(* Title: Pure/sorts.ML
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
The order-sorted algebra of type classes.
Classes denote (possibly empty) collections of types that are
partially ordered by class inclusion. They are represented
symbolically by strings.
Sorts are intersections of finitely many classes. They are represented
by lists of classes. Normal forms of sorts are sorted lists of
minimal classes (wrt. current class inclusion).
*)
signature SORTS =
sig
val make: sort list -> sort Ord_List.T
val subset: sort Ord_List.T * sort Ord_List.T -> bool
val union: sort Ord_List.T -> sort Ord_List.T -> sort Ord_List.T
val subtract: sort Ord_List.T -> sort Ord_List.T -> sort Ord_List.T
val remove_sort: sort -> sort Ord_List.T -> sort Ord_List.T
val insert_sort: sort -> sort Ord_List.T -> sort Ord_List.T
val insert_typ: typ -> sort Ord_List.T -> sort Ord_List.T
val insert_typs: typ list -> sort Ord_List.T -> sort Ord_List.T
val insert_term: term -> sort Ord_List.T -> sort Ord_List.T
val insert_terms: term list -> sort Ord_List.T -> sort Ord_List.T
type algebra
val classes_of: algebra -> serial Graph.T
val arities_of: algebra -> (class * sort list) list Symtab.table
val all_classes: algebra -> class list
val super_classes: algebra -> class -> class list
val class_less: algebra -> class * class -> bool
val class_le: algebra -> class * class -> bool
val sort_eq: algebra -> sort * sort -> bool
val sort_le: algebra -> sort * sort -> bool
val sorts_le: algebra -> sort list * sort list -> bool
val inter_sort: algebra -> sort * sort -> sort
val minimize_sort: algebra -> sort -> sort
val complete_sort: algebra -> sort -> sort
val minimal_sorts: algebra -> sort list -> sort Ord_List.T
val certify_class: algebra -> class -> class (*exception TYPE*)
val certify_sort: algebra -> sort -> sort (*exception TYPE*)
val add_class: Context.pretty -> class * class list -> algebra -> algebra
val add_classrel: Context.pretty -> class * class -> algebra -> algebra
val add_arities: Context.pretty -> string * (class * sort list) list -> algebra -> algebra
val empty_algebra: algebra
val merge_algebra: Context.pretty -> algebra * algebra -> algebra
val subalgebra: Context.pretty -> (class -> bool) -> (class * string -> sort list option)
-> algebra -> (sort -> sort) * algebra
type class_error
val class_error: Proof.context -> class_error -> string
exception CLASS_ERROR of class_error
val mg_domain: algebra -> string -> sort -> sort list (*exception CLASS_ERROR*)
val meet_sort: algebra -> typ * sort
-> sort Vartab.table -> sort Vartab.table (*exception CLASS_ERROR*)
val meet_sort_typ: algebra -> typ * sort -> typ -> typ (*exception CLASS_ERROR*)
val of_sort: algebra -> typ * sort -> bool
val of_sort_derivation: algebra ->
{class_relation: typ -> 'a * class -> class -> 'a,
type_constructor: string * typ list -> ('a * class) list list -> class -> 'a,
type_variable: typ -> ('a * class) list} ->
typ * sort -> 'a list (*exception CLASS_ERROR*)
val classrel_derivation: algebra ->
('a * class -> class -> 'a) -> 'a * class -> class -> 'a (*exception CLASS_ERROR*)
val witness_sorts: algebra -> string list -> (typ * sort) list -> sort list -> (typ * sort) list
end;
structure Sorts: SORTS =
struct
(** ordered lists of sorts **)
val make = Ord_List.make Term_Ord.sort_ord;
val subset = Ord_List.subset Term_Ord.sort_ord;
val union = Ord_List.union Term_Ord.sort_ord;
val subtract = Ord_List.subtract Term_Ord.sort_ord;
val remove_sort = Ord_List.remove Term_Ord.sort_ord;
val insert_sort = Ord_List.insert Term_Ord.sort_ord;
fun insert_typ (TFree (_, S)) Ss = insert_sort S Ss
| insert_typ (TVar (_, S)) Ss = insert_sort S Ss
| insert_typ (Type (_, Ts)) Ss = insert_typs Ts Ss
and insert_typs [] Ss = Ss
| insert_typs (T :: Ts) Ss = insert_typs Ts (insert_typ T Ss);
fun insert_term (Const (_, T)) Ss = insert_typ T Ss
| insert_term (Free (_, T)) Ss = insert_typ T Ss
| insert_term (Var (_, T)) Ss = insert_typ T Ss
| insert_term (Bound _) Ss = Ss
| insert_term (Abs (_, T, t)) Ss = insert_term t (insert_typ T Ss)
| insert_term (t $ u) Ss = insert_term t (insert_term u Ss);
fun insert_terms [] Ss = Ss
| insert_terms (t :: ts) Ss = insert_terms ts (insert_term t Ss);
(** order-sorted algebra **)
(*
classes: graph representing class declarations together with proper
subclass relation, which needs to be transitive and acyclic.
arities: table of association lists of all type arities; (t, ars)
means that type constructor t has the arities ars; an element
(c, Ss) of ars represents the arity t::(Ss)c. "Coregularity" of
the arities structure requires that for any two declarations
t::(Ss1)c1 and t::(Ss2)c2 such that c1 <= c2 holds Ss1 <= Ss2.
