use "fold_map" instead of "fold (fn .. => .. (ts @ [t], ..)) .."
(* Title: Tools/subtyping.ML
Author: Dmitriy Traytel, TU Muenchen
Coercive subtyping via subtype constraints.
*)
signature SUBTYPING =
sig
datatype variance = COVARIANT | CONTRAVARIANT | INVARIANT
val infer_types: Proof.context -> (string -> typ option) -> (indexname -> typ option) ->
term list -> term list
val add_type_map: term -> Context.generic -> Context.generic
val add_coercion: term -> Context.generic -> Context.generic
val gen_coercion: Proof.context -> typ Vartab.table -> (typ * typ) -> term
val setup: theory -> theory
end;
structure Subtyping: SUBTYPING =
struct
(** coercions data **)
datatype variance = COVARIANT | CONTRAVARIANT | INVARIANT
datatype data = Data of
{coes: term Symreltab.table, (*coercions table*)
coes_graph: unit Graph.T, (*coercions graph*)
tmaps: (term * variance list) Symtab.table}; (*map functions*)
fun make_data (coes, coes_graph, tmaps) =
Data {coes = coes, coes_graph = coes_graph, tmaps = tmaps};
structure Data = Generic_Data
(
type T = data;
val empty = make_data (Symreltab.empty, Graph.empty, Symtab.empty);
val extend = I;
fun merge
(Data {coes = coes1, coes_graph = coes_graph1, tmaps = tmaps1},
Data {coes = coes2, coes_graph = coes_graph2, tmaps = tmaps2}) =
make_data (Symreltab.merge (op aconv) (coes1, coes2),
Graph.merge (op =) (coes_graph1, coes_graph2),
Symtab.merge (eq_pair (op aconv) (op =)) (tmaps1, tmaps2));
);
fun map_data f =
Data.map (fn Data {coes, coes_graph, tmaps} =>
make_data (f (coes, coes_graph, tmaps)));
fun map_coes f =
map_data (fn (coes, coes_graph, tmaps) =>
(f coes, coes_graph, tmaps));
fun map_coes_graph f =
map_data (fn (coes, coes_graph, tmaps) =>
(coes, f coes_graph, tmaps));
fun map_coes_and_graph f =
map_data (fn (coes, coes_graph, tmaps) =>
let val (coes', coes_graph') = f (coes, coes_graph);
in (coes', coes_graph', tmaps) end);
fun map_tmaps f =
map_data (fn (coes, coes_graph, tmaps) =>
(coes, coes_graph, f tmaps));
val rep_data = (fn Data args => args) o Data.get o Context.Proof;
val coes_of = #coes o rep_data;
val coes_graph_of = #coes_graph o rep_data;
val tmaps_of = #tmaps o rep_data;
(** utils **)
fun nameT (Type (s, [])) = s;
fun t_of s = Type (s, []);
fun sort_of (TFree (_, S)) = SOME S
| sort_of (TVar (_, S)) = SOME S
| sort_of _ = NONE;
val is_typeT = fn (Type _) => true | _ => false;
val is_compT = fn (Type (_, _ :: _)) => true | _ => false;
val is_freeT = fn (TFree _) => true | _ => false;
val is_fixedvarT = fn (TVar (xi, _)) => not (Type_Infer.is_param xi) | _ => false;
(* unification *)
exception TYPE_INFERENCE_ERROR of unit -> string;
exception NO_UNIFIER of string * typ Vartab.table;
fun unify weak ctxt =
let
val thy = ProofContext.theory_of ctxt;
val pp = Syntax.pp ctxt;
val arity_sorts = Type.arity_sorts pp (Sign.tsig_of thy);
(* adjust sorts of parameters *)
fun not_of_sort x S' S =
"Variable " ^ x ^ "::" ^ Syntax.string_of_sort ctxt S' ^ " not of sort " ^
Syntax.string_of_sort ctxt S;
fun meet (_, []) tye_idx = tye_idx
| meet (Type (a, Ts), S) (tye_idx as (tye, _)) =
meets (Ts, arity_sorts a S handle ERROR msg => raise NO_UNIFIER (msg, tye)) tye_idx
