(* 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_ELIM_RULE =
sig
exception ELIMR2FOL of string
val elimRule_tac : Thm.thm -> Tactical.tactic
val elimR2Fol : Thm.thm -> Term.term
val transform_elim : Thm.thm -> Thm.thm
end;
structure ResElimRule: RES_ELIM_RULE =
struct
(* 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_imp (prem,concl) = Const("op -->", Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ prem $ concl;
fun make_disjs [x] = x
| make_disjs (x :: xs) = Const("op |",Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ x $ (make_disjs xs)
fun make_conjs [x] = x
| make_conjs (x :: xs) = Const("op &", Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ x $ (make_conjs xs)
fun add_EX term [] = term
| add_EX term ((x,xtp)::xs) = add_EX (Const ("Ex",Type("fun",[Type("fun",[xtp,Type("bool",[])]),Type("bool",[])])) $ Abs (x,xtp,term)) xs;
exception TRUEPROP of string;
fun strip_trueprop (Const ("Trueprop", Type("fun",[Type("bool",[]),Type("prop",[])])) $ P) = P
| strip_trueprop _ = raise TRUEPROP("not a prop!");
fun neg P = Const ("Not", Type("fun",[Type("bool",[]),Type("bool",[])])) $ P;
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' = strip_trueprop P
val prems' = P'::prems
in
strip_concl' prems' bvs Q
end
| strip_concl' prems bvs P =
let val P' = neg (strip_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' = strip_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
make_imp(strip_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 neg (strip_trueprop main)
else trans_elim (main, tl prems, concl)
end;
fun trueprop term = Const ("Trueprop", Type("fun",[Type("bool",[]),Type("prop",[])])) $ term;
(* 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",[]))
=> trueprop (elimR2Fol_aux prems concl)
| Free(x,Type("prop",[])) => trueprop(elimR2Fol_aux prems concl)
| _ => raise ELIMR2FOL("Not an elimination rule!")
end;
(**** use prove_goalw_cterm to prove ****)
(* convert an elim-rule into an equivalent theorem that does not have the predicate variable. *)
fun transform_elim thm =
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;
end;
signature RES_AXIOMS =
sig
val clausify_axiom : Thm.thm -> ResClause.clause list
val cnf_axiom : Thm.thm -> Thm.thm list
val meta_cnf_axiom : Thm.thm -> Thm.thm list
val cnf_elim : Thm.thm -> Thm.thm list
val cnf_rule : Thm.thm -> Thm.thm list
val cnf_classical_rules_thy : Theory.theory -> Thm.thm list list * Thm.thm list
val clausify_classical_rules_thy
: Theory.theory -> ResClause.clause list list * Thm.thm list
val cnf_simpset_rules_thy
: Theory.theory -> Thm.thm list list * Thm.thm list
val clausify_simpset_rules_thy
: Theory.theory -> ResClause.clause list list * Thm.thm list
val rm_Eps
: (Term.term * Term.term) list -> Thm.thm list -> Term.term list
val claset_rules_of_thy : Theory.theory -> Thm.thm list
val simpset_rules_of_thy : Theory.theory -> (string * Thm.thm) list
val clausify_rules : Thm.thm list -> Thm.thm list -> ResClause.clause list list * Thm.thm list
end;
structure ResAxioms : RES_AXIOMS =
struct
open ResElimRule;
(* to be fixed: cnf_intro, cnf_rule, is_introR *)
(* 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;
(* 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 thm]
fun cnf_elim thm = cnf_rule (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;
(* remove "True" clause *)
fun rm_redundant_cls [] = []
| rm_redundant_cls (thm::thms) =
let val t = prop_of thm
in
case t of (Const ("Trueprop", _) $ Const ("True", _)) => rm_redundant_cls thms
| _ => thm::(rm_redundant_cls thms)
end;
(* transform an Isabelle thm into CNF *)
fun cnf_axiom thm =
let val thm' = transfer_to_Reconstruction thm
val thm'' = if (is_elimR thm') then (cnf_elim thm') else cnf_rule thm'
in
map Thm.varifyT (rm_redundant_cls thm'')
end;
fun meta_cnf_axiom thm =
map (zero_var_indexes o Meson.make_meta_clause) (cnf_axiom thm);
(* 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 =
let val vars' = univ_vars_of_aux P vars
in
univ_vars_of_aux Q vars'
end
| 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 =
let val sk_fun = sk_lookup epss t
in
case sk_fun of NONE => get_new_skolem epss t
| SOME sk => (sk,epss)
end;
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 =
let val tm = prop_of thm
in
rm_Eps_cls_aux epss tm
end;
(* 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;
(* changed, now it also finds out the name of the theorem. *)
(* convert a theorem into CNF and then into Clause.clause format. *)
fun clausify_axiom thm =
let val isa_clauses = cnf_axiom thm (*"isa_clauses" are already "standard"ed. *)
val isa_clauses' = rm_Eps [] isa_clauses
val thm_name = Thm.name_of_thm thm
val clauses_n = length isa_clauses
fun make_axiom_clauses _ [] = []
| make_axiom_clauses i (cls::clss) = (ResClause.make_axiom_clause cls (thm_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
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 (thm::thms) err_list =
let val (ts,es) = cnf_rules thms err_list
in (cnf_axiom 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 (map #2 (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 (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;