*)
datatype algebra = Algebra of
{classes: serial Graph.T,
arities: (class * sort list) list Symtab.table};
fun classes_of (Algebra {classes, ...}) = classes;
fun arities_of (Algebra {arities, ...}) = arities;
fun make_algebra (classes, arities) =
Algebra {classes = classes, arities = arities};
fun map_classes f (Algebra {classes, arities}) = make_algebra (f classes, arities);
fun map_arities f (Algebra {classes, arities}) = make_algebra (classes, f arities);
(* classes *)
fun all_classes (Algebra {classes, ...}) = Graph.all_preds classes (Graph.maximals classes);
val super_classes = Graph.imm_succs o classes_of;
(* class relations *)
val class_less = Graph.is_edge o classes_of;
fun class_le algebra (c1, c2) = c1 = c2 orelse class_less algebra (c1, c2);
(* sort relations *)
fun sort_le algebra (S1, S2) =
S1 = S2 orelse forall (fn c2 => exists (fn c1 => class_le algebra (c1, c2)) S1) S2;
fun sorts_le algebra (Ss1, Ss2) =
ListPair.all (sort_le algebra) (Ss1, Ss2);
fun sort_eq algebra (S1, S2) =
sort_le algebra (S1, S2) andalso sort_le algebra (S2, S1);
(* intersection *)
fun inter_class algebra c S =
let
fun intr [] = [c]
| intr (S' as c' :: c's) =
if class_le algebra (c', c) then S'
else if class_le algebra (c, c') then intr c's
else c' :: intr c's
in intr S end;
fun inter_sort algebra (S1, S2) =
sort_strings (fold (inter_class algebra) S1 S2);
(* normal forms *)
fun minimize_sort _ [] = []
| minimize_sort _ (S as [_]) = S
| minimize_sort algebra S =
filter (fn c => not (exists (fn c' => class_less algebra (c', c)) S)) S
|> sort_distinct string_ord;
fun complete_sort algebra =
Graph.all_succs (classes_of algebra) o minimize_sort algebra;
fun minimal_sorts algebra raw_sorts =
let
fun le S1 S2 = sort_le algebra (S1, S2);
val sorts = make (map (minimize_sort algebra) raw_sorts);
in sorts |> filter_out (fn S => exists (fn S' => le S' S andalso not (le S S')) sorts) end;
(* certify *)
fun certify_class algebra c =
if can (Graph.get_node (classes_of algebra)) c then c
else raise TYPE ("Undeclared class: " ^ quote c, [], []);
fun certify_sort classes = map (certify_class classes);
(** build algebras **)
(* classes *)
fun err_dup_class c = error ("Duplicate declaration of class: " ^ quote c);
fun err_cyclic_classes pp css =
error (cat_lines (map (fn cs =>
"Cycle in class relation: " ^ Syntax.string_of_classrel (Syntax.init_pretty pp) cs) css));
fun add_class pp (c, cs) = map_classes (fn classes =>
let
val classes' = classes |> Graph.new_node (c, serial ())
handle Graph.DUP dup => err_dup_class dup;
val classes'' = classes' |> fold Graph.add_edge_trans_acyclic (map (pair c) cs)
handle Graph.CYCLES css => err_cyclic_classes pp css;
in classes'' end);
(* arities *)
local
fun for_classes _ NONE = ""
| for_classes pp (SOME (c1, c2)) =
" for classes " ^ Syntax.string_of_classrel (Syntax.init_pretty pp) [c1, c2];
fun err_conflict pp t cc (c, Ss) (c', Ss') =
error ("Conflict of type arities" ^ for_classes pp cc ^ ":\n " ^
Syntax.string_of_arity (Syntax.init_pretty pp) (t, Ss, [c]) ^ " and\n " ^
Syntax.string_of_arity (Syntax.init_pretty pp) (t, Ss', [c']));
fun coregular pp algebra t (c, Ss) ars =
let
fun conflict (c', Ss') =
if class_le algebra (c, c') andalso not (sorts_le algebra (Ss, Ss')) then
SOME ((c, c'), (c', Ss'))
else if class_le algebra (c', c) andalso not (sorts_le algebra (Ss', Ss)) then
SOME ((c', c), (c', Ss'))
else NONE;
in
(case get_first conflict ars of
SOME ((c1, c2), (c', Ss')) => err_conflict pp t (SOME (c1, c2)) (c, Ss) (c', Ss')
| NONE => (c, Ss) :: ars)
end;
