(* 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
exception ELIMR2FOL of string
val tagging_enabled : bool
val elimRule_tac : thm -> Tactical.tactic
val elimR2Fol : thm -> term
val transform_elim : thm -> thm
val clausify_axiom_pairs : (string*thm) -> (ResClause.clause*thm) list
val clausify_axiom_pairsH : (string*thm) -> (ResHolClause.clause*thm) list
val cnf_axiom : (string * thm) -> thm list
val meta_cnf_axiom : thm -> thm list
val claset_rules_of_thy : theory -> (string * thm) list
val simpset_rules_of_thy : theory -> (string * thm) list
val claset_rules_of_ctxt: Proof.context -> (string * thm) list
val simpset_rules_of_ctxt : Proof.context -> (string * thm) list
val clausify_rules_pairs : (string*thm) list -> (ResClause.clause*thm) list list
val clausify_rules_pairsH : (string*thm) list -> (ResHolClause.clause*thm) list list
val pairname : thm -> (string * thm)
val skolem_thm : thm -> thm list
val cnf_rules1 : (string * thm) list -> string list -> (string * thm list) list * string list
val cnf_rules2 : (string * thm) list -> string list -> (string * term list) list * string list
val meson_method_setup : (theory -> theory) list (*Reconstruction.thy*)
val setup : (theory -> theory) list (*Main.thy*)
end;
structure ResAxioms : RES_AXIOMS =
struct
val tagging_enabled = false (*compile_time option*)
(**** Transformation of Elimination Rules into First-Order Formulas****)
(* a tactic used to prove an elim-rule. *)
fun elimRule_tac th =
((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac th 1) THEN
REPEAT(fast_tac HOL_cs 1);
exception ELIMR2FOL of string;
(* functions used to construct a formula *)
fun make_disjs [x] = x
| make_disjs (x :: xs) = HOLogic.mk_disj(x, make_disjs xs)
fun make_conjs [x] = x
| make_conjs (x :: xs) = HOLogic.mk_conj(x, make_conjs xs)
fun add_EX tm [] = tm
| add_EX tm ((x,xtp)::xs) = add_EX (HOLogic.exists_const xtp $ Abs(x,xtp,tm)) xs;
fun is_neg (Const("Trueprop",_) $ (Const("Not",_) $ Free(p,_))) (Const("Trueprop",_) $ Free(q,_)) = (p = q)
| is_neg _ _ = false;
exception STRIP_CONCL;
fun strip_concl' prems bvs (Const ("==>",_) $ P $ Q) =
let val P' = HOLogic.dest_Trueprop P
val prems' = P'::prems
in
strip_concl' prems' bvs Q
end
| strip_concl' prems bvs P =
let val P' = HOLogic.Not $ (HOLogic.dest_Trueprop P)
in
add_EX (make_conjs (P'::prems)) bvs
end;
fun strip_concl prems bvs concl (Const ("all", _) $ Abs (x,xtp,body)) =
strip_concl prems ((x,xtp)::bvs) concl body
| strip_concl prems bvs concl (Const ("==>",_) $ P $ Q) =
if (is_neg P concl) then (strip_concl' prems bvs Q)
else strip_concl (HOLogic.dest_Trueprop P::prems) bvs concl Q
| strip_concl prems bvs concl _ = add_EX (make_conjs prems) bvs;
fun trans_elim (main,others,concl) =
let val others' = map (strip_concl [] [] concl) others
val disjs = make_disjs others'
in
HOLogic.mk_imp (HOLogic.dest_Trueprop main, disjs)
end;
(* aux function of elim2Fol, take away predicate variable. *)
fun elimR2Fol_aux prems concl =
let val nprems = length prems
val main = hd prems
in
if (nprems = 1) then HOLogic.Not $ (HOLogic.dest_Trueprop main)
else trans_elim (main, tl prems, concl)
end;
(* convert an elim rule into an equivalent formula, of type term. *)
fun elimR2Fol elimR =
let val elimR' = Drule.freeze_all elimR
val (prems,concl) = (prems_of elimR', concl_of elimR')
in
case concl of Const("Trueprop",_) $ Free(_,Type("bool",[]))
=> HOLogic.mk_Trueprop (elimR2Fol_aux prems concl)
| Free(x,Type("prop",[])) => HOLogic.mk_Trueprop(elimR2Fol_aux prems concl)
| _ => raise ELIMR2FOL("Not an elimination rule!")
