(* Title: HOL/Tools/set_comprehension_pointfree.ML
Author: Felix Kuperjans, Lukas Bulwahn, TU Muenchen
Author: Rafal Kolanski, NICTA
Simproc for rewriting set comprehensions to pointfree expressions.
*)
signature SET_COMPREHENSION_POINTFREE =
sig
val base_simproc : simpset -> cterm -> thm option
val code_simproc : simpset -> cterm -> thm option
val simproc : simpset -> cterm -> thm option
end
structure Set_Comprehension_Pointfree : SET_COMPREHENSION_POINTFREE =
struct
(* syntactic operations *)
fun mk_inf (t1, t2) =
let
val T = fastype_of t1
in
Const (@{const_name Lattices.inf_class.inf}, T --> T --> T) $ t1 $ t2
end
fun mk_sup (t1, t2) =
let
val T = fastype_of t1
in
Const (@{const_name Lattices.sup_class.sup}, T --> T --> T) $ t1 $ t2
end
fun mk_Compl t =
let
val T = fastype_of t
in
Const (@{const_name "Groups.uminus_class.uminus"}, T --> T) $ t
end
fun mk_image t1 t2 =
let
val T as Type (@{type_name fun}, [_ , R]) = fastype_of t1
in
Const (@{const_name image},
T --> fastype_of t2 --> HOLogic.mk_setT R) $ t1 $ t2
end;
fun mk_sigma (t1, t2) =
let
val T1 = fastype_of t1
val T2 = fastype_of t2
val setT = HOLogic.dest_setT T1
val resT = HOLogic.mk_setT (HOLogic.mk_prodT (setT, HOLogic.dest_setT T2))
in
Const (@{const_name Sigma},
T1 --> (setT --> T2) --> resT) $ t1 $ absdummy setT t2
end;
fun mk_vimage f s =
let
val T as Type (@{type_name fun}, [T1, T2]) = fastype_of f
in
Const (@{const_name vimage}, T --> HOLogic.mk_setT T2 --> HOLogic.mk_setT T1) $ f $ s
end;
fun dest_Collect (Const (@{const_name Collect}, _) $ Abs (x, T, t)) = ((x, T), t)
| dest_Collect t = raise TERM ("dest_Collect", [t])
(* Copied from predicate_compile_aux.ML *)
fun strip_ex (Const (@{const_name Ex}, _) $ Abs (x, T, t)) =
let
val (xTs, t') = strip_ex t
in
((x, T) :: xTs, t')
end
| strip_ex t = ([], t)
fun mk_prod1 Ts (t1, t2) =
let
val (T1, T2) = pairself (curry fastype_of1 Ts) (t1, t2)
in
HOLogic.pair_const T1 T2 $ t1 $ t2
end;
fun mk_split_abs vs (Bound i) t = let val (x, T) = nth vs i in Abs (x, T, t) end
| mk_split_abs vs (Const ("Product_Type.Pair", _) $ u $ v) t =
HOLogic.mk_split (mk_split_abs vs u (mk_split_abs vs v t))
| mk_split_abs _ t _ = raise TERM ("mk_split_abs: bad term", [t]);
(* a variant of HOLogic.strip_psplits *)
val strip_psplits =
let
fun strip [] qs vs t = (t, rev vs, qs)
| strip (p :: ps) qs vs (Const ("Product_Type.prod.prod_case", _) $ t) =
strip ((1 :: p) :: (2 :: p) :: ps) (p :: qs) vs t
| strip (_ :: ps) qs vs (Abs (s, T, t)) = strip ps qs ((s, T) :: vs) t
| strip (_ :: ps) qs vs t = strip ps qs
((Name.uu_, hd (binder_types (fastype_of1 (map snd vs, t)))) :: vs)
(incr_boundvars 1 t $ Bound 0)
in strip [[]] [] [] end;
(* patterns *)
datatype pattern = Pattern of int list
fun dest_Pattern (Pattern bs) = bs
fun dest_bound (Bound i) = i
| dest_bound t = raise TERM("dest_bound", [t]);
fun type_of_pattern Ts (Pattern bs) = HOLogic.mk_tupleT (map (nth Ts) bs)
fun term_of_pattern Ts (Pattern bs) =
let
fun mk [b] = Bound b
| mk (b :: bs) = HOLogic.pair_const (nth Ts b) (type_of_pattern Ts (Pattern bs))
$ Bound b $ mk bs
in mk bs end;
(* formulas *)
datatype formula = Atom of (pattern * term) | Int of formula * formula | Un of formula * formula
fun map_atom f (Atom a) = Atom (f a)
| map_atom _ x = x
fun is_collect_atom (Atom (_, Const(@{const_name Collect}, _) $ _)) = true
| is_collect_atom (Atom (_, Const (@{const_name "Groups.uminus_class.uminus"}, _) $ (Const(@{const_name Collect}, _) $ _))) = true
| is_collect_atom _ = false
fun mk_split _ [(x, T)] t = (T, Abs (x, T, t))
| mk_split rT ((x, T) :: vs) t =
let
val (T', t') = mk_split rT vs t
val t'' = HOLogic.split_const (T, T', rT) $ (Abs (x, T, t'))
in (domain_type (fastype_of t''), t'') end
fun mk_term vs t =
let
val bs = loose_bnos t
val vs' = map (nth (rev vs)) bs
val subst = map_index (fn (i, j) => (j, Bound i)) (rev bs)
|> sort (fn (p1, p2) => int_ord (fst p1, fst p2))
|> (fn subst' => map (fn i => the_default (Bound i) (AList.lookup (op =) subst' i)) (0 upto (fst (snd (split_last subst')))))
val t' = subst_bounds (subst, t)
val tuple = Pattern bs
in (tuple, (vs', t')) end
fun default_atom vs t =
let
val (tuple, (vs', t')) = mk_term vs t
val T = HOLogic.mk_tupleT (map snd vs')
val s = HOLogic.Collect_const T $ (snd (mk_split @{typ bool} vs' t'))
in
(tuple, Atom (tuple, s))
end
fun mk_atom vs (t as Const (@{const_name "Set.member"}, _) $ x $ s) =
if not (null (loose_bnos s)) then
default_atom vs t
else
(case try ((map dest_bound) o HOLogic.strip_tuple) x of
SOME pat => (Pattern pat, Atom (Pattern pat, s))
| NONE =>
let
val (tuple, (vs', x')) = mk_term vs x
val rT = HOLogic.dest_setT (fastype_of s)
val s = mk_vimage (snd (mk_split rT vs' x')) s
in (tuple, Atom (tuple, s)) end)
| mk_atom vs (Const (@{const_name "HOL.Not"}, _) $ t) = apsnd (map_atom (apsnd mk_Compl)) (mk_atom vs t)
| mk_atom vs t = default_atom vs t
fun merge' [] (pats1, pats2) = ([], (pats1, pats2))
| merge' pat (pats, []) = (pat, (pats, []))
| merge' pat (pats1, pats) =
let
fun disjoint_to_pat p = null (inter (op =) pat p)
val overlap_pats = filter_out disjoint_to_pat pats
val rem_pats = filter disjoint_to_pat pats
val (pat, (pats', pats1')) = merge' (distinct (op =) (flat overlap_pats @ pat)) (rem_pats, pats1)
in
(pat, (pats1', pats'))
end
fun merge ([], pats) = pats
| merge (pat :: pats', pats) =
let val (pat', (pats1', pats2')) = merge' pat (pats', pats)
in pat' :: merge (pats1', pats2') end;
fun restricted_merge ([], pats) = pats
| restricted_merge (pat :: pats', pats) =
let
fun disjoint_to_pat p = null (inter (op =) pat p)
val overlap_pats = filter_out disjoint_to_pat pats
val rem_pats = filter disjoint_to_pat pats
in
case overlap_pats of
[] => pat :: restricted_merge (pats', rem_pats)
| [pat'] => if subset (op =) (pat, pat') then
pat' :: restricted_merge (pats', rem_pats)
else if subset (op =) (pat', pat) then
pat :: restricted_merge (pats', rem_pats)
else error "restricted merge: two patterns require relational join"
| _ => error "restricted merge: multiple patterns overlap"
