(* 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.term
val transform_elim : thm -> thm
val clausify_axiom : thm -> ResClause.clause list
val cnf_axiom : (string * thm) -> thm list
val meta_cnf_axiom : thm -> thm list
val cnf_rule : thm -> thm list
val cnf_classical_rules_thy : theory -> thm list list * thm list
val clausify_classical_rules_thy : theory -> ResClause.clause list list * thm list
val cnf_simpset_rules_thy : theory -> thm list list * thm list
val clausify_simpset_rules_thy : theory -> ResClause.clause list list * thm list
val rm_Eps
: (Term.term * Term.term) list -> thm list -> Term.term list
val claset_rules_of_thy : theory -> (string * thm) list
val simpset_rules_of_thy : theory -> (string * thm) list
val clausify_rules : thm list -> thm list -> ResClause.clause list list * thm 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 thm =
((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac thm 1) THEN
REPEAT(Fast_tac 1);
(* This following version fails sometimes, need to investigate, do not use it now. *)
fun elimRule_tac' thm =
((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac thm 1) THEN
REPEAT(SOLVE((etac exI 1) ORELSE (rtac conjI 1) ORELSE (rtac disjI1 1) ORELSE (rtac disjI2 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.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 thm =
case (concl_of thm) 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 thm =
if is_elimR thm then
let val tm = elimR2Fol thm
val ctm = cterm_of (sign_of_thm thm) tm
in
prove_goalw_cterm [] ctm (fn prems => [elimRule_tac thm])
end
else thm;
(**** 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. *)
fun skolem_axiom thm =
if Term.is_first_order (prop_of thm) then
let val thm' = (skolemize o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) thm
in
repeat_RS thm' someI_ex
end
else raise THM ("skolem_axiom: not first-order", 0, [thm]);
fun cnf_rule thm = make_clauses [skolem_axiom (transform_elim thm)];
(*Transfer a theorem in to theory Reconstruction.thy if it is not already
inside that theory -- because it's needed for Skolemization *)
val recon_thy = ThyInfo.get_theory"Reconstruction";
fun transfer_to_Reconstruction thm =
transfer recon_thy thm handle THM _ => thm;
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 thm =
map (zero_var_indexes o Thm.varifyT)
(rm_redundant_cls (cnf_rule (transfer_to_Reconstruction thm)));
(*Cache for clauses: could be a hash table if we provided them.*)
val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table)
fun cnf_axiom (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 clause_cache := Symtab.update ((s, (th,cls)), !clause_cache); 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 clause_cache := Symtab.update ((s, (th,cls)), !clause_cache); 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 thm = rm_Eps_cls_aux epss (prop_of thm);
(* remove the epsilon terms in a formula, by skolem terms. *)
fun rm_Eps _ [] = []
| rm_Eps epss (thm::thms) =
let val (thm',epss') = rm_Eps_cls epss thm
in
thm' :: (rm_Eps epss' thms)
end;
(* convert a theorem into CNF and then into Clause.clause format. *)
fun clausify_axiom thm =
let val name = Thm.name_of_thm thm
val isa_clauses = cnf_axiom (name, thm)
(*"isa_clauses" are already in "standard" form. *)
val isa_clauses' = rm_Eps [] isa_clauses
val clauses_n = length isa_clauses
fun make_axiom_clauses _ [] = []
| make_axiom_clauses i (cls::clss) =
(ResClause.make_axiom_clause cls (name,i)) :: make_axiom_clauses (i+1) clss
in
make_axiom_clauses 0 isa_clauses'
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.dest 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,thm) :: thms) err_list =
let val (ts,es) = cnf_rules thms err_list
in (cnf_axiom (name,thm) :: ts,es) handle _ => (ts, (thm::es)) end;
(* CNF all rules from a given theory's classical reasoner *)
fun cnf_classical_rules_thy thy =
cnf_rules (claset_rules_of_thy thy) [];
(* CNF all simplifier rules from a given theory's simpset *)
fun cnf_simpset_rules_thy thy =
cnf_rules (simpset_rules_of_thy thy) [];
(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****)
(* classical rules *)
fun clausify_rules [] err_list = ([],err_list)
| clausify_rules (thm::thms) err_list =
let val (ts,es) = clausify_rules thms err_list
in
((clausify_axiom thm)::ts,es) handle _ => (ts,(thm::es))
end;
(* convert all classical rules from a given theory into Clause.clause format. *)
fun clausify_classical_rules_thy thy =
clausify_rules (map #2 (claset_rules_of_thy thy)) [];
(* convert all simplifier rules from a given theory into Clause.clause format. *)
fun clausify_simpset_rules_thy thy =
clausify_rules (map #2 (simpset_rules_of_thy thy)) [];
end;