(* 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 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 cnf_axiom : (string * thm) -> thm list
val meta_cnf_axiom : thm -> thm list
val cnf_rule : thm -> thm list
val cnf_rules : (string*thm) list -> thm list -> thm list list * thm list
val rm_Eps : (term * term) list -> thm list -> term list
val claset_rules_of_thy : theory -> (string * thm) list
val simpset_rules_of_thy : theory -> (string * thm) list
val clausify_rules_pairs : (string * thm) list -> thm list -> (ResClause.clause * thm) list list * thm list
val clause_setup : (theory -> theory) list
val meson_method_setup : (theory -> theory) list
end;
structure ResAxioms : RES_AXIOMS =
struct
(**** 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
(let val P' = HOLogic.dest_Trueprop P
val prems' = P'::prems
in
strip_concl prems' bvs concl Q
end)
| 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
prove_goalw_cterm [] ctm (fn prems => [elimRule_tac th])
end
else th;
(**** Transformation of Clasets and Simpsets into First-Order Axioms ****)
(* to be fixed: cnf_intro, cnf_rule, is_introR *)
(* repeated resolution *)
fun repeat_RS thm1 thm2 =
let val thm1' = thm1 RS thm2 handle THM _ => thm1
in
if eq_thm(thm1,thm1') then thm1' else (repeat_RS thm1' thm2)
end;
(*Convert a theorem into NNF and also skolemize it. Original version, using
Hilbert's epsilon in the resulting clauses.*)
fun skolem_axiom th =
let val th' = (skolemize o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) th
in repeat_RS th' someI_ex
end;
fun cnf_rule th = make_clauses [skolem_axiom (transform_elim th)];
(*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);
(* transform an Isabelle thm into CNF *)
fun cnf_axiom_aux th =
map zero_var_indexes
(rm_redundant_cls (cnf_rule (transfer_to_Reconstruction th)));
(**** SKOLEMIZATION BY INFERENCE (lcp) ****)
(*Traverse a term, accumulating Skolem function definitions.*)
fun declare_skofuns s t thy =
let fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (n, thy) =
(*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 (Theory.sign_of 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 thy'' = Theory.add_defs_i false [(cname ^ "_def", def)] thy'
in dec_sko (subst_bound (list_comb(c,args), p)) (n+1, thy'') end
| dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) (n, thy) =
(*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, thy) 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 ("Trueprop", _) $ p) nthy =
dec_sko p nthy
| dec_sko t (n,thy) = (n,thy) (*Do nothing otherwise*)
in #2 (dec_sko t (1,thy)) end;
(*cterms are used throughout for efficiency*)
val cTrueprop = Thm.cterm_of (Theory.sign_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.*)
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 (chil,cabs) = Thm.dest_comb ch
val {sign,t, ...} = rep_cterm chil
val (Const ("Hilbert_Choice.Eps", Type("fun",[_,T]))) = t
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)));
in prove_goalw_cterm [def] (Drule.mk_implies (ex_tm, conc))
(fn [prem] => [ rtac (prem RS someI_ex) 1 ])
end;
(*Converts an Isabelle theorem (intro, elim or simp format) into nnf.*)
fun to_nnf thy th =
th |> Thm.transfer thy
|> 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!? *)
val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table)
(*Declare Skolem functions for a theorem, supplied in nnf and with its name*)
fun skolem thy (name,th) =
let val cname = (case name of "" => gensym "sko" | s => Sign.base_name s)
val thy' = declare_skofuns cname (#prop (rep_thm th)) thy
in (map (skolem_of_def o #2) (axioms_of thy'), thy') end;
(*Populate the clause cache using the supplied theorems*)
fun skolemlist [] thy = thy
| skolemlist ((name,th)::nths) thy =
(case Symtab.curried_lookup (!clause_cache) name of
NONE =>
let val (nnfth,ok) = (to_nnf thy th, true)
handle THM _ => (asm_rl, false)
in
if ok then
let val (skoths,thy') = skolem thy (name, nnfth)
val cls = Meson.make_cnf skoths nnfth
in change clause_cache (Symtab.curried_update (name, (th, cls)));
skolemlist nths thy'