end;
(* check if a rule is an elim rule *)
fun is_elimR th =
case (concl_of th) of (Const ("Trueprop", _) $ Var (idx,_)) => true
| Var(indx,Type("prop",[])) => true
| _ => false;
(* convert an elim-rule into an equivalent theorem that does not have the
predicate variable. Leave other theorems unchanged.*)
fun transform_elim th =
if is_elimR th then
let val tm = elimR2Fol th
val ctm = cterm_of (sign_of_thm th) tm
in Goal.prove_raw [] ctm (fn _ => elimRule_tac th) end
else th;
(**** Transformation of Clasets and Simpsets into First-Order Axioms ****)
(*Transfer a theorem into theory Reconstruction.thy if it is not already
inside that theory -- because it's needed for Skolemization *)
(*This will refer to the final version of theory Reconstruction.*)
val recon_thy_ref = Theory.self_ref (the_context ());
(*If called while Reconstruction is being created, it will transfer to the
current version. If called afterward, it will transfer to the final version.*)
fun transfer_to_Reconstruction th =
transfer (Theory.deref recon_thy_ref) th handle THM _ => th;
fun is_taut th =
case (prop_of th) of
(Const ("Trueprop", _) $ Const ("True", _)) => true
| _ => false;
(* remove tautologous clauses *)
val rm_redundant_cls = List.filter (not o is_taut);
(**** 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 fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (n, (thy, axs)) =
(*Existential: declare a Skolem function, then insert into body and continue*)
let val cname = s ^ "_" ^ Int.toString n
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 def = equals cT $ c $ rhs
val thy' = Theory.add_consts_i [(cname, cT, NoSyn)] thy
(*Theory is augmented with the constant, then its def*)
val cdef = cname ^ "_def"
val thy'' = Theory.add_defs_i false [(cdef, def)] thy'
in dec_sko (subst_bound (list_comb(c,args), p))
(n+1, (thy'', get_axiom thy'' cdef :: axs))
end
| dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) (n, thx) =
(*Universal quant: insert a free variable into body and continue*)
let val fname = variant (add_term_names (p,[])) a
in dec_sko (subst_bound (Free(fname,T), p)) (n, thx) end
| dec_sko (Const ("op &", _) $ p $ q) nthy = dec_sko q (dec_sko p nthy)
| dec_sko (Const ("op |", _) $ p $ q) nthy = dec_sko q (dec_sko p nthy)
| dec_sko (Const ("HOL.tag", _) $ p) nthy = dec_sko p nthy
| dec_sko (Const ("Trueprop", _) $ p) nthy = dec_sko p nthy
| dec_sko t nthx = nthx (*Do nothing otherwise*)
in #2 (dec_sko (#prop (rep_thm th)) (1, (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 name = variant (add_term_names (p,[])) (gensym "sko_")
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 (name, 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 = 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 ("HOL.tag", _) $ p) defs = 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 (rep_thm th)) [] 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 (Drule.freeze_all def))
val (ch, frees) = c_variant_abs_multi (rhs, [])
val (chilbert,cabs) = Thm.dest_comb ch
val {sign,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 sign (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) into nnf.*)
(*It now works for HOL too. *)
fun to_nnf th =
th |> transfer_to_Reconstruction
|> transform_elim |> Drule.freeze_all
|> ObjectLogic.atomize_thm |> make_nnf;
(*The cache prevents repeated clausification of a theorem,
and also repeated declaration of Skolem functions*)
(* FIXME better use Termtab!? No, we MUST use theory data!!*)
val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table)
(*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);
(*Skolemize a named theorem, with Skolem functions as additional premises.*)
(*also works for HOL*)
fun skolem_thm th =
let val nnfth = to_nnf th
in Meson.make_cnf (skolem_of_nnf nnfth) nnfth
end
handle THM _ => [];
(*Declare Skolem functions for a theorem, supplied in nnf and with its name.