end;
fun map_atoms f (Atom a) = Atom (f a)
| map_atoms f (Un (fm1, fm2)) = Un (pairself (map_atoms f) (fm1, fm2))
| map_atoms f (Int (fm1, fm2)) = Int (pairself (map_atoms f) (fm1, fm2))
fun extend Ts bs t = foldr1 mk_sigma (t :: map (fn b => HOLogic.mk_UNIV (nth Ts b)) bs)
fun rearrange vs (pat, pat') t =
let
val subst = map_index (fn (i, b) => (b, i)) (rev pat)
val vs' = map (nth (rev vs)) pat
val Ts' = map snd (rev vs')
val bs = map (fn b => the (AList.lookup (op =) subst b)) pat'
val rt = term_of_pattern Ts' (Pattern bs)
val rT = type_of_pattern Ts' (Pattern bs)
val (_, f) = mk_split rT vs' rt
in
mk_image f t
end;
fun adjust vs pats (Pattern pat, t) =
let
val SOME p = find_first (fn p => not (null (inter (op =) pat p))) pats
val missing = subtract (op =) pat p
val Ts = rev (map snd vs)
val t' = extend Ts missing t
in (Pattern p, rearrange vs (pat @ missing, p) t') end
fun adjust_atoms vs pats fm = map_atoms (adjust vs pats) fm
fun merge_inter vs (pats1, fm1) (pats2, fm2) =
let
val pats = restricted_merge (map dest_Pattern pats1, map dest_Pattern pats2)
val (fm1', fm2') = pairself (adjust_atoms vs pats) (fm1, fm2)
in
(map Pattern pats, Int (fm1', fm2'))
end;
fun merge_union vs (pats1, fm1) (pats2, fm2) =
let
val pats = merge (map dest_Pattern pats1, map dest_Pattern pats2)
val (fm1', fm2') = pairself (adjust_atoms vs pats) (fm1, fm2)
in
(map Pattern pats, Un (fm1', fm2'))
end;
fun mk_formula vs (@{const HOL.conj} $ t1 $ t2) = merge_inter vs (mk_formula vs t1) (mk_formula vs t2)
| mk_formula vs (@{const HOL.disj} $ t1 $ t2) = merge_union vs (mk_formula vs t1) (mk_formula vs t2)
| mk_formula vs t = apfst single (mk_atom vs t)
fun strip_Int (Int (fm1, fm2)) = fm1 :: (strip_Int fm2)
| strip_Int fm = [fm]
(* term construction *)
fun reorder_bounds pats t =
let
val bounds = maps dest_Pattern pats
val bperm = bounds ~~ ((length bounds - 1) downto 0)
|> sort (fn (i,j) => int_ord (fst i, fst j)) |> map snd
in
subst_bounds (map Bound bperm, t)
end;
fun is_reordering t =
let val (t', _, _) = HOLogic.strip_psplits t
in forall (fn Bound _ => true) (HOLogic.strip_tuple t') end
fun mk_pointfree_expr t =
let
val ((x, T), (vs, t'')) = apsnd strip_ex (dest_Collect t)
val Ts = map snd (rev vs)
fun mk_mem_UNIV n = HOLogic.mk_mem (Bound n, HOLogic.mk_UNIV (nth Ts n))
fun lookup (pat', t) pat = if pat = pat' then t else HOLogic.mk_UNIV (type_of_pattern Ts pat)
val conjs = HOLogic.dest_conj t''
val refl = HOLogic.eq_const T $ Bound (length vs) $ Bound (length vs)
val is_the_eq =
the_default false o (try (fn eq => fst (HOLogic.dest_eq eq) = Bound (length vs)))
val eq = the_default refl (find_first is_the_eq conjs)
val f = snd (HOLogic.dest_eq eq)
val conjs' = filter_out (fn t => eq = t) conjs
val unused_bounds = subtract (op =) (distinct (op =) (maps loose_bnos conjs'))
(0 upto (length vs - 1))
val (pats, fm) =
mk_formula ((x, T) :: vs) (foldr1 HOLogic.mk_conj (conjs' @ map mk_mem_UNIV unused_bounds))
fun mk_set (Atom pt) = foldr1 mk_sigma (map (lookup pt) pats)