| meet (TFree (x, S'), S) (tye_idx as (tye, _)) =
if Sign.subsort thy (S', S) then tye_idx
else raise NO_UNIFIER (not_of_sort x S' S, tye)
| meet (TVar (xi, S'), S) (tye_idx as (tye, idx)) =
if Sign.subsort thy (S', S) then tye_idx
else if Type_Infer.is_param xi then
(Vartab.update_new
(xi, Type_Infer.mk_param idx (Sign.inter_sort thy (S', S))) tye, idx + 1)
else raise NO_UNIFIER (not_of_sort (Term.string_of_vname xi) S' S, tye)
and meets (T :: Ts, S :: Ss) (tye_idx as (tye, _)) =
meets (Ts, Ss) (meet (Type_Infer.deref tye T, S) tye_idx)
| meets _ tye_idx = tye_idx;
val weak_meet = if weak then fn _ => I else meet
(* occurs check and assignment *)
fun occurs_check tye xi (TVar (xi', _)) =
if xi = xi' then raise NO_UNIFIER ("Occurs check!", tye)
else
(case Vartab.lookup tye xi' of
NONE => ()
| SOME T => occurs_check tye xi T)
| occurs_check tye xi (Type (_, Ts)) = List.app (occurs_check tye xi) Ts
| occurs_check _ _ _ = ();
fun assign xi (T as TVar (xi', _)) S env =
if xi = xi' then env
else env |> weak_meet (T, S) |>> Vartab.update_new (xi, T)
| assign xi T S (env as (tye, _)) =
(occurs_check tye xi T; env |> weak_meet (T, S) |>> Vartab.update_new (xi, T));
(* unification *)
fun show_tycon (a, Ts) =
quote (Syntax.string_of_typ ctxt (Type (a, replicate (length Ts) dummyT)));
fun unif (T1, T2) (env as (tye, _)) =
(case pairself (`Type_Infer.is_paramT o Type_Infer.deref tye) (T1, T2) of
((true, TVar (xi, S)), (_, T)) => assign xi T S env
| ((_, T), (true, TVar (xi, S))) => assign xi T S env
| ((_, Type (a, Ts)), (_, Type (b, Us))) =>
if weak andalso null Ts andalso null Us then env
else if a <> b then
raise NO_UNIFIER
("Clash of types " ^ show_tycon (a, Ts) ^ " and " ^ show_tycon (b, Us), tye)
else fold unif (Ts ~~ Us) env
| ((_, T), (_, U)) => if T = U then env else raise NO_UNIFIER ("", tye));
in unif end;
val weak_unify = unify true;
val strong_unify = unify false;
(* Typ_Graph shortcuts *)
val add_edge = Typ_Graph.add_edge_acyclic;
fun get_preds G T = Typ_Graph.all_preds G [T];
fun get_succs G T = Typ_Graph.all_succs G [T];
fun maybe_new_typnode T G = perhaps (try (Typ_Graph.new_node (T, ()))) G;
fun maybe_new_typnodes Ts G = fold maybe_new_typnode Ts G;
fun new_imm_preds G Ts =
subtract (op =) Ts (distinct (op =) (maps (Typ_Graph.imm_preds G) Ts));
fun new_imm_succs G Ts =
subtract op= Ts (distinct (op =) (maps (Typ_Graph.imm_succs G) Ts));
(* Graph shortcuts *)
fun maybe_new_node s G = perhaps (try (Graph.new_node (s, ()))) G
fun maybe_new_nodes ss G = fold maybe_new_node ss G
(** error messages **)
fun gen_msg err msg =
err () ^ "\nNow trying to infer coercions:\n\nCoercion inference failed" ^
(if msg = "" then "" else ": " ^ msg) ^ "\n";
fun prep_output ctxt tye bs ts Ts =
let
val (Ts_bTs', ts') = Type_Infer.finish ctxt tye (Ts @ map snd bs, ts);
val (Ts', Ts'') = chop (length Ts) Ts_bTs';
fun prep t =
let val xs = rev (Term.variant_frees t (rev (map fst bs ~~ Ts'')))
in Term.subst_bounds (map Syntax.mark_boundT xs, t) end;
in (map prep ts', Ts') end;
fun err_loose i = error ("Loose bound variable: B." ^ string_of_int i);
fun unif_failed msg =
"Type unification failed" ^ (if msg = "" then "" else ": " ^ msg) ^ "\n\n";
