(* Author: Jia Meng, Cambridge University Computer Laboratory
Transformation of axiom rules (elim/intro/etc) into CNF forms.
*)
signature RES_AXIOMS =
sig
val trace: bool Unsynchronized.ref
val trace_msg: (unit -> string) -> unit
val cnf_axiom: theory -> thm -> thm list
val pairname: thm -> string * thm
val multi_base_blacklist: string list
val bad_for_atp: thm -> bool
val type_has_empty_sort: typ -> bool
val cnf_rules_pairs: theory -> (string * thm) list -> (thm * (string * int)) list
val neg_clausify: thm list -> thm list
val expand_defs_tac: thm -> tactic
val combinators: thm -> thm
val neg_conjecture_clauses: Proof.context -> thm -> int -> thm list * (string * typ) list
val atpset_rules_of: Proof.context -> (string * thm) list
val suppress_endtheory: bool Unsynchronized.ref
(*for emergency use where endtheory causes problems*)
val setup: theory -> theory
end;
structure ResAxioms: RES_AXIOMS =
struct
val trace = Unsynchronized.ref false;
fun trace_msg msg = if ! trace then tracing (msg ()) else ();
(* FIXME legacy *)
fun freeze_thm th = #1 (Drule.freeze_thaw th);
fun type_has_empty_sort (TFree (_, [])) = true
| type_has_empty_sort (TVar (_, [])) = true
| type_has_empty_sort (Type (_, Ts)) = exists type_has_empty_sort Ts
| type_has_empty_sort _ = 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(_,Type("bool",[]))) =>
Thm.instantiate ([], [(cterm_of @{theory HOL} v, cfalse)]) th
| v as Var(_, Type("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 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;
(*Traverse a theorem, declaring Skolem function definitions. String s is the suggested
prefix for the Skolem constant.*)
fun declare_skofuns s th =
let
val nref = Unsynchronized.ref 0
fun dec_sko (Const ("Ex",_) $ (xtp as Abs (_, T, p))) (axs, thy) =
(*Existential: declare a Skolem function, then insert into body and continue*)
let
val cname = "sko_" ^ s ^ "_" ^ Int.toString (Unsynchronized.inc nref)
val args0 = OldTerm.term_frees xtp (*get the formal parameter list*)
val Ts = map type_of args0
val extraTs = rhs_extra_types (Ts ---> T) xtp
val argsx = map (fn T => Free (gensym "vsk", T)) extraTs
val args = argsx @ args0
val cT = extraTs ---> Ts ---> T
val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp)
(*Forms a lambda-abstraction over the formal parameters*)
val (c, thy') =
Sign.declare_const [Markup.property_internal] ((Binding.name cname, cT), NoSyn) thy
val cdef = cname ^ "_def"
val thy'' = Theory.add_defs_i true false [(Binding.name cdef, Logic.mk_equals (c, rhs))] thy'
val ax = Thm.axiom thy'' (Sign.full_bname thy'' cdef)
in dec_sko (subst_bound (list_comb (c, args), p)) (ax :: axs, thy'') end
| dec_sko (Const ("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 t thx = thx (*Do nothing otherwise*)
in fn thy => dec_sko (Thm.prop_of th) ([], thy) end;
(*Traverse a theorem, accumulating Skolem function definitions.*)
fun assume_skofuns s th =
let val sko_count = Unsynchronized.ref 0
fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,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 = OldTerm.term_frees xtp \\ skos (*the formal parameters*)
val Ts = map type_of args
val cT = Ts ---> T
val id = "sko_" ^ s ^ "_" ^ Int.toString (Unsynchronized.inc sko_count)
val c = Free (id, cT)
val rhs = list_abs_free (map dest_Free args,
HOLogic.choice_const T $ xtp)
(*Forms a lambda-abstraction over the formal parameters*)
val def = Logic.mk_equals (c, rhs)
in dec_sko (subst_bound (list_comb(c,args), p))