It returns a modified theory, unless skolemization fails.*)
fun skolem thy (name,th) =
let val cname = (case name of "" => gensym "sko" | s => Sign.base_name s)
in Option.map
(fn nnfth =>
let val (thy',defs) = declare_skofuns cname nnfth thy
val skoths = map skolem_of_def defs
in (thy', Meson.make_cnf skoths nnfth) end)
(SOME (to_nnf th) handle THM _ => NONE)
end;
(*Populate the clause cache using the supplied theorem. Return the clausal form
and modified theory.*)
fun skolem_cache_thm ((name,th), thy) =
case Symtab.lookup (!clause_cache) name of
NONE =>
(case skolem thy (name, Thm.transfer thy th) of
NONE => ([th],thy)
| SOME (thy',cls) =>
(change clause_cache (Symtab.update (name, (th, cls))); (cls,thy')))
| SOME (th',cls) =>
if eq_thm(th,th') then (cls,thy)
else (warning ("skolem_cache: Ignoring variant of theorem " ^ name);
warning (string_of_thm th);
warning (string_of_thm th');
([th],thy));
fun skolem_cache ((name,th), thy) = #2 (skolem_cache_thm ((name,th), thy));
(*Exported function to convert Isabelle theorems into axiom clauses*)
fun cnf_axiom_g cnf (name,th) =
case name of
"" => cnf th (*no name, so can't cache*)
| s => case Symtab.lookup (!clause_cache) s of
NONE =>
let val cls = cnf th
in change clause_cache (Symtab.update (s, (th, cls))); cls end
| SOME(th',cls) =>
if eq_thm(th,th') then cls
else (warning ("cnf_axiom: duplicate or variant of theorem " ^ name);
warning (string_of_thm th);
warning (string_of_thm th');
cls);
fun pairname th = (Thm.name_of_thm th, th);
val cnf_axiom = cnf_axiom_g skolem_thm;
fun meta_cnf_axiom th =
map Meson.make_meta_clause (cnf_axiom (pairname th));
(**** Extract and Clausify theorems from a theory's claset and simpset ****)
(*Preserve the name of "th" after the transformation "f"*)
fun preserve_name f th = Thm.name_thm (Thm.name_of_thm th, f th);
(*Tags identify the major premise or conclusion, as hints to resolution provers.
However, they don't appear to help in recent tests, and they complicate the code.*)
val tagI = thm "tagI";
val tagD = thm "tagD";
val tag_intro = preserve_name (fn th => th RS tagI);
val tag_elim = preserve_name (fn th => tagD RS th);
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
debug ("rules_of_claset intros: " ^ Int.toString(length intros) ^
" elims: " ^ Int.toString(length elims));
if tagging_enabled
then map pairname (map tag_intro intros @ map tag_elim elims)
else map pairname (intros @ elims)
end;
fun rules_of_simpset ss =
let val ({rules,...}, _) = rep_ss ss
val simps = Net.entries rules
in
debug ("rules_of_simpset: " ^ Int.toString(length simps));
map (fn r => (#name r, #thm r)) simps
end;
fun claset_rules_of_thy thy = rules_of_claset (claset_of thy);
fun simpset_rules_of_thy thy = rules_of_simpset (simpset_of thy);
fun claset_rules_of_ctxt ctxt = rules_of_claset (local_claset_of ctxt);
fun simpset_rules_of_ctxt ctxt = rules_of_simpset (local_simpset_of ctxt);
(**** Translate a set of classical/simplifier rules into CNF (still as type "thm") ****)
(* classical rules *)
fun cnf_rules_g cnf_axiom [] err_list = ([],err_list)
| cnf_rules_g cnf_axiom ((name,th) :: ths) err_list =
let val (ts,es) = cnf_rules_g cnf_axiom ths err_list
in (cnf_axiom (name,th) :: ts,es) handle _ => (ts, (th::es)) end;
(*works for both FOL and HOL*)
val cnf_rules = cnf_rules_g cnf_axiom;
fun cnf_rules1 [] err_list = ([],err_list)
| cnf_rules1 ((name,th) :: ths) err_list =
let val (ts,es) = cnf_rules1 ths err_list
in
((name,cnf_axiom (name,th)) :: ts,es) handle _ => (ts,(name::es)) end;
fun cnf_rules2 [] err_list = ([],err_list)
| cnf_rules2 ((name,th) :: ths) err_list =
let val (ts,es) = cnf_rules2 ths err_list
in
((name,map prop_of (cnf_axiom (name,th))) :: ts,es) handle _ => (ts,(name::es)) end;
(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause, or ResHolClause.clause) ****)
fun make_axiom_clauses _ _ [] = []
| make_axiom_clauses name i (cls::clss) =
(ResClause.make_axiom_clause (prop_of cls) (name,i), cls) ::
(make_axiom_clauses name (i+1) clss)
(* outputs a list of (clause,theorem) pairs *)
fun clausify_axiom_pairs (name,th) =
filter (fn (c,th) => not (ResClause.isTaut c))
(make_axiom_clauses name 0 (cnf_axiom (name,th)));
fun make_axiom_clausesH _ _ [] = []
| make_axiom_clausesH name i (cls::clss) =
(ResHolClause.make_axiom_clause (prop_of cls) (name,i), cls) ::
(make_axiom_clausesH name (i+1) clss)
fun clausify_axiom_pairsH (name,th) =
filter (fn (c,th) => not (ResHolClause.isTaut c))
(make_axiom_clausesH name 0 (cnf_axiom (name,th)));
fun clausify_rules_pairs_aux result [] = result
| clausify_rules_pairs_aux result ((name,th)::ths) =
clausify_rules_pairs_aux (clausify_axiom_pairs (name,th) :: result) ths
handle THM (msg,_,_) =>
(debug ("Cannot clausify " ^ name ^ ": " ^ msg);
clausify_rules_pairs_aux result ths)
| ResClause.CLAUSE (msg,t) =>
(debug ("Cannot clausify " ^ name ^ ": " ^ msg ^
": " ^ TermLib.string_of_term t);
clausify_rules_pairs_aux result ths)
fun clausify_rules_pairs_auxH result [] = result
| clausify_rules_pairs_auxH result ((name,th)::ths) =
clausify_rules_pairs_auxH (clausify_axiom_pairsH (name,th) :: result) ths
handle THM (msg,_,_) =>
(debug ("Cannot clausify " ^ name ^ ": " ^ msg);
clausify_rules_pairs_auxH result ths)
| ResHolClause.LAM2COMB (t) =>
(debug ("Cannot clausify " ^ TermLib.string_of_term t);
clausify_rules_pairs_auxH result ths)
val clausify_rules_pairs = clausify_rules_pairs_aux [];
val clausify_rules_pairsH = clausify_rules_pairs_auxH [];
(*These should include any plausibly-useful theorems, especially if they need
Skolem functions. FIXME: this list is VERY INCOMPLETE*)
val default_initial_thms = map pairname
[refl_def, antisym_def, sym_def, trans_def, single_valued_def,
subset_refl, Union_least, Inter_greatest];
(*Setup function: takes a theory and installs ALL simprules and claset rules
into the clause cache*)
fun clause_cache_setup thy =
let val simps = simpset_rules_of_thy thy
and clas = claset_rules_of_thy thy
and thy0 = List.foldl skolem_cache thy default_initial_thms
val thy1 = List.foldl skolem_cache thy0 clas
in List.foldl skolem_cache thy1 simps end;
(*Could be duplicate theorem names, due to multiple attributes*)
(*** meson proof methods ***)
fun cnf_rules_of_ths ths = List.concat (#1 (cnf_rules (map pairname ths) []));
fun meson_meth ths ctxt =
Method.SIMPLE_METHOD' HEADGOAL
(CHANGED_PROP o Meson.meson_claset_tac (cnf_rules_of_ths ths) (local_claset_of ctxt));
val meson_method_setup =
[Method.add_methods
[("meson", Method.thms_ctxt_args meson_meth,
"The MESON resolution proof procedure")]];
(*** The Skolemization attribute ***)
fun conj2_rule (th1,th2) = conjI OF [th1,th2];
(*Conjoin a list of clauses to recreate a single theorem*)
val conj_rule = foldr1 conj2_rule;
fun skolem_global_fun (thy, th) =
let val name = Thm.name_of_thm th
val (cls,thy') = skolem_cache_thm ((name,th), thy)
in (thy', conj_rule cls) end;
val skolem_global = Attrib.no_args skolem_global_fun;
fun skolem_local x = Attrib.no_args (Attrib.rule (K (conj_rule o skolem_thm))) x;
val setup_attrs = Attrib.add_attributes
[("skolem", (skolem_global, skolem_local),
"skolemization of a theorem")];
val setup = [clause_cache_setup, setup_attrs];
end;