fun subtyping_err_appl_msg ctxt msg tye bs t T u U () =
let val ([t', u'], [T', U']) = prep_output ctxt tye bs [t, u] [T, U]
in msg ^ Type.appl_error (Syntax.pp ctxt) t' T' u' U' ^ "\n" end;
fun err_appl_msg ctxt msg tye bs t T u U () =
let val ([t', u'], [T', U']) = prep_output ctxt tye bs [t, u] [T, U]
in unif_failed msg ^ Type.appl_error (Syntax.pp ctxt) t' T' u' U' ^ "\n" end;
fun err_list ctxt msg tye Ts =
let
val (_, Ts') = prep_output ctxt tye [] [] Ts;
val text = cat_lines ([msg,
"Cannot unify a list of types that should be the same:",
(Pretty.string_of (Pretty.list "[" "]" (map (Pretty.typ (Syntax.pp ctxt)) Ts')))]);
in
error text
end;
fun err_bound ctxt msg tye packs =
let
val pp = Syntax.pp ctxt;
val (ts, Ts) = fold
(fn (bs, t $ u, U, _, U') => fn (ts, Ts) =>
let val (t', T') = prep_output ctxt tye bs [t, u] [U', U]
in (t' :: ts, T' :: Ts) end)
packs ([], []);
val text = cat_lines ([msg, "Cannot fulfil subtype constraints:"] @
(map2 (fn [t, u] => fn [T, U] => Pretty.string_of (
Pretty.block [
Pretty.typ pp T, Pretty.brk 2, Pretty.str "<:", Pretty.brk 2, Pretty.typ pp U,
Pretty.brk 3, Pretty.str "from function application", Pretty.brk 2,
Pretty.block [Pretty.term pp (t $ u)]]))
ts Ts))
in
error text
end;
(** constraint generation **)
fun generate_constraints ctxt err =
let
fun gen cs _ (Const (_, T)) tye_idx = (T, tye_idx, cs)
| gen cs _ (Free (_, T)) tye_idx = (T, tye_idx, cs)
| gen cs _ (Var (_, T)) tye_idx = (T, tye_idx, cs)
| gen cs bs (Bound i) tye_idx =
(snd (nth bs i handle Subscript => err_loose i), tye_idx, cs)
| gen cs bs (Abs (x, T, t)) tye_idx =
let val (U, tye_idx', cs') = gen cs ((x, T) :: bs) t tye_idx
in (T --> U, tye_idx', cs') end
| gen cs bs (t $ u) tye_idx =
let
val (T, tye_idx', cs') = gen cs bs t tye_idx;
val (U', (tye, idx), cs'') = gen cs' bs u tye_idx';
val U = Type_Infer.mk_param idx [];
val V = Type_Infer.mk_param (idx + 1) [];
val tye_idx''= strong_unify ctxt (U --> V, T) (tye, idx + 2)
handle NO_UNIFIER (msg, tye') => error (gen_msg err msg);
val error_pack = (bs, t $ u, U, V, U');
in (V, tye_idx'', ((U', U), error_pack) :: cs'') end;
in
gen [] []
end;
(** constraint resolution **)
exception BOUND_ERROR of string;
fun process_constraints ctxt err cs tye_idx =
let
val coes_graph = coes_graph_of ctxt;
val tmaps = tmaps_of ctxt;
val tsig = Sign.tsig_of (ProofContext.theory_of ctxt);
val pp = Syntax.pp ctxt;
val arity_sorts = Type.arity_sorts pp tsig;
val subsort = Type.subsort tsig;
fun split_cs _ [] = ([], [])
| split_cs f (c :: cs) =
(case pairself f (fst c) of
(false, false) => apsnd (cons c) (split_cs f cs)
| _ => apfst (cons c) (split_cs f cs));
(* check whether constraint simplification will terminate using weak unification *)
val _ = fold (fn (TU, error_pack) => fn tye_idx =>
weak_unify ctxt TU tye_idx handle NO_UNIFIER (msg, tye) =>
error (gen_msg err ("weak unification of subtype constraints fails\n" ^ msg))) cs tye_idx;
(* simplify constraints *)
fun simplify_constraints cs tye_idx =
let
fun contract a Ts Us error_pack done todo tye idx =
let
val arg_var =
(case Symtab.lookup tmaps a of
(*everything is invariant for unknown constructors*)