(def :: defs)
end
| dec_sko (Const ("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 t defs = defs (*Do nothing otherwise*)
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 []);
(*Make a version of fun_cong with a given variable name*)
local
val fun_cong' = fun_cong RS asm_rl; (*renumber f, g to prevent clashes with (a,0)*)
val cx = hd (vars_of_thm fun_cong');
val ty = typ_of (ctyp_of_term cx);
val thy = theory_of_thm fun_cong;
fun mkvar a = cterm_of thy (Var((a,0),ty));
in
fun xfun_cong x = Thm.instantiate ([], [(cx, mkvar x)]) fun_cong'
end;
(*Removes the lambdas from an equation of the form t = (%x. u). A non-negative n,
serves as an upper bound on how many to remove.*)
fun strip_lambdas 0 th = th
| strip_lambdas n th =
case prop_of th of
_ $ (Const ("op =", _) $ _ $ Abs (x,_,_)) =>
strip_lambdas (n-1) (freeze_thm (th RS xfun_cong x))
| _ => th;
val lambda_free = not o Term.has_abs;
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("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 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()
| _ => error "abstract: Bad term"
end;
(*Traverse a theorem, declaring abstraction function definitions. String s is the suggested
prefix for the constants.*)
fun combinators_aux ct =
if lambda_free (term_of ct) then 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 = combinators_aux cta
val cu = Thm.rhs_of u_th
val comb_eq = abstract (Thm.cabs cv cu)
in transitive (abstract_rule v cv u_th) comb_eq end
| _ $ _ =>
let val (ct1, ct2) = Thm.dest_comb ct
in combination (combinators_aux ct1) (combinators_aux ct2) end;
fun combinators th =
if lambda_free (prop_of th) then th
else
let val th = Drule.eta_contraction_rule th
val eqth = combinators_aux (cprop_of th)
in equal_elim eqth th end
handle THM (msg,_,_) =>
(warning (cat_lines
["Error in the combinator translation of " ^ Display.string_of_thm_without_context th,
" Exception message: " ^ msg]);
TrueI); (*A type variable of sort {} will cause make abstraction fail.*)
(*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_of_def def =
let val (c,rhs) = Thm.dest_equals (cprop_of (freeze_thm def))
val (ch, frees) = c_variant_abs_multi (rhs, [])
val (chilbert,cabs) = Thm.dest_comb ch
val thy = Thm.theory_of_cterm chilbert
val t = Thm.term_of chilbert
val T = case t of Const ("Hilbert_Choice.Eps", Type("fun",[_,T])) => T
| _ => raise THM ("skolem_of_def: expected Eps", 0, [def])
val cex = Thm.cterm_of thy (HOLogic.exists_const T)
val ex_tm = c_mkTrueprop (Thm.capply cex cabs)
and conc = c_mkTrueprop (Drule.beta_conv cabs (Drule.list_comb(c,frees)));
fun tacf [prem] = rewrite_goals_tac [def] THEN rtac (prem 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
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 ObjectLogic.atomize
|> Meson.make_nnf ctxt |> strip_lambdas ~1
in (th3, ctxt) end;
(*Generate Skolem functions for a theorem supplied in nnf*)
fun assume_skolem_of_def s th =
map (skolem_of_def o assume o (cterm_of (theory_of_thm th))) (assume_skofuns s th);
(*** Blacklisting (duplicated in ResAtp?) ***)
val max_lambda_nesting = 3;
fun excessive_lambdas (f$t, k) = excessive_lambdas (f,k) orelse excessive_lambdas (t,k)
| excessive_lambdas (Abs(_,_,t), k) = k=0 orelse excessive_lambdas (t,k-1)
| excessive_lambdas _ = false;
fun is_formula_type T = (T = HOLogic.boolT orelse T = propT);
(*Don't count nested lambdas at the level of formulas, as they are quantifiers*)