end
else skolemlist nths thy
end
| SOME _ => skolemlist nths thy) (*FIXME: check for duplicate names?*)
(*Exported function to convert Isabelle theorems into axiom clauses*)
fun cnf_axiom (name,th) =
case name of
"" => cnf_axiom_aux th (*no name, so can't cache*)
| s => case Symtab.curried_lookup (!clause_cache) s of
NONE =>
let val cls = cnf_axiom_aux th
in change clause_cache (Symtab.curried_update (s, (th, cls))); cls end
| SOME(th',cls) =>
if eq_thm(th,th') then cls
else (*New theorem stored under the same name? Possible??*)
let val cls = cnf_axiom_aux th
in change clause_cache (Symtab.curried_update (s, (th, cls))); cls end;
fun pairname th = (Thm.name_of_thm th, th);
fun meta_cnf_axiom th =
map Meson.make_meta_clause (cnf_axiom (pairname th));
(* changed: with one extra case added *)
fun univ_vars_of_aux (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,body)) vars =
univ_vars_of_aux body vars
| univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars =
univ_vars_of_aux body vars (* EX x. body *)
| univ_vars_of_aux (P $ Q) vars =
univ_vars_of_aux Q (univ_vars_of_aux P vars)
| univ_vars_of_aux (t as Var(_,_)) vars =
if (t mem vars) then vars else (t::vars)
| univ_vars_of_aux _ vars = vars;
fun univ_vars_of t = univ_vars_of_aux t [];
fun get_new_skolem epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,tp,_))) =
let val all_vars = univ_vars_of t
val sk_term = ResSkolemFunction.gen_skolem all_vars tp
in
(sk_term,(t,sk_term)::epss)
end;
fun sk_lookup [] t = NONE
| sk_lookup ((tm,sk_tm)::tms) t = if (t = tm) then SOME (sk_tm) else (sk_lookup tms t);
(* get the proper skolem term to replace epsilon term *)
fun get_skolem epss t =
case (sk_lookup epss t) of NONE => get_new_skolem epss t
| SOME sk => (sk,epss);
fun rm_Eps_cls_aux epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,_))) =
get_skolem epss t
| rm_Eps_cls_aux epss (P $ Q) =
let val (P',epss') = rm_Eps_cls_aux epss P
val (Q',epss'') = rm_Eps_cls_aux epss' Q
in (P' $ Q',epss'') end
| rm_Eps_cls_aux epss t = (t,epss);
fun rm_Eps_cls epss th = rm_Eps_cls_aux epss (prop_of th);
(* remove the epsilon terms in a formula, by skolem terms. *)
fun rm_Eps _ [] = []
| rm_Eps epss (th::thms) =
let val (th',epss') = rm_Eps_cls epss th
in th' :: (rm_Eps epss' thms) end;
(**** Extract and Clausify theorems from a theory's claset and simpset ****)
fun claset_rules_of_thy thy =
let val clsset = rep_cs (claset_of thy)
val safeEs = #safeEs clsset
val safeIs = #safeIs clsset
val hazEs = #hazEs clsset
val hazIs = #hazIs clsset
in
map pairname (safeEs @ safeIs @ hazEs @ hazIs)
end;
fun simpset_rules_of_thy thy =
let val rules = #rules (fst (rep_ss (simpset_of thy)))
in map (fn r => (#name r, #thm r)) (Net.entries rules) end;
(**** Translate a set of classical/simplifier rules into CNF (still as type "thm") ****)
(* classical rules *)
fun cnf_rules [] err_list = ([],err_list)
| cnf_rules ((name,th) :: thms) err_list =
let val (ts,es) = cnf_rules thms err_list
in (cnf_axiom (name,th) :: ts,es) handle _ => (ts, (th::es)) end;
(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****)
(* outputs a list of (clause,thm) pairs *)
fun clausify_axiom_pairs (thm_name,thm) =
let val isa_clauses = cnf_axiom (thm_name,thm) (*"isa_clauses" are already "standard"ed. *)
val isa_clauses' = rm_Eps [] isa_clauses
val clauses_n = length isa_clauses
val thy = theory_of_thm thm
fun make_axiom_clauses _ [] []= []
| make_axiom_clauses i (cls::clss) (cls'::clss') =
(ResClause.make_axiom_clause cls (thm_name,i), cls') ::
make_axiom_clauses (i+1) clss clss'
in
make_axiom_clauses 0 isa_clauses' isa_clauses
end;
fun clausify_rules_pairs [] err_list = ([],err_list)
| clausify_rules_pairs ((name,thm)::thms) err_list =
let val (ts,es) = clausify_rules_pairs thms err_list
in
((clausify_axiom_pairs (name,thm))::ts, es)
handle _ => (ts, (thm::es))
end;
(*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
in skolemlist clas (skolemlist simps thy) end;
val clause_setup = [clause_cache_setup];
(*** 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")]];
end;