| mk_set (Un (f1, f2)) = mk_sup (mk_set f1, mk_set f2)
| mk_set (Int (f1, f2)) = mk_inf (mk_set f1, mk_set f2)
val pat = foldr1 (mk_prod1 Ts) (map (term_of_pattern Ts) pats)
val t = mk_split_abs (rev ((x, T) :: vs)) pat (reorder_bounds pats f)
in
if the_default false (try is_reordering t) andalso is_collect_atom fm then
error "mk_pointfree_expr: trivial case"
else (fm, mk_image t (mk_set fm))
end;
val rewrite_term = try mk_pointfree_expr
(* proof tactic *)
val prod_case_distrib = @{lemma "(prod_case g x) z = prod_case (% x y. (g x y) z) x" by (simp add: prod_case_beta)}
(* FIXME: another clone *)
fun eq_conv cv1 cv2 ct =
(case Thm.term_of ct of
Const (@{const_name HOL.eq}, _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv1) cv2 ct
| _ => raise CTERM ("eq_conv", [ct]))
val vimageI2' = @{lemma "f a \<notin> A ==> a \<notin> f -` A" by simp}
val vimageE' =
@{lemma "a \<notin> f -` B ==> (\<And> x. f a = x ==> x \<notin> B ==> P) ==> P" by simp}
val collectI' = @{lemma "\<not> P a ==> a \<notin> {x. P x}" by auto}
val collectE' = @{lemma "a \<notin> {x. P x} ==> (\<not> P a ==> Q) ==> Q" by auto}
fun elim_Collect_tac ss = dtac @{thm iffD1[OF mem_Collect_eq]}
THEN' (REPEAT_DETERM o (eresolve_tac @{thms exE}))
THEN' REPEAT_DETERM o etac @{thm conjE}
THEN' TRY o hyp_subst_tac' ss;
fun intro_image_tac ctxt = rtac @{thm image_eqI}
THEN' (REPEAT_DETERM1 o
(rtac @{thm refl}
ORELSE' rtac
@{thm arg_cong2[OF refl, where f="op =", OF prod.cases, THEN iffD2]}
ORELSE' CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1
(HOLogic.Trueprop_conv (eq_conv Conv.all_conv (Conv.rewr_conv (mk_meta_eq prod_case_distrib)))))) ctxt)))
fun elim_image_tac ss = etac @{thm imageE}
THEN' REPEAT_DETERM o CHANGED o
(TRY o Simplifier.full_simp_tac (Simplifier.inherit_context ss (HOL_basic_ss addsimps @{thms split_paired_all prod.cases}))
THEN' REPEAT_DETERM o etac @{thm Pair_inject}
THEN' TRY o hyp_subst_tac' ss)
fun tac1_of_formula ss (Int (fm1, fm2)) =
TRY o etac @{thm conjE}
THEN' rtac @{thm IntI}
THEN' (fn i => tac1_of_formula ss fm2 (i + 1))
THEN' tac1_of_formula ss fm1
| tac1_of_formula ss (Un (fm1, fm2)) =
etac @{thm disjE} THEN' rtac @{thm UnI1}
THEN' tac1_of_formula ss fm1
THEN' rtac @{thm UnI2}
THEN' tac1_of_formula ss fm2
| tac1_of_formula ss (Atom _) =
REPEAT_DETERM1 o (atac
ORELSE' rtac @{thm SigmaI}
ORELSE' ((rtac @{thm CollectI} ORELSE' rtac collectI') THEN'
TRY o Simplifier.simp_tac (Simplifier.inherit_context ss (HOL_basic_ss addsimps [@{thm prod.cases}])))
ORELSE' ((rtac @{thm vimageI2} ORELSE' rtac vimageI2') THEN'
TRY o Simplifier.simp_tac (Simplifier.inherit_context ss (HOL_basic_ss addsimps [@{thm prod.cases}])))
ORELSE' (rtac @{thm image_eqI} THEN'
(REPEAT_DETERM o
(rtac @{thm refl}
ORELSE' rtac @{thm arg_cong2[OF refl, where f="op =", OF prod.cases, THEN iffD2]})))
ORELSE' rtac @{thm UNIV_I}
ORELSE' rtac @{thm iffD2[OF Compl_iff]}
ORELSE' atac)
fun tac2_of_formula ss (Int (fm1, fm2)) =
TRY o etac @{thm IntE}
THEN' TRY o rtac @{thm conjI}
THEN' (fn i => tac2_of_formula ss fm2 (i + 1))
THEN' tac2_of_formula ss fm1
| tac2_of_formula ss (Un (fm1, fm2)) =
etac @{thm UnE} THEN' rtac @{thm disjI1}
THEN' tac2_of_formula ss fm1
THEN' rtac @{thm disjI2}
THEN' tac2_of_formula ss fm2
| tac2_of_formula ss (Atom _) =
REPEAT_DETERM o
(atac
ORELSE' dtac @{thm iffD1[OF mem_Sigma_iff]}
ORELSE' etac @{thm conjE}
ORELSE' ((etac @{thm CollectE} ORELSE' etac collectE') THEN'
TRY o Simplifier.full_simp_tac (Simplifier.inherit_context ss (HOL_basic_ss addsimps [@{thm prod.cases}])) THEN'
REPEAT_DETERM o etac @{thm Pair_inject} THEN' TRY o hyp_subst_tac' ss THEN' TRY o rtac @{thm refl})
ORELSE' (etac @{thm imageE}
THEN' (REPEAT_DETERM o CHANGED o
(TRY o Simplifier.full_simp_tac (Simplifier.inherit_context ss (HOL_basic_ss addsimps @{thms split_paired_all prod.cases}))
THEN' REPEAT_DETERM o etac @{thm Pair_inject}
THEN' TRY o hyp_subst_tac' ss THEN' TRY o rtac @{thm refl})))
ORELSE' etac @{thm ComplE}
ORELSE' ((etac @{thm vimageE} ORELSE' etac vimageE')
THEN' TRY o Simplifier.full_simp_tac (Simplifier.inherit_context ss (HOL_basic_ss addsimps [@{thm prod.cases}]))
THEN' TRY o hyp_subst_tac' ss THEN' TRY o rtac @{thm refl}))
fun tac ss ctxt fm =
let
val subset_tac1 = rtac @{thm subsetI}
THEN' elim_Collect_tac ss
THEN' intro_image_tac ctxt
THEN' tac1_of_formula ss fm
val subset_tac2 = rtac @{thm subsetI}
THEN' elim_image_tac ss
THEN' rtac @{thm iffD2[OF mem_Collect_eq]}
THEN' REPEAT_DETERM o resolve_tac @{thms exI}
THEN' (TRY o REPEAT_ALL_NEW (rtac @{thm conjI}))
THEN' (K (TRY (FIRSTGOAL ((TRY o hyp_subst_tac' ss) THEN' rtac @{thm refl}))))
THEN' (fn i => EVERY (rev (map_index (fn (j, f) =>
REPEAT_DETERM (etac @{thm IntE} (i + j)) THEN tac2_of_formula ss f (i + j)) (strip_Int fm))))
in
rtac @{thm subset_antisym} THEN' subset_tac1 THEN' subset_tac2
end;
(* preprocessing conversion:
rewrites {(x1, ..., xn). P x1 ... xn} to {(x1, ..., xn) | x1 ... xn. P x1 ... xn} *)
fun comprehension_conv ss ct =
let
val ctxt = Simplifier.the_context ss
fun dest_Collect (Const (@{const_name Collect}, T) $ t) = (HOLogic.dest_setT (body_type T), t)
| dest_Collect t = raise TERM ("dest_Collect", [t])
fun list_ex vs t = fold_rev (fn (x, T) => fn t => HOLogic.exists_const T $ Abs (x, T, t)) vs t
fun mk_term t =
let
val (T, t') = dest_Collect t
val (t'', vs, fp) = case strip_psplits t' of
(_, [_], _) => raise TERM("mk_term", [t'])
| (t'', vs, fp) => (t'', vs, fp)
val Ts = map snd vs
val eq = HOLogic.eq_const T $ Bound (length Ts) $
(HOLogic.mk_ptuple fp (HOLogic.mk_ptupleT fp Ts) (rev (map_index (fn (i, _) => Bound i) Ts)))
in
HOLogic.Collect_const T $ absdummy T (list_ex vs (HOLogic.mk_conj (eq, t'')))
end;
fun is_eq th = is_some (try (HOLogic.dest_eq o HOLogic.dest_Trueprop) (prop_of th))
val unfold_thms = @{thms split_paired_all mem_Collect_eq prod.cases}
fun tac ctxt =
rtac @{thm set_eqI}
THEN' Simplifier.simp_tac
(Simplifier.inherit_context ss (HOL_basic_ss addsimps unfold_thms))
THEN' rtac @{thm iffI}
THEN' REPEAT_DETERM o rtac @{thm exI}