fun excessive_lambdas_fm Ts (Abs(_,T,t)) = excessive_lambdas_fm (T::Ts) t
| excessive_lambdas_fm Ts t =
if is_formula_type (fastype_of1 (Ts, t))
then exists (excessive_lambdas_fm Ts) (#2 (strip_comb t))
else excessive_lambdas (t, max_lambda_nesting);
(*The max apply_depth of any metis call in MetisExamples (on 31-10-2007) was 11.*)
val max_apply_depth = 15;
fun apply_depth (f$t) = Int.max (apply_depth f, apply_depth t + 1)
| apply_depth (Abs(_,_,t)) = apply_depth t
| apply_depth _ = 0;
fun too_complex t =
apply_depth t > max_apply_depth orelse
Meson.too_many_clauses NONE t orelse
excessive_lambdas_fm [] t;
fun is_strange_thm th =
case head_of (concl_of th) of
Const (a,_) => (a <> "Trueprop" andalso a <> "==")
| _ => false;
fun bad_for_atp th =
Thm.is_internal th
orelse too_complex (prop_of th)
orelse exists_type type_has_empty_sort (prop_of th)
orelse is_strange_thm th;
val multi_base_blacklist =
["defs","select_defs","update_defs","induct","inducts","split","splits","split_asm",
"cases","ext_cases"]; (*FIXME: put other record thms here, or use the "Internal" marker*)
(*Keep the full complexity of the original name*)
fun flatten_name s = space_implode "_X" (Long_Name.explode s);
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 skolem_thm (s, th) =
if member (op =) multi_base_blacklist (Long_Name.base_name s) orelse bad_for_atp th then []
else
let
val ctxt0 = Variable.thm_context th
val (nnfth, ctxt1) = to_nnf th ctxt0
val (cnfs, ctxt2) = Meson.make_cnf (assume_skolem_of_def s nnfth) nnfth ctxt1
in cnfs |> map combinators |> Variable.export ctxt2 ctxt0 |> Meson.finish_cnf end
handle THM _ => [];
(*The cache prevents repeated clausification of a theorem, and also repeated declaration of
Skolem functions.*)
structure ThmCache = TheoryDataFun
(
type T = thm list Thmtab.table * unit Symtab.table;
val empty = (Thmtab.empty, Symtab.empty);
val copy = I;
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, ())));
(*Exported function to 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
NONE => map Thm.close_derivation (skolem_thm (fake_name th, th))
| SOME cls => cls
end;
(**** Rules from the context ****)
fun pairname th = (Thm.get_name_hint th, th);
fun atpset_rules_of ctxt = map pairname (ResAtpset.get ctxt);
(**** Translate a set of theorems into CNF ****)
fun pair_name_cls k (n, []) = []
| pair_name_cls k (n, cls::clss) = (cls, (n,k)) :: pair_name_cls (k+1) (n, clss)
fun cnf_rules_pairs_aux _ pairs [] = pairs
| cnf_rules_pairs_aux thy pairs ((name,th)::ths) =
let val pairs' = (pair_name_cls 0 (name, cnf_axiom thy th)) @ pairs
handle THM _ => pairs | ResClause.CLAUSE _ => pairs
in cnf_rules_pairs_aux thy pairs' ths end;
(*The combination of rev and tail recursion preserves the original order*)
fun cnf_rules_pairs thy l = cnf_rules_pairs_aux thy [] (rev l);
(**** Convert all facts of the theory into clauses (ResClause.clause, or ResHolClause.clause) ****)
local
fun skolem_def (name, th) thy =
let val ctxt0 = Variable.thm_context th in
(case try (to_nnf th) ctxt0 of
NONE => (NONE, thy)
| SOME (nnfth, ctxt1) =>
let val (defs, thy') = declare_skofuns (flatten_name name) nnfth thy
in (SOME (th, ctxt0, ctxt1, nnfth, defs), thy') end)
end;
fun skolem_cnfs (th, ctxt0, ctxt1, nnfth, defs) =
let
val (cnfs, ctxt2) = Meson.make_cnf (map skolem_of_def defs) nnfth ctxt1;
val cnfs' = cnfs
|> map combinators
|> Variable.export ctxt2 ctxt0
|> Meson.finish_cnf
|> map Thm.close_derivation;
in (th, cnfs') end;
in
fun saturate_skolem_cache thy =
let
val new_facts = (PureThy.facts_of thy, []) |-> Facts.fold_static (fn (name, ths) =>
if 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 bad_for_atp 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;
val suppress_endtheory = Unsynchronized.ref false;
fun clause_cache_endtheory thy =
if ! suppress_endtheory then NONE
else saturate_skolem_cache thy;
(*The cache can be kept smaller by inspecting the prop of each thm. Can ignore all that are
lambda_free, but then the individual theory caches become much bigger.*)
(*** meson proof methods ***)
(*Expand all new definitions of abstraction or Skolem functions in a proof state.*)
fun is_absko (Const ("==", _) $ Free (a,_) $ u) = String.isPrefix "sko_" a
| is_absko _ = false;
fun is_okdef xs (Const ("==", _) $ t $ u) = (*Definition of Free, not in certain terms*)
is_Free t andalso not (member (op aconv) xs t)
| is_okdef _ _ = false
(*This function tries to cope with open locales, which introduce hypotheses of the form
Free == t, conjecture clauses, which introduce various hypotheses, and also definitions
of sko_ functions. *)
fun expand_defs_tac st0 st =
let val hyps0 = #hyps (rep_thm st0)
val hyps = #hyps (crep_thm st)
val newhyps = filter_out (member (op aconv) hyps0 o Thm.term_of) hyps
val defs = filter (is_absko o Thm.term_of) newhyps
val remaining_hyps = filter_out (member (op aconv) (map Thm.term_of defs))
(map Thm.term_of hyps)
val fixed = OldTerm.term_frees (concl_of st) @
List.foldl (union (op aconv)) [] (map OldTerm.term_frees remaining_hyps)
in Seq.of_list [LocalDefs.expand (filter (is_okdef fixed o Thm.term_of) defs) st] end;
fun meson_general_tac ctxt ths i st0 =
let
val thy = ProofContext.theory_of ctxt
val ctxt0 = Classical.put_claset HOL_cs ctxt
in (Meson.meson_tac ctxt0 (maps (cnf_axiom thy) ths) i THEN expand_defs_tac st0) st0 end;
val meson_method_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";
(*** Converting a subgoal into negated conjecture clauses. ***)
fun neg_skolemize_tac ctxt =
EVERY' [rtac ccontr, ObjectLogic.atomize_prems_tac, Meson.skolemize_tac ctxt];
val neg_clausify = Meson.make_clauses #> map combinators #> Meson.finish_cnf;
fun neg_conjecture_clauses ctxt st0 n =
let
val st = Seq.hd (neg_skolemize_tac ctxt n st0)
val ({params, prems, ...}, _) = Subgoal.focus (Variable.set_body false ctxt) n st
in (neg_clausify prems, map (Term.dest_Free o Thm.term_of o #2) params) end;
(*Conversion of a subgoal to conjecture clauses. Each clause has
leading !!-bound universal variables, to express generality. *)
fun neg_clausify_tac ctxt =
neg_skolemize_tac ctxt THEN'
SUBGOAL (fn (prop, i) =>
let val ts = Logic.strip_assums_hyp prop in
EVERY'
[Subgoal.FOCUS
(fn {prems, ...} =>
(Method.insert_tac
(map forall_intr_vars (neg_clausify prems)) i)) ctxt,
REPEAT_DETERM_N (length ts) o etac thin_rl] i
end);
val neg_clausify_setup =
Method.setup @{binding neg_clausify} (Scan.succeed (SIMPLE_METHOD' o neg_clausify_tac))
"conversion of goal to conjecture clauses";
(** Attribute for converting a theorem into clauses **)
val clausify_setup =
Attrib.setup @{binding clausify}
(Scan.lift OuterParse.nat >>
(fn i => Thm.rule_attribute (fn context => fn th =>
Meson.make_meta_clause (nth (cnf_axiom (Context.theory_of context) th) i))))
"conversion of theorem to clauses";
(** setup **)
val setup =
meson_method_setup #>
neg_clausify_setup #>
clausify_setup #>
perhaps saturate_skolem_cache #>
Theory.at_end clause_cache_endtheory;
end;