(* Author: Jia Meng, Cambridge University Computer Laboratory
ID: $Id$
Copyright 2004 University of Cambridge
Transformation of axiom rules (elim/intro/etc) into CNF forms.
*)
signature RES_AXIOMS =
sig
val trace_abs: bool ref
val cnf_axiom : string * thm -> thm list
val cnf_name : string -> thm list
val meta_cnf_axiom : thm -> thm list
val pairname : thm -> string * thm
val skolem_thm : thm -> thm list
val to_nnf : thm -> thm
val cnf_rules_pairs : (string * thm) list -> (thm * (string * int)) list
val meson_method_setup : theory -> theory
val setup : theory -> theory
val assume_abstract_list: thm list -> thm list
val neg_conjecture_clauses: thm -> int -> thm list * (string * typ) list
val claset_rules_of: Proof.context -> (string * thm) list (*FIXME DELETE*)
val simpset_rules_of: Proof.context -> (string * thm) list (*FIXME DELETE*)
val atpset_rules_of: Proof.context -> (string * thm) list
end;
structure ResAxioms =
struct
(*For running the comparison between combinators and abstractions.
CANNOT be a ref, as the setting is used while Isabelle is built.
Currently FALSE, i.e. all the "abstraction" code below is unused, but so far
it seems to be inferior to combinators...*)
val abstract_lambdas = true;
val trace_abs = ref false;
(* FIXME legacy *)
fun freeze_thm th = #1 (Drule.freeze_thaw th);
val lhs_of = #1 o Logic.dest_equals o Thm.prop_of;
val rhs_of = #2 o Logic.dest_equals o Thm.prop_of;
(*Store definitions of abstraction functions, ensuring that identical right-hand
sides are denoted by the same functions and thereby reducing the need for
extensionality in proofs.
FIXME! Store in theory data!!*)
(*Populate the abstraction cache with common combinators.*)
fun seed th net =
let val (_,ct) = Thm.dest_abs NONE (Drule.rhs_of th)
val t = Logic.legacy_varify (term_of ct)
in Net.insert_term Thm.eq_thm (t, th) net end;
val abstraction_cache = ref
(seed (thm"ATP_Linkup.I_simp")
(seed (thm"ATP_Linkup.B_simp")
(seed (thm"ATP_Linkup.K_simp") Net.empty)));
(**** Transformation of Elimination Rules into First-Order Formulas****)
val cfalse = cterm_of HOL.thy HOLogic.false_const;
val ctp_false = cterm_of HOL.thy (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 HOL.thy v, cfalse)]) th
| v as Var(_, Type("prop",[])) =>
Thm.instantiate ([], [(cterm_of HOL.thy v, ctp_false)]) th
| _ => th;
(**** Transformation of Clasets and Simpsets into First-Order Axioms ****)
(*Transfer a theorem into theory ATP_Linkup.thy if it is not already
inside that theory -- because it's needed for Skolemization *)
(*This will refer to the final version of theory ATP_Linkup.*)
val recon_thy_ref = Theory.self_ref (the_context ());
(*If called while ATP_Linkup is being created, it will transfer to the
current version. If called afterward, it will transfer to the final version.*)
fun transfer_to_ATP_Linkup th =
transfer (Theory.deref recon_thy_ref) th handle THM _ => th;
(**** SKOLEMIZATION BY INFERENCE (lcp) ****)
(*Traverse a theorem, declaring Skolem function definitions. String s is the suggested
prefix for the Skolem constant. Result is a new theory*)
fun declare_skofuns s th thy =
let val nref = ref 0
fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (thy, axs) =
(*Existential: declare a Skolem function, then insert into body and continue*)
let val cname = Name.internal (s ^ "_sko" ^ Int.toString (inc nref))
val args = term_frees xtp (*get the formal parameter list*)
val Ts = map type_of args
val cT = Ts ---> T
val c = Const (Sign.full_name thy cname, cT)
val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp)
(*Forms a lambda-abstraction over the formal parameters*)
val thy' = Sign.add_consts_authentic [(cname, cT, NoSyn)] thy
(*Theory is augmented with the constant, then its def*)
val cdef = cname ^ "_def"
val thy'' = Theory.add_defs_i false false [(cdef, equals cT $ c $ rhs)] thy'
in dec_sko (subst_bound (list_comb(c,args), p))
(thy'', get_axiom thy'' cdef :: axs)