fun complete algebra (c, Ss) = map (rpair Ss) (c :: super_classes algebra c);
fun insert pp algebra t (c, Ss) ars =
(case AList.lookup (op =) ars c of
NONE => coregular pp algebra t (c, Ss) ars
| SOME Ss' =>
if sorts_le algebra (Ss, Ss') then ars
else if sorts_le algebra (Ss', Ss)
then coregular pp algebra t (c, Ss) (remove (op =) (c, Ss') ars)
else err_conflict pp t NONE (c, Ss) (c, Ss'));
in
fun insert_ars pp algebra t = fold_rev (insert pp algebra t);
fun insert_complete_ars pp algebra (t, ars) arities =
let val ars' =
Symtab.lookup_list arities t
|> fold_rev (insert_ars pp algebra t) (map (complete algebra) ars);
in Symtab.update (t, ars') arities end;
fun add_arities pp arg algebra =
algebra |> map_arities (insert_complete_ars pp algebra arg);
fun add_arities_table pp algebra =
Symtab.fold (fn (t, ars) => insert_complete_ars pp algebra (t, ars));
end;
(* classrel *)
fun rebuild_arities pp algebra = algebra |> map_arities (fn arities =>
Symtab.empty
|> add_arities_table pp algebra arities);
fun add_classrel pp rel = rebuild_arities pp o map_classes (fn classes =>
classes |> Graph.add_edge_trans_acyclic rel
handle Graph.CYCLES css => err_cyclic_classes pp css);
(* empty and merge *)
val empty_algebra = make_algebra (Graph.empty, Symtab.empty);
fun merge_algebra pp
(Algebra {classes = classes1, arities = arities1},
Algebra {classes = classes2, arities = arities2}) =
let
val classes' = Graph.merge_trans_acyclic (op =) (classes1, classes2)
handle Graph.DUP c => err_dup_class c
| Graph.CYCLES css => err_cyclic_classes pp css;
val algebra0 = make_algebra (classes', Symtab.empty);
val arities' =
(case (pointer_eq (classes1, classes2), pointer_eq (arities1, arities2)) of
(true, true) => arities1
| (true, false) => (*no completion*)
(arities1, arities2) |> Symtab.join (fn t => fn (ars1, ars2) =>
if pointer_eq (ars1, ars2) then raise Symtab.SAME
else insert_ars pp algebra0 t ars2 ars1)
| (false, true) => (*unary completion*)
Symtab.empty
|> add_arities_table pp algebra0 arities1
| (false, false) => (*binary completion*)
Symtab.empty
|> add_arities_table pp algebra0 arities1
|> add_arities_table pp algebra0 arities2);
in make_algebra (classes', arities') end;
(* algebra projections *) (* FIXME potentially violates abstract type integrity *)
fun subalgebra pp P sargs (algebra as Algebra {classes, arities}) =
let
val restrict_sort = minimize_sort algebra o filter P o Graph.all_succs classes;
fun restrict_arity t (c, Ss) =
if P c then
(case sargs (c, t) of
SOME sorts =>
SOME (c, Ss |> map2 (curry (inter_sort algebra)) sorts |> map restrict_sort)
| NONE => NONE)
else NONE;
val classes' = classes |> Graph.subgraph P;
val arities' = arities |> Symtab.map (map_filter o restrict_arity);
in (restrict_sort, rebuild_arities pp (make_algebra (classes', arities'))) end;
(** sorts of types **)
(* errors -- performance tuning via delayed message composition *)
datatype class_error =
No_Classrel of class * class |
No_Arity of string * class |
No_Subsort of sort * sort;
fun class_error ctxt (No_Classrel (c1, c2)) =
"No class relation " ^ Syntax.string_of_classrel ctxt [c1, c2]
| class_error ctxt (No_Arity (a, c)) =
"No type arity " ^ Syntax.string_of_arity ctxt (a, [], [c])
| class_error ctxt (No_Subsort (S1, S2)) =
"Cannot derive subsort relation " ^
Syntax.string_of_sort ctxt S1 ^ " < " ^ Syntax.string_of_sort ctxt S2;
exception CLASS_ERROR of class_error;
(* mg_domain *)
fun mg_domain algebra a S =
let
val arities = arities_of algebra;
fun dom c =
(case AList.lookup (op =) (Symtab.lookup_list arities a) c of
NONE => raise CLASS_ERROR (No_Arity (a, c))
