typ) list
val setup: theory -> theory
end;
structure Clausifier : CLAUSIFIER =
struct
type cnf_thm = thm * ((string * int) * thm)
val trace = Unsynchronized.ref false;
fun trace_msg msg = if !trace then tracing (msg ()) else ();
val skolem_theory_name = "Sledgehammer" ^ Long_Name.separator ^ "Sko"
val skolem_prefix = "sko_"
val skolem_infix = "$"
val type_has_topsort = Term.exists_subtype
(fn TFree (_, []) => true
| TVar (_, []) => true
| _ => false);
(**** 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.*)
fun transform_elim 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) ****)
(*Keep the full complexity of the original name*)
fun flatten_name s = space_implode "_X" (Long_Name.explode s);
fun skolem_name thm_name j var_name =
skolem_prefix ^ thm_name ^ "_" ^ Int.toString j ^
skolem_infix ^ (if var_name = "" then "g" else flatten_name var_name)
(* Hack: Could return false positives (e.g., a user happens to declare a
constant called "SomeTheory.sko_means_shoe_in_$wedish". *)
val is_skolem_const_name =
Long_Name.base_name
#> String.isPrefix skolem_prefix andf String.isSubstring skolem_infix
(* The number of type arguments of a constant, zero if it's monomorphic. For
(instances of) Skolem pseudoconstants, this information is encoded in the
constant name. *)
fun num_type_args thy s =
if String.isPrefix skolem_theory_name s then
s |> unprefix skolem_theory_name
|> space_explode skolem_infix |> hd
|> space_explode "_" |> List.last |> Int.fromString |> the
else
(s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length
fun rhs_extra_types lhsT rhs =
let val lhs_vars = Term.add_tfreesT lhsT []
fun add_new_TFrees (TFree v) =
if member (op =) lhs_vars v then I else insert (op =) (TFree v)
| add_new_TFrees _ = I
val rhs_consts = fold_aterms (fn Const c => insert (op =) c | _ => I) rhs []
in fold (#2 #> Term.fold_atyps add_new_TFrees) rhs_consts [] end;
fun skolem_type_and_args bound_T body =
let
val args1 = OldTerm.term_frees body
val Ts1 = map type_of args1
val Ts2 = rhs_extra_types (Ts1 ---> bound_T) body
val args2 = map (fn T => Free (gensym "vsk", T)) Ts2
in (Ts2 ---> Ts1 ---> bound_T, args2 @ args1) end
(* Traverse a theorem, declaring Skolem function definitions. String "s" is the
suggested prefix for the Skolem constants. *)
fun declare_skolem_funs s th thy =
let
val skolem_count = Unsynchronized.ref 0 (* FIXME ??? *)
fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (s', T, p)))
(axs, thy) =
(*Existential: declare a Skolem function, then insert into body and continue*)
let
val id = skolem_name s (Unsynchronized.inc skolem_count) s'
val (cT, args) = skolem_type_and_args T body
val rhs = list_abs_free (map dest_Free args,
HOLogic.choice_const T $ body)
(*Forms a lambda-abstraction over the formal parameters*)
val (c, thy) =
Sign.declare_const ((Binding.conceal (Binding.name id), cT), NoSyn) thy
val cdef = id ^ "_def"
val ((_, ax), thy) =
Thm.add_def true false (Binding.name cdef, Logic.mk_equals (c, rhs)) thy
val ax' = Drule.export_without_context ax
in dec_sko (subst_bound (list_comb (c, args), p)) (ax' :: axs, thy) end
| dec_sko (Const (@{const_name All}, _) $ (Abs (a, T, p))) thx =
(*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)) thx end
| dec_sko (@{const "op &"} $ p $ q) thx = dec_sko q (dec_sko p thx)
| dec_sko (@{const "op |"} $ p $ q) thx = dec_sko q (dec_sko p thx)