end
| dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) thx =
(*Universal quant: insert a free variable into body and continue*)
let val fname = Name.variant (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 dec_sko (prop_of th) (thy,[]) end;
(*Traverse a theorem, accumulating Skolem function definitions.*)
fun assume_skofuns th =
let 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 = term_frees xtp \\ skos (*the formal parameters*)
val Ts = map type_of args
val cT = Ts ---> T
val c = Free (gensym "sko_", 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 = equals cT $ c $ rhs
in dec_sko (subst_bound (list_comb(c,args), p))
(def :: defs)
end
| dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) defs =
(*Universal quant: insert a free variable into body and continue*)
let val fname = Name.variant (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 FUNCTION DEFINITIONS ****)
(*Returns the vars of a theorem*)
fun vars_of_thm th =
map (Thm.cterm_of (theory_of_thm th) o Var) (Drule.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;
(*Convert meta- to object-equality. Fails for theorems like split_comp_eq,
where some types have the empty sort.*)
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
fun mk_object_eq th = th RS meta_eq_to_obj_eq
handle THM _ => error ("Theorem contains empty sort: " ^ string_of_thm th);
(*Apply a function definition to an argument, beta-reducing the result.*)
fun beta_comb cf x =
let val th1 = combination cf (reflexive x)
val th2 = beta_conversion false (Drule.rhs_of th1)
in transitive th1 th2 end;
(*Apply a function definition to arguments, beta-reducing along the way.*)
fun list_combination cf [] = cf
| list_combination cf (x::xs) = list_combination (beta_comb cf x) xs;
fun list_cabs ([] , t) = t
| list_cabs (v::vars, t) = Thm.cabs v (list_cabs(vars,t));
fun assert_eta_free ct =
let val t = term_of ct
in if (t aconv Envir.eta_contract t) then ()
else error ("Eta redex in term: " ^ string_of_cterm ct)
end;
fun eq_absdef (th1, th2) =
Context.joinable (theory_of_thm th1, theory_of_thm th2) andalso
rhs_of th1 aconv rhs_of th2;
fun lambda_free (Abs _) = false
| lambda_free (t $ u) = lambda_free t andalso lambda_free u
| lambda_free _ = true;
fun monomorphic t =
Term.fold_types (Term.fold_atyps (fn TVar _ => K false | _ => I)) t true;
fun dest_abs_list ct =
let val (cv,ct') = Thm.dest_abs NONE ct
val (cvs,cu) = dest_abs_list ct'
in (cv::cvs, cu) end
handle CTERM _ => ([],ct);
fun lambda_list [] u = u
| lambda_list (v::vs) u = lambda v (lambda_list vs u);
fun abstract_rule_list [] [] th = th
| abstract_rule_list (v::vs) (ct::cts) th = abstract_rule v ct (abstract_rule_list vs cts th)
| abstract_rule_list _ _ th = raise THM ("abstract_rule_list", 0, [th]);
val Envir.Envir {asol = tenv0, iTs = tyenv0, ...} = Envir.empty 0
(*Does an existing abstraction definition have an RHS that matches the one we need now?
thy is the current theory, which must extend that of theorem th.*)
fun match_rhs thy t th =
let val _ = if !trace_abs then warning ("match_rhs: " ^ string_of_cterm (cterm_of thy t) ^
" against\n" ^ string_of_thm th) else ();
val (tyenv,tenv) = Pattern.first_order_match thy (rhs_of th, t) (tyenv0,tenv0)
val term_insts = map Meson.term_pair_of (Vartab.dest tenv)
val ct_pairs = if subthy (theory_of_thm th, thy) andalso
forall lambda_free (map #2 term_insts)
then map (pairself (cterm_of thy)) term_insts
else raise Pattern.MATCH (*Cannot allow lambdas in the instantiation*)
fun ctyp2 (ixn, (S, T)) = (ctyp_of thy (TVar (ixn, S)), ctyp_of thy T)
val th' = cterm_instantiate ct_pairs th
in SOME (th, instantiate (map ctyp2 (Vartab.dest tyenv), []) th') end
handle _ => NONE;
(*Traverse a theorem, declaring abstraction function definitions. String s is the suggested
prefix for the constants. Resulting theory is returned in the first theorem. *)
fun declare_absfuns th =
let fun abstract thy ct =
if lambda_free (term_of ct) then (transfer thy (reflexive ct), [])
else
case term_of ct of
Abs _ =>
let val cname = Name.internal (gensym "abs_");
val _ = assert_eta_free ct;
val (cvs,cta) = dest_abs_list ct
val (vs,Tvs) = ListPair.unzip (map (dest_Free o term_of) cvs)
val _ = if !trace_abs then warning ("Nested lambda: " ^ string_of_cterm cta) else ();
val (u'_th,defs) = abstract thy cta
val _ = if !trace_abs then warning ("Returned " ^ string_of_thm u'_th) else ();
val cu' = Drule.rhs_of u'_th
val u' = term_of cu'
val abs_v_u = lambda_list (map term_of cvs) u'
(*get the formal parameters: ALL variables free in the term*)
val args = term_frees abs_v_u
val _ = if !trace_abs then warning (Int.toString (length args) ^ " arguments") else ();
val rhs = list_abs_free (map dest_Free args, abs_v_u)
(*Forms a lambda-abstraction over the formal parameters*)
val _ = if !trace_abs then warning ("Looking up " ^ string_of_cterm cu') else ();
val thy = theory_of_thm u'_th
val (ax,ax',thy) =
case List.mapPartial (match_rhs thy abs_v_u)
(Net.match_term (!abstraction_cache) u') of
(ax,ax')::_ =>
(if !trace_abs then warning ("Re-using axiom " ^ string_of_thm ax) else ();
(ax,ax',thy))
| [] =>
let val _ = if !trace_abs then warning "Lookup was empty" else ();
val Ts = map type_of args
val cT = Ts ---> (Tvs ---> typ_of (ctyp_of_term cu'))
val c = Const (Sign.full_name thy cname, cT)
val thy = Sign.add_consts_authentic [(cname, cT, NoSyn)] thy
(*Theory is augmented with the constant,
then its definition*)
val cdef = cname ^ "_def"
val thy = Theory.add_defs_i false false
[(cdef, equals cT $ c $ rhs)] thy
val _ = if !trace_abs then (warning ("Definition is " ^
string_of_thm (get_axiom thy cdef)))
else ();
val ax = get_axiom thy cdef |> freeze_thm
|> mk_object_eq |> strip_lambdas (length args)
|> mk_meta_eq |> Meson.generalize
val (_,ax') = Option.valOf (match_rhs thy abs_v_u ax)
val _ = if !trace_abs then
(warning ("Declaring: " ^ string_of_thm ax);
warning ("Instance: " ^ string_of_thm ax'))
else ();
val _ = abstraction_cache := Net.insert_term eq_absdef
((Logic.varify u'), ax) (!abstraction_cache)
handle Net.INSERT =>
raise THM ("declare_absfuns: INSERT", 0, [th,u'_th,ax])
in (ax,ax',thy) end
in if !trace_abs then warning ("Lookup result: " ^ string_of_thm ax') else ();
(transitive (abstract_rule_list vs cvs u'_th) (symmetric ax'), ax::defs) end
| (t1$t2) =>
let val (ct1,ct2) = Thm.dest_comb ct
val (th1,defs1) = abstract thy ct1
val (th2,defs2) = abstract (theory_of_thm th1) ct2
in (combination th1 th2, defs1@defs2) end
val _ = if !trace_abs then warning ("declare_absfuns, Abstracting: " ^ string_of_thm th) else ();
val (eqth,defs) = abstract (theory_of_thm th) (cprop_of th)
val ths = equal_elim eqth th :: map (strip_lambdas ~1 o mk_object_eq o freeze_thm) defs
val _ = if !trace_abs then warning ("declare_absfuns, Result: " ^ string_of_thm (hd ths)) else ();
in (theory_of_thm eqth, map Drule.eta_contraction_rule ths) end;
fun name_of def = try (#1 o dest_Free o lhs_of) def;
(*A name is valid provided it isn't the name of a defined abstraction.*)
fun valid_name defs (Free(x,T)) = not (x mem_string (List.mapPartial name_of defs))
| valid_name defs _ = false;
fun assume_absfuns th =
let val thy = theory_of_thm th
val cterm = cterm_of thy
fun abstract ct =
if lambda_free (term_of ct) then (reflexive ct, [])
else
case term_of ct of
Abs (_,T,u) =>
let val _ = assert_eta_free ct;
val (cvs,cta) = dest_abs_list ct
val (vs,Tvs) = ListPair.unzip (map (dest_Free o term_of) cvs)
val (u'_th,defs) = abstract cta
val cu' = Drule.rhs_of u'_th
val u' = term_of cu'
(*Could use Thm.cabs instead of lambda to work at level of cterms*)
val abs_v_u = lambda_list (map term_of cvs) (term_of cu')
(*get the formal parameters: free variables not present in the defs
(to avoid taking abstraction function names as parameters) *)
val args = filter (valid_name defs) (term_frees abs_v_u)