THEN' rtac @{thm conjI} THEN' rtac @{thm refl} THEN' atac
THEN' REPEAT_DETERM o etac @{thm exE}
THEN' etac @{thm conjE}
THEN' REPEAT_DETERM o etac @{thm Pair_inject}
THEN' Subgoal.FOCUS (fn {prems, ...} =>
Simplifier.simp_tac
(Simplifier.inherit_context ss (HOL_basic_ss addsimps (filter is_eq prems))) 1) ctxt
THEN' TRY o atac
in
case try mk_term (term_of ct) of
NONE => Thm.reflexive ct
| SOME t' =>
Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (term_of ct, t')))
(fn {context, ...} => tac context 1)
RS @{thm eq_reflection}
end
(* main simprocs *)
val prep_thms =
map mk_meta_eq ([@{thm Bex_def}, @{thm Pow_iff[symmetric]}] @ @{thms ex_simps[symmetric]})
val post_thms =
map mk_meta_eq [@{thm Times_Un_distrib1[symmetric]},
@{lemma "A \<times> B \<union> A \<times> C = A \<times> (B \<union> C)" by auto},
@{lemma "(A \<times> B \<inter> C \<times> D) = (A \<inter> C) \<times> (B \<inter> D)" by auto}]
fun conv ss t =
let
val ctxt = Simplifier.the_context ss
val ([t'], ctxt') = Variable.import_terms true [t] (Variable.declare_term t ctxt)
val ss' = Simplifier.context ctxt' ss
val ct = cterm_of (Proof_Context.theory_of ctxt') t'
fun unfold_conv thms =
Raw_Simplifier.rewrite_cterm (false, false, false) (K (K NONE))
(Raw_Simplifier.inherit_context ss' empty_ss addsimps thms)
val prep_eq = (comprehension_conv ss' then_conv unfold_conv prep_thms) ct
val t'' = term_of (Thm.rhs_of prep_eq)
fun mk_thm (fm, t''') = Goal.prove ctxt' [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq (t'', t'''))) (fn {context, ...} => tac ss' context fm 1)
fun unfold th = th RS ((prep_eq RS meta_eq_to_obj_eq) RS @{thm trans})
val post = Conv.fconv_rule (HOLogic.Trueprop_conv (eq_conv Conv.all_conv (unfold_conv post_thms)))
val export = singleton (Variable.export ctxt' ctxt)
in
Option.map (export o post o unfold o mk_thm) (rewrite_term t'')
end;
fun base_simproc ss redex =
let
val set_compr = term_of redex
in
conv ss set_compr
|> Option.map (fn thm => thm RS @{thm eq_reflection})
end;
fun instantiate_arg_cong ctxt pred =
let
val certify = cterm_of (Proof_Context.theory_of ctxt)
val arg_cong = Thm.incr_indexes (maxidx_of_term pred + 1) @{thm arg_cong}
val f $ _ = fst (HOLogic.dest_eq (HOLogic.dest_Trueprop (concl_of arg_cong)))
in
cterm_instantiate [(certify f, certify pred)] arg_cong
end;
fun simproc ss redex =
let
val ctxt = Simplifier.the_context ss
val pred $ set_compr = term_of redex
val arg_cong' = instantiate_arg_cong ctxt pred
in
conv ss set_compr
|> Option.map (fn thm => thm RS arg_cong' RS @{thm eq_reflection})
end;
fun code_simproc ss redex =
let
fun unfold_conv thms =
Raw_Simplifier.rewrite_cterm (false, false, false) (K (K NONE))
(Raw_Simplifier.inherit_context ss empty_ss addsimps thms)
val prep_thm = unfold_conv @{thms eq_equal[symmetric]} redex
in
case base_simproc ss (Thm.rhs_of prep_thm) of
SOME rewr_thm => SOME (transitive_thm OF [transitive_thm OF [prep_thm, rewr_thm],
unfold_conv @{thms eq_equal} (Thm.rhs_of rewr_thm)])
| NONE => NONE
end;
end;