| SOME Ss => Ss);
fun dom_inter c Ss = ListPair.map (inter_sort algebra) (dom c, Ss);
in
(case S of
[] => raise Fail "Unknown domain of empty intersection"
| c :: cs => fold dom_inter cs (dom c))
end;
(* meet_sort *)
fun meet_sort algebra =
let
fun inters S S' = inter_sort algebra (S, S');
fun meet _ [] = I
| meet (TFree (_, S)) S' =
if sort_le algebra (S, S') then I
else raise CLASS_ERROR (No_Subsort (S, S'))
| meet (TVar (v, S)) S' =
if sort_le algebra (S, S') then I
else Vartab.map_default (v, S) (inters S')
| meet (Type (a, Ts)) S = fold2 meet Ts (mg_domain algebra a S);
in uncurry meet end;
fun meet_sort_typ algebra (T, S) =
let val tab = meet_sort algebra (T, S) Vartab.empty;
in Term.map_type_tvar (fn (v, _) => TVar (v, (the o Vartab.lookup tab) v)) end;
(* of_sort *)
fun of_sort algebra =
let
fun ofS (_, []) = true
| ofS (TFree (_, S), S') = sort_le algebra (S, S')
| ofS (TVar (_, S), S') = sort_le algebra (S, S')
| ofS (Type (a, Ts), S) =
let val Ss = mg_domain algebra a S in
ListPair.all ofS (Ts, Ss)
end handle CLASS_ERROR _ => false;
in ofS end;
(* animating derivations *)
fun of_sort_derivation algebra {class_relation, type_constructor, type_variable} =
let
val arities = arities_of algebra;
fun weaken T D1 S2 =
let val S1 = map snd D1 in
if S1 = S2 then map fst D1
else
S2 |> map (fn c2 =>
(case D1 |> find_first (fn (_, c1) => class_le algebra (c1, c2)) of
SOME d1 => class_relation T d1 c2
| NONE => raise CLASS_ERROR (No_Subsort (S1, S2))))
end;
fun derive (_, []) = []
| derive (T as Type (a, Us), S) =
let
val Ss = mg_domain algebra a S;
val dom = map2 (fn U => fn S => derive (U, S) ~~ S) Us Ss;
in
S |> map (fn c =>
let
val Ss' = the (AList.lookup (op =) (Symtab.lookup_list arities a) c);
val dom' = map (fn ((U, d), S') => weaken U d S' ~~ S') ((Us ~~ dom) ~~ Ss');
in type_constructor (a, Us) dom' c end)
end
| derive (T, S) = weaken T (type_variable T) S;
in derive end;
fun classrel_derivation algebra class_relation =
let
fun path (x, c1 :: c2 :: cs) = path (class_relation (x, c1) c2, c2 :: cs)
| path (x, _) = x;
in
fn (x, c1) => fn c2 =>
(case Graph.irreducible_paths (classes_of algebra) (c1, c2) of
[] => raise CLASS_ERROR (No_Classrel (c1, c2))
| cs :: _ => path (x, cs))
end;
(* witness_sorts *)
fun witness_sorts algebra types hyps sorts =
let
fun le S1 S2 = sort_le algebra (S1, S2);
fun get S2 (T, S1) = if le S1 S2 then SOME (T, S2) else NONE;
fun mg_dom t S = SOME (mg_domain algebra t S) handle CLASS_ERROR _ => NONE;
fun witn_sort _ [] solved_failed = (SOME (propT, []), solved_failed)
| witn_sort path S (solved, failed) =
if exists (le S) failed then (NONE, (solved, failed))
else
(case get_first (get S) solved of
SOME w => (SOME w, (solved, failed))
| NONE =>
(case get_first (get S) hyps of
SOME w => (SOME w, (w :: solved, failed))
| NONE => witn_types path types S (solved, failed)))
and witn_sorts path x = fold_map (witn_sort path) x
and witn_types _ [] S (solved, failed) = (NONE, (solved, S :: failed))
| witn_types path (t :: ts) S solved_failed =
(case mg_dom t S of
SOME SS =>
(*do not descend into stronger args (achieving termination)*)
if exists (fn D => le D S orelse exists (le D) path) SS then
witn_types path ts S solved_failed
else
let val (ws, (solved', failed')) = witn_sorts (S :: path) SS solved_failed in
if forall is_some ws then
let val w = (Type (t, map (#1 o the) ws), S)
in (SOME w, (w :: solved', failed')) end
else witn_types path ts S (solved', failed')
end
| NONE => witn_types path ts S solved_failed);
in map_filter I (#1 (witn_sorts [] sorts ([], []))) end;
end;