val crhs = list_cabs (map cterm args, cterm abs_v_u)
(*Forms a lambda-abstraction over the formal parameters*)
val rhs = term_of crhs
val (ax,ax') =
case List.mapPartial (match_rhs thy abs_v_u)
(Net.match_term (!abstraction_cache) u') of
(ax,ax')::_ =>
(if !trace_abs then warning ("Re-using axiom " ^ string_of_thm ax) else ();
(ax,ax'))
| [] =>
let val Ts = map type_of args
val const_ty = Ts ---> (Tvs ---> typ_of (ctyp_of_term cu'))
val c = Free (gensym "abs_", const_ty)
val ax = assume (Thm.capply (cterm (equals const_ty $ c)) crhs)
|> mk_object_eq |> strip_lambdas (length args)
|> mk_meta_eq |> Meson.generalize
val (_,ax') = Option.valOf (match_rhs thy abs_v_u ax)
val _ = abstraction_cache := Net.insert_term eq_absdef (rhs,ax)
(!abstraction_cache)
handle Net.INSERT =>
raise THM ("assume_absfuns: INSERT", 0, [th,u'_th,ax])
in (ax,ax') end
in if !trace_abs then warning ("Lookup result: " ^ string_of_thm ax') else ();
(transitive (abstract_rule_list vs cvs u'_th) (symmetric ax'), ax::defs) end
| (t1$t2) =>
let val (ct1,ct2) = Thm.dest_comb ct
val (t1',defs1) = abstract ct1
val (t2',defs2) = abstract ct2
in (combination t1' t2', defs1@defs2) end
val _ = if !trace_abs then warning ("assume_absfuns, Abstracting: " ^ string_of_thm th) else ();
val (eqth,defs) = abstract (cprop_of th)
val ths = equal_elim eqth th :: map (strip_lambdas ~1 o mk_object_eq o freeze_thm) defs
val _ = if !trace_abs then warning ("assume_absfuns, Result: " ^ string_of_thm (hd ths)) else ();
in map Drule.eta_contraction_rule ths end;
(*cterms are used throughout for efficiency*)
val cTrueprop = Thm.cterm_of HOL.thy 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) = Drule.dest_equals (cprop_of (freeze_thm def))
val (ch, frees) = c_variant_abs_multi (rhs, [])
val (chilbert,cabs) = Thm.dest_comb ch
val {thy,t, ...} = rep_cterm 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 someI_ex) 1
in Goal.prove_raw [ex_tm] conc tacf
|> forall_intr_list frees
|> 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 =
th |> transfer_to_ATP_Linkup
|> transform_elim |> zero_var_indexes |> freeze_thm
|> ObjectLogic.atomize_thm |> make_nnf |> strip_lambdas ~1;
(*Generate Skolem functions for a theorem supplied in nnf*)
fun skolem_of_nnf th =
map (skolem_of_def o assume o (cterm_of (theory_of_thm th))) (assume_skofuns th);
fun assert_lambda_free ths msg =
case filter (not o lambda_free o prop_of) ths of
[] => ()
| ths' => error (msg ^ "\n" ^ space_implode "\n" (map string_of_thm ths'));
fun assume_abstract th =
if lambda_free (prop_of th) then [th]
else th |> Drule.eta_contraction_rule |> assume_absfuns
|> tap (fn ths => assert_lambda_free ths "assume_abstract: lambdas")
(*Replace lambdas by assumed function definitions in the theorems*)
fun assume_abstract_list ths =
if abstract_lambdas then List.concat (map assume_abstract ths)
else map Drule.eta_contraction_rule ths;
(*Replace lambdas by declared function definitions in the theorems*)
fun declare_abstract' (thy, []) = (thy, [])
| declare_abstract' (thy, th::ths) =
let val (thy', th_defs) =
if lambda_free (prop_of th) then (thy, [th])
else
th |> zero_var_indexes |> freeze_thm
|> Drule.eta_contraction_rule |> transfer thy |> declare_absfuns
val _ = assert_lambda_free th_defs "declare_abstract: lambdas"
val (thy'', ths') = declare_abstract' (thy', ths)
in (thy'', th_defs @ ths') end;
fun declare_abstract (thy, ths) =
if abstract_lambdas then declare_abstract' (thy, ths)
else (thy, map Drule.eta_contraction_rule ths);
(*Skolemize a named theorem, with Skolem functions as additional premises.*)
fun skolem_thm th =
let val nnfth = to_nnf th
in Meson.make_cnf (skolem_of_nnf nnfth) nnfth |> assume_abstract_list |> Meson.finish_cnf
end
handle THM _ => [];
(*Keep the full complexity of the original name*)
fun flatten_name s = space_implode "_X" (NameSpace.explode s);
(*Declare Skolem functions for a theorem, supplied in nnf and with its name.
It returns a modified theory, unless skolemization fails.*)
fun skolem thy th =
let val cname = (case PureThy.get_name_hint th of "" => gensym "" | s => flatten_name s)
val _ = Output.debug (fn () => "skolemizing " ^ cname ^ ": ")
in Option.map
(fn nnfth =>
let val (thy',defs) = declare_skofuns cname nnfth thy
val cnfs = Meson.make_cnf (map skolem_of_def defs) nnfth
val (thy'',cnfs') = declare_abstract (thy',cnfs)
in (map Goal.close_result (Meson.finish_cnf cnfs'), thy'')
end)
(SOME (to_nnf th) handle THM _ => NONE)
end;
structure ThmCache = TheoryDataFun
(struct
val name = "ATP/thm_cache";
type T = (thm list) Thmtab.table ref;
val empty : T = ref Thmtab.empty;
fun copy (ref tab) : T = ref tab;
val extend = copy;
fun merge _ (ref tab1, ref tab2) : T = ref (Thmtab.merge (K true) (tab1, tab2));
fun print _ _ = ();
end);
(*The cache prevents repeated clausification of a theorem, and also repeated declaration of
Skolem functions. The global one holds theorems proved prior to this point. Theory data
holds the remaining ones.*)
val global_clause_cache = ref (Thmtab.empty : (thm list) Thmtab.table);
(*Populate the clause cache using the supplied theorem. Return the clausal form
and modified theory.*)
fun skolem_cache_thm clause_cache th thy =
case Thmtab.lookup (!clause_cache) th of
NONE =>
(case skolem thy (Thm.transfer thy th) of
NONE => ([th],thy)
| SOME (cls,thy') =>
(if null cls
then warning ("skolem_cache: empty clause set for " ^ string_of_thm th)
else ();
change clause_cache (Thmtab.update (th, cls));
(cls,thy')))
| SOME cls => (cls,thy);
(*Exported function to convert Isabelle theorems into axiom clauses*)
fun cnf_axiom th =
let val cache = ThmCache.get (Thm.theory_of_thm th)
handle ERROR _ => global_clause_cache
val in_cache = if cache = global_clause_cache then NONE else Thmtab.lookup (!cache) th
in
case in_cache of
NONE =>
(case Thmtab.lookup (!global_clause_cache) th of
NONE =>
let val cls = map Goal.close_result (skolem_thm th)
in Output.debug (fn () => "inserted into cache: " ^ PureThy.get_name_hint th);
change cache (Thmtab.update (th, cls)); cls
end
| SOME cls => cls)
| SOME cls => cls
end;
fun pairname th = (PureThy.get_name_hint th, th);
(**** Extract and Clausify theorems from a theory's claset and simpset ****)
fun rules_of_claset cs =
let val {safeIs,safeEs,hazIs,hazEs,...} = rep_cs cs
val intros = safeIs @ hazIs
val elims = map Classical.classical_rule (safeEs @ hazEs)
in
Output.debug (fn () => "rules_of_claset intros: " ^ Int.toString(length intros) ^
" elims: " ^ Int.toString(length elims));
map pairname (intros @ elims)
end;
fun rules_of_simpset ss =
let val ({rules,...}, _) = rep_ss ss
val simps = Net.entries rules
in
Output.debug (fn () => "rules_of_simpset: " ^ Int.toString(length simps));
map (fn r => (#name r, #thm r)) simps
end;
fun claset_rules_of ctxt = rules_of_claset (local_claset_of ctxt);
fun simpset_rules_of ctxt = rules_of_simpset (local_simpset_of ctxt);
fun atpset_rules_of ctxt = map pairname (ResAtpset.get_atpset ctxt);
(**** Translate a set of theorems into CNF ****)
(* classical rules: works for both FOL and HOL *)
fun cnf_rules [] err_list = ([],err_list)
| cnf_rules ((name,th) :: ths) err_list =
let val (ts,es) = cnf_rules ths err_list
in (cnf_axiom th :: ts,es) handle _ => (ts, (th::es)) end;
