(* 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 cnf_axiomH : (string * thm) -> thm list
val meta_cnf_axiom : thm -> thm list
val meta_cnf_axiomH : thm -> 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 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 clause_setup : (theory -> theory) list
val meson_method_setup : (theory -> theory) list
val pairname : thm -> (string * thm)
val repeat_RS : thm -> thm -> thm
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
(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 Goal.prove_raw [] ctm (fn _ => elimRule_tac th) end
else th;
(**** Transformation of Clasets and Simpsets into First-Order Axioms ****)
(* 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_g mk_nnf th =
let val th' = (skolemize o mk_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) th
in repeat_RS th' someI_ex
end;
val skolem_axiom = skolem_axiom_g make_nnf;
val skolem_axiomH = skolem_axiom_g make_nnf1;
fun cnf_rule th = make_clauses [skolem_axiom (transform_elim th)];
fun cnf_ruleH th = make_clauses [skolem_axiomH (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 theorem into CNF *)
fun cnf_axiom_aux_g cnf_rule th =
map zero_var_indexes
(rm_redundant_cls (cnf_rule (transfer_to_Reconstruction th)));
val cnf_axiom_aux = cnf_axiom_aux_g cnf_rule;
val cnf_axiom_auxH = cnf_axiom_aux_g cnf_ruleH;
(**** 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, 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 (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 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 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 OldGoals.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',axs) = declare_skofuns cname (#prop (rep_thm th)) thy
in (map skolem_of_def axs, thy') end;
(*Populate the clause cache using the supplied theorems*)
fun skolemlist [] thy = thy
| skolemlist ((name,th)::nths) thy =
(case Symtab.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.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_g cnf_axiom_aux (name,th) =
case name of
"" => cnf_axiom_aux th (*no name, so can't cache*)
| s => case Symtab.lookup (!clause_cache) s of
NONE =>
let val cls = cnf_axiom_aux th
in change clause_cache (Symtab.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.update (s, (th, cls))); cls end;
val cnf_axiom = cnf_axiom_g cnf_axiom_aux;
val cnf_axiomH = cnf_axiom_g cnf_axiom_auxH;
fun pairname th = (Thm.name_of_thm th, th);
fun meta_cnf_axiom th =
map Meson.make_meta_clause (cnf_axiom (pairname th));
fun meta_cnf_axiomH th =
map Meson.make_meta_clause (cnf_axiomH (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 [];
(**** Generating Skoklem Functions ****)
val skolem_prefix = "sk_";
val skolem_id = ref 0;
fun gen_skolem_name () =
let val sk_fun = skolem_prefix ^ string_of_int (! skolem_id);
in inc skolem_id; sk_fun end;
fun gen_skolem univ_vars tp =
let
val skolem_type = map Term.fastype_of univ_vars ---> tp
val skolem_term = Const (gen_skolem_name (), skolem_type)
in Term.list_comb (skolem_term, univ_vars) end;
fun get_new_skolem epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,tp,_))) =
let val all_vars = univ_vars_of t
val sk_term = gen_skolem all_vars tp
in
(sk_term,(t,sk_term)::epss)
end;
(*FIXME: use a-list lookup!!*)
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);
(* replace the epsilon terms in a formula by skolem terms. *)
fun rm_Eps _ [] = []
| rm_Eps epss (th::ths) =
let val (th',epss') = rm_Eps_cls epss th
in th' :: (rm_Eps epss' ths) end;
(**** 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 = 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;
val cnf_rules = cnf_rules_g cnf_axiom;
val cnf_rulesH = cnf_rules_g cnf_axiomH;
(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****)
fun addclause (c,th) l =
if ResClause.isTaut c then l else (c,th) :: l;
fun addclauseH (c,th) l =
if ResHolClause.isTaut c then l else (c,th)::l;
(* outputs a list of (clause,theorem) pairs *)
fun clausify_axiom_pairs (thm_name,th) =
let val isa_clauses = cnf_axiom (thm_name,th)
val isa_clauses' = rm_Eps [] isa_clauses
val clauses_n = length isa_clauses
fun make_axiom_clauses _ [] []= []
| make_axiom_clauses i (cls::clss) (cls'::clss') =
addclause (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_axiom_pairsH (thm_name,th) =
let val isa_clauses = cnf_axiomH (thm_name,th)
val isa_clauses' = rm_Eps [] isa_clauses
val clauses_n = length isa_clauses
fun make_axiom_clauses _ [] []= []
| make_axiom_clauses i (cls::clss) (cls'::clss') =
addclauseH (ResHolClause.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_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 [];
(*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;