NONE => replicate (length Ts) INVARIANT
| SOME av => snd av);
fun new_constraints (variance, constraint) (cs, tye_idx) =
(case variance of
COVARIANT => (constraint :: cs, tye_idx)
| CONTRAVARIANT => (swap constraint :: cs, tye_idx)
| INVARIANT => (cs, strong_unify ctxt constraint tye_idx
handle NO_UNIFIER (msg, tye) =>
error (gen_msg err ("failed to unify invariant arguments\n" ^ msg))));
val (new, (tye', idx')) = apfst (fn cs => (cs ~~ replicate (length cs) error_pack))
(fold new_constraints (arg_var ~~ (Ts ~~ Us)) ([], (tye, idx)));
val test_update = is_compT orf is_freeT orf is_fixedvarT;
val (ch, done') =
if not (null new) then ([], done)
else split_cs (test_update o Type_Infer.deref tye') done;
val todo' = ch @ todo;
in
simplify done' (new @ todo') (tye', idx')
end
(*xi is definitely a parameter*)
and expand varleq xi S a Ts error_pack done todo tye idx =
let
val n = length Ts;
val args = map2 Type_Infer.mk_param (idx upto idx + n - 1) (arity_sorts a S);
val tye' = Vartab.update_new (xi, Type(a, args)) tye;
val (ch, done') = split_cs (is_compT o Type_Infer.deref tye') done;
val todo' = ch @ todo;
val new =
if varleq then (Type(a, args), Type (a, Ts))
else (Type (a, Ts), Type (a, args));
in
simplify done' ((new, error_pack) :: todo') (tye', idx + n)
end
(*TU is a pair of a parameter and a free/fixed variable*)
and eliminate TU error_pack done todo tye idx =
let
val [TVar (xi, S)] = filter Type_Infer.is_paramT TU;
val [T] = filter_out Type_Infer.is_paramT TU;
val SOME S' = sort_of T;
val test_update = if is_freeT T then is_freeT else is_fixedvarT;
val tye' = Vartab.update_new (xi, T) tye;
val (ch, done') = split_cs (test_update o Type_Infer.deref tye') done;
val todo' = ch @ todo;
in
if subsort (S', S) (*TODO check this*)
then simplify done' todo' (tye', idx)
else error (gen_msg err "sort mismatch")
end
and simplify done [] tye_idx = (done, tye_idx)
| simplify done (((T, U), error_pack) :: todo) (tye_idx as (tye, idx)) =
(case (Type_Infer.deref tye T, Type_Infer.deref tye U) of
(Type (a, []), Type (b, [])) =>
if a = b then simplify done todo tye_idx
else if Graph.is_edge coes_graph (a, b) then simplify done todo tye_idx
else error (gen_msg err (a ^ " is not a subtype of " ^ b))
| (Type (a, Ts), Type (b, Us)) =>
if a <> b then error (gen_msg err "different constructors")
(fst tye_idx) error_pack
else contract a Ts Us error_pack done todo tye idx
| (TVar (xi, S), Type (a, Ts as (_ :: _))) =>
expand true xi S a Ts error_pack done todo tye idx
| (Type (a, Ts as (_ :: _)), TVar (xi, S)) =>
expand false xi S a Ts error_pack done todo tye idx
| (T, U) =>
if T = U then simplify done todo tye_idx
else if exists (is_freeT orf is_fixedvarT) [T, U] andalso
exists Type_Infer.is_paramT [T, U]
then eliminate [T, U] error_pack done todo tye idx
else if exists (is_freeT orf is_fixedvarT) [T, U]
then error (gen_msg err "not eliminated free/fixed variables")
else simplify (((T, U), error_pack) :: done) todo tye_idx);
in
simplify [] cs tye_idx
end;
(* do simplification *)
val (cs', tye_idx') = simplify_constraints cs tye_idx;
fun find_error_pack lower T' = map_filter