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 pairs ((name,th)::ths) =
let val pairs' = (pair_name_cls 0 (name, cnf_axiom th)) @ pairs
handle THM _ => pairs | ResClause.CLAUSE _ => pairs
in cnf_rules_pairs_aux pairs' ths end;
(*The combination of rev and tail recursion preserves the original order*)
fun cnf_rules_pairs l = cnf_rules_pairs_aux [] (rev l);
(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause, or ResHolClause.clause) ****)
(*Setup function: takes a theory and installs ALL known theorems into the clause cache*)
fun skolem_cache clause_cache th thy = #2 (skolem_cache_thm clause_cache th 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.*)
fun clause_cache_setup thy =
fold (skolem_cache global_clause_cache) (map #2 (PureThy.all_thms_of thy)) thy;
(*** meson proof methods ***)
fun cnf_rules_of_ths ths = List.concat (map cnf_axiom ths);
val meson_method_setup = Method.add_methods
[("meson", Method.thms_args (fn ths =>
Method.SIMPLE_METHOD' (CHANGED_PROP o Meson.meson_claset_tac (cnf_rules_of_ths ths) HOL_cs)),
"MESON resolution proof procedure")];
(** Attribute for converting a theorem into clauses **)
fun meta_cnf_axiom th = map Meson.make_meta_clause (cnf_axiom th);
fun clausify_rule (th,i) = List.nth (meta_cnf_axiom th, i)
val clausify = Attrib.syntax (Scan.lift Args.nat
>> (fn i => Thm.rule_attribute (fn _ => fn th => clausify_rule (th, i))));
(*** Converting a subgoal into negated conjecture clauses. ***)
val neg_skolemize_tac = EVERY' [rtac ccontr, ObjectLogic.atomize_tac, skolemize_tac];
(*finish_cnf removes tautologies and functional reflexivity axioms, but by calling Thm.varifyT
it can introduce TVars, which we don't want in conjecture clauses.*)
val neg_clausify = map Thm.freezeT o Meson.finish_cnf o assume_abstract_list o make_clauses;
fun neg_conjecture_clauses st0 n =
let val st = Seq.hd (neg_skolemize_tac n st0)
val (params,_,_) = strip_context (Logic.nth_prem (n, Thm.prop_of st))
in (neg_clausify (Option.valOf (metahyps_thms n st)), params) end
handle Option => raise ERROR "unable to Skolemize subgoal";
(*Conversion of a subgoal to conjecture clauses. Each clause has
leading !!-bound universal variables, to express generality. *)
val neg_clausify_tac =
neg_skolemize_tac THEN'
SUBGOAL
(fn (prop,_) =>
let val ts = Logic.strip_assums_hyp prop
in EVERY1
[METAHYPS
(fn hyps =>
(Method.insert_tac
(map forall_intr_vars (neg_clausify hyps)) 1)),
REPEAT_DETERM_N (length ts) o (etac thin_rl)]
end);
(** The Skolemization attribute **)
fun conj2_rule (th1,th2) = conjI OF [th1,th2];
(*Conjoin a list of theorems to form a single theorem*)
fun conj_rule [] = TrueI
| conj_rule ths = foldr1 conj2_rule ths;
fun skolem_attr (Context.Theory thy, th) =
let val (cls, thy') = skolem_cache_thm (ThmCache.get thy) th thy
in (Context.Theory thy', conj_rule cls) end
| skolem_attr (context, th) = (context, conj_rule (cnf_axiom th));
val setup_attrs = Attrib.add_attributes
[("skolem", Attrib.no_args skolem_attr, "skolemization of a theorem"),
("clausify", clausify, "conversion of theorem to clauses")];
val setup_methods = Method.add_methods
[("neg_clausify", Method.no_args (Method.SIMPLE_METHOD' neg_clausify_tac),
"conversion of goal to conjecture clauses")];
val setup = clause_cache_setup #> ThmCache.init #> setup_attrs #> setup_methods;
end;