| dec_sko (@{const Trueprop} $ p) thx = dec_sko p thx
| dec_sko _ thx = thx
in dec_sko (prop_of th) ([], thy) end
fun mk_skolem_id t =
let val T = fastype_of t in
Const (@{const_name skolem_id}, T --> T) $ t
end
fun quasi_beta_eta_contract (Abs (s, T, t')) =
Abs (s, T, quasi_beta_eta_contract t')
| quasi_beta_eta_contract t = Envir.beta_eta_contract t
(*Traverse a theorem, accumulating Skolem function definitions.*)
fun assume_skolem_funs s th =
let
val skolem_count = Unsynchronized.ref 0 (* FIXME ??? *)
fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (s', T, p))) defs =
(*Existential: declare a Skolem function, then insert into body and continue*)
let
val skos = map (#1 o Logic.dest_equals) defs (*existing sko fns*)
val args = subtract (op =) skos (OldTerm.term_frees body) (*the formal parameters*)
val Ts = map type_of args
val cT = Ts ---> T (* FIXME: use "skolem_type_and_args" *)
val id = skolem_name s (Unsynchronized.inc skolem_count) s'
val c = Free (id, cT) (* FIXME: needed? ### *)
(* Forms a lambda-abstraction over the formal parameters *)
val rhs =
list_abs_free (map dest_Free args,
HOLogic.choice_const T
$ quasi_beta_eta_contract body)
|> mk_skolem_id
val def = Logic.mk_equals (c, rhs)
val comb = list_comb (rhs, args)
in dec_sko (subst_bound (comb, p)) (def :: defs) end
| dec_sko (Const (@{const_name All},_) $ Abs (a, T, p)) defs =
(*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)) defs end
| dec_sko (@{const "op &"} $ p $ q) defs = dec_sko q (dec_sko p defs)
| dec_sko (@{const "op |"} $ p $ q) defs = dec_sko q (dec_sko p defs)
| dec_sko (@{const Trueprop} $ p) defs = dec_sko p defs
| dec_sko _ defs = defs
in dec_sko (prop_of th) [] end;
(**** REPLACING ABSTRACTIONS BY COMBINATORS ****)
(*Returns the vars of a theorem*)
fun vars_of_thm th =
map (Thm.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). *)
fun extensionalize th =
case prop_of th of
_ $ (Const (@{const_name "op ="}, Type (_, [Type (@{type_name fun}, _), _]))
$ _ $ Abs (s, _, _)) => extensionalize (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 and 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 do_introduce_combinators 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 = do_introduce_combinators 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 (do_introduce_combinators ct1)
(do_introduce_combinators ct2)
end
fun introduce_combinators th =
if is_quasi_lambda_free (prop_of th) then
th
else
let
val th = Drule.eta_contraction_rule th
val eqth = do_introduce_combinators (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 = Thm.cterm_of @{theory HOL} HOLogic.Trueprop;
(*cterm version of mk_cTrueprop*)
fun c_mkTrueprop A = Thm.capply cTrueprop A;
(*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);
(*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 inline def =
let
val (c, rhs) = def |> Drule.legacy_freeze_thaw |> #1 |> cprop_of
|> Thm.dest_equals
val rhs' = rhs |> inline ? (snd o Thm.dest_comb)
val (ch, frees) = c_variant_abs_multi (rhs', [])
val (chilbert, cabs) = ch |> Thm.dest_comb
val thy = Thm.theory_of_cterm chilbert
val t = Thm.term_of chilbert
val T =
case t of
Const (@{const_name Eps}, Type (@{type_name fun}, [_, T])) => T
| _ => raise TERM ("skolem_theorem_of_def: expected \"Eps\"", [t])
val cex = Thm.cterm_of thy (HOLogic.exists_const T)
val ex_tm = c_mkTrueprop (Thm.capply cex cabs)
and conc =
Drule.list_comb (if inline then rhs else c, frees)
|> Drule.beta_conv cabs |> c_mkTrueprop
fun tacf [prem] =
(if inline then rewrite_goals_tac @{thms skolem_id_def_raw}
else rewrite_goals_tac [def])
THEN rtac ((prem |> inline ? rewrite_rule @{thms 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 |> zero_var_indexes
val ((_, [th2]), ctxt) = Variable.import true [th1] ctxt0
val th3 = th2 |> Conv.fconv_rule Object_Logic.atomize
|> extensionalize
|> Meson.make_nnf ctxt
in (th3, ctxt) end;
(*Generate Skolem functions for a theorem supplied in nnf*)
fun skolem_theorems_of_assume s th =
map (skolem_theorem_of_def true o Thm.assume o cterm_of (theory_of_thm th))
(assume_skolem_funs s th)
(*** Blacklisting (more in "Sledgehammer_Fact_Filter") ***)
val max_lambda_nesting = 3
fun term_has_too_many_lambdas max (t1 $ t2) =
exists (term_has_too_many_lambdas max) [t1, t2]
| term_has_too_many_lambdas max (Abs (_, _, t)) =
max = 0 orelse term_has_too_many_lambdas (max - 1) t
| term_has_too_many_lambdas _ _ = false
fun is_formula_type T = (T = HOLogic.boolT orelse T = propT)
(* Don't count nested lambdas at the level of formulas, since they are
quantifiers. *)
fun formula_has_too_many_lambdas Ts (Abs (_, T, t)) =
formula_has_too_many_lambdas (T :: Ts) t
| formula_has_too_many_lambdas Ts t =
if is_formula_type (fastype_of1 (Ts, t)) then
exists (formula_has_too_many_lambdas Ts) (#2 (strip_comb t))
else
term_has_too_many_lambdas max_lambda_nesting t
(* The max apply depth of any "metis" call in "Metis_Examples" (on 31-10-2007)
was 11. *)
val max_apply_depth = 15
fun apply_depth (f $ t) = Int.max (apply_depth f, apply_depth t + 1)
| apply_depth (Abs (_, _, t)) = apply_depth t
| apply_depth _ = 0
fun is_formula_too_complex t =
apply_depth t > max_apply_depth orelse Meson.too_many_clauses NONE t orelse
formula_has_too_many_lambdas [] t
fun is_strange_thm th =
case head_of (concl_of th) of
Const (a, _) => (a <> @{const_name Trueprop} andalso
a <> @{const_name "=="})
| _ => false;
fun is_theorem_bad_for_atps thm =
let val t = prop_of thm in
is_formula_too_complex t orelse exists_type type_has_topsort t orelse
is_strange_thm thm
end
(* FIXME: put other record thms here, or declare as "no_atp" *)
(* FIXME: move to "Sledgehammer_Fact_Filter" *)
val multi_base_blacklist =
["defs", "select_defs", "update_defs", "induct", "inducts", "split", "splits",
"split_asm", "cases", "ext_cases"];
fun fake_name th =
if Thm.has_name_hint th then flatten_name (Thm.get_name_hint th)
else gensym "unknown_thm_";
(*Skolemize a named theorem, with Skolem functions as additional premises.*)
fun skolemize_theorem s th =
if member (op =) multi_base_blacklist (Long_Name.base_name s) orelse
is_theorem_bad_for_atps th then
[]
else
let
val ctxt0 = Variable.global_thm_context th
val (nnfth, ctxt) = to_nnf th ctxt0
val defs = skolem_theorems_of_assume s nnfth
val (cnfs, ctxt) = Meson.make_cnf defs nnfth ctxt
in
cnfs |> map introduce_combinators
|> Variable.export ctxt ctxt0
|> Meson.finish_cnf
end
handle THM _ => []
(*The cache prevents repeated clausification of a theorem, and also repeated declaration of
Skolem functions.*)
structure ThmCache = Theory_Data
(
type T = thm list Thmtab.table * unit Symtab.table;
val empty = (Thmtab.empty, Symtab.empty);
val extend = I;
fun merge ((cache1, seen1), (cache2, seen2)) : T =
(Thmtab.merge (K true) (cache1, cache2), Symtab.merge (K true) (seen1, seen2));
);
val lookup_cache = Thmtab.lookup o #1 o ThmCache.get;
val already_seen = Symtab.defined o #2 o ThmCache.get;
val update_cache = ThmCache.map o apfst o Thmtab.update;
fun mark_seen name = ThmCache.map (apsnd (Symtab.update (name, ())));
(* Convert Isabelle theorems into axiom clauses. *)
fun cnf_axiom thy th0 =
let val th = Thm.transfer thy th0 in
case lookup_cache thy th of
SOME cls => cls
| NONE => map Thm.close_derivation (skolemize_theorem (fake_name th) th)
end;
(**** Translate a set of theorems into CNF ****)
(*The combination of rev and tail recursion preserves the original order*)
fun cnf_rules_pairs thy =
let
fun do_one _ [] = []
| do_one ((name, k), th) (cls :: clss) =
(cls, ((name, k), th)) :: do_one ((name, k + 1), th) clss
fun do_all pairs [] = pairs
| do_all pairs ((name, th) :: ths) =
let
val new_pairs = do_one ((name, 0), th) (cnf_axiom thy th)
handle THM _ => []
in do_all (new_pairs @ pairs) ths end
in do_all [] o rev end
(**** Convert all facts of the theory into FOL or HOL clauses ****)
local
fun skolem_def (name, th) thy =
let val ctxt0 = Variable.global_thm_context th in
case try (to_nnf th) ctxt0 of
NONE => (NONE, thy)
| SOME (nnfth, ctxt) =>
let val (defs, thy') = declare_skolem_funs (flatten_name name) nnfth thy
in (SOME (th, ctxt0, ctxt, nnfth, defs), thy') end
end;
fun skolem_cnfs (th, ctxt0, ctxt, nnfth, defs) =
let
val (cnfs, ctxt) =
Meson.make_cnf (map (skolem_theorem_of_def false) defs) nnfth ctxt
val cnfs' = cnfs
|> map introduce_combinators
|> Variable.export ctxt ctxt0
|> Meson.finish_cnf
|> map Thm.close_derivation;
in (th, cnfs') end;
in
fun saturate_cache thy =
let
val facts = PureThy.facts_of thy;
val new_facts = (facts, []) |-> Facts.fold_static (fn (name, ths) =>
if Facts.is_concealed facts name orelse already_seen thy name then I
else cons (name, ths));
val new_thms = (new_facts, []) |-> fold (fn (name, ths) =>
if member (op =) multi_base_blacklist (Long_Name.base_name name) then
I
else
fold_index (fn (i, th) =>
if is_theorem_bad_for_atps th orelse
is_some (lookup_cache thy th) then
I
else
cons (name ^ "_" ^ string_of_int (i + 1), Thm.transfer thy th)) ths)
in
if null new_facts then
NONE
else
let
val (defs, thy') = thy
|> fold (mark_seen o #1) new_facts
|> fold_map skolem_def (sort_distinct (Thm.thm_ord o pairself snd) new_thms)
|>> map_filter I;
val cache_entries = Par_List.map skolem_cnfs defs;
in SOME (fold update_cache cache_entries thy') end
end;
end;
(* For emergency use where the Skolem cache causes problems. *)
val auto_saturate_cache = Unsynchronized.ref true
fun conditionally_saturate_cache thy =
if !auto_saturate_cache then saturate_cache thy else NONE
(*** Converting a subgoal into negated conjecture clauses. ***)
fun neg_skolemize_tac ctxt =
EVERY' [rtac ccontr, Object_Logic.atomize_prems_tac, Meson.skolemize_tac ctxt]
val neg_clausify =
single
#> Meson.make_clauses_unsorted
#> map introduce_combinators
#> Meson.finish_cnf
fun neg_conjecture_clauses ctxt st0 n =
let
(* "Option" is thrown if the assumptions contain schematic variables. *)
val st = Seq.hd (neg_skolemize_tac ctxt n st0) handle Option.Option => st0
val ({params, prems, ...}, _) =
Subgoal.focus (Variable.set_body false ctxt) n st
in (map neg_clausify prems, map (dest_Free o term_of o #2) params) end
(** setup **)
val setup =
perhaps conditionally_saturate_cache
#> Theory.at_end conditionally_saturate_cache
end;