(fn ((T, U), pack) => if if lower then T' = U else T' = T then SOME pack else NONE) cs';
fun find_cycle_packs nodes =
let
val (but_last, last) = split_last nodes
val pairs = (last, hd nodes) :: (but_last ~~ tl nodes);
in
map_filter
(fn (TU, pack) => if member (op =) pairs TU then SOME pack else NONE)
cs'
end;
fun unify_list (T :: Ts) tye_idx =
fold (fn U => fn tye_idx' => strong_unify ctxt (T, U) tye_idx') Ts tye_idx;
(*styps stands either for supertypes or for subtypes of a type T
in terms of the subtype-relation (excluding T itself)*)
fun styps super T =
(if super then Graph.imm_succs else Graph.imm_preds) coes_graph T
handle Graph.UNDEF _ => [];
fun minmax sup (T :: Ts) =
let
fun adjust T U = if sup then (T, U) else (U, T);
fun extract T [] = T
| extract T (U :: Us) =
if Graph.is_edge coes_graph (adjust T U) then extract T Us
else if Graph.is_edge coes_graph (adjust U T) then extract U Us
else raise BOUND_ERROR "uncomparable types in type list";
in
t_of (extract T Ts)
end;
fun ex_styp_of_sort super T styps_and_sorts =
let
fun adjust T U = if super then (T, U) else (U, T);
fun styp_test U Ts = forall
(fn T => T = U orelse Graph.is_edge coes_graph (adjust U T)) Ts;
fun fitting Ts S U = Type.of_sort tsig (t_of U, S) andalso styp_test U Ts
in
forall (fn (Ts, S) => exists (fitting Ts S) (T :: styps super T)) styps_and_sorts
end;
(* computes the tightest possible, correct assignment for 'a::S
e.g. in the supremum case (sup = true):
------- 'a::S---
/ / \ \
/ / \ \
'b::C1 'c::C2 ... T1 T2 ...
sorts - list of sorts [C1, C2, ...]
T::Ts - non-empty list of base types [T1, T2, ...]
*)
fun tightest sup S styps_and_sorts (T :: Ts) =
let
fun restriction T = Type.of_sort tsig (t_of T, S)
andalso ex_styp_of_sort (not sup) T styps_and_sorts;
fun candidates T = inter (op =) (filter restriction (T :: styps sup T));
in
(case fold candidates Ts (filter restriction (T :: styps sup T)) of
[] => raise BOUND_ERROR ("no " ^ (if sup then "supremum" else "infimum"))
| [T] => t_of T
| Ts => minmax sup Ts)
end;
fun build_graph G [] tye_idx = (G, tye_idx)
| build_graph G ((T, U) :: cs) tye_idx =
if T = U then build_graph G cs tye_idx
else
let
val G' = maybe_new_typnodes [T, U] G;
val (G'', tye_idx') = (add_edge (T, U) G', tye_idx)
handle Typ_Graph.CYCLES cycles =>
let
val (tye, idx) =
fold
(fn cycle => fn tye_idx' => (unify_list cycle tye_idx'
handle NO_UNIFIER (msg, tye) =>
err_bound ctxt
(gen_msg err ("constraint cycle not unifiable" ^ msg)) (fst tye_idx)
(find_cycle_packs cycle)))
cycles tye_idx
in
collapse (tye, idx) cycles G
end
in
build_graph G'' cs tye_idx'
end
and collapse (tye, idx) cycles G = (*nodes non-empty list*)
let
(*all cycles collapse to one node,
because all of them share at least the nodes x and y*)
val nodes = (distinct (op =) (flat cycles));
val T = Type_Infer.deref tye (hd nodes);
val P = new_imm_preds G nodes;
val S = new_imm_succs G nodes;
val G' = Typ_Graph.del_nodes (tl nodes) G;
fun check_and_gen super T' =
let val U = Type_Infer.deref tye T';
in
if not (is_typeT T) orelse not (is_typeT U) orelse T = U
then if super then (hd nodes, T') else (T', hd nodes)
else
if super andalso
Graph.is_edge coes_graph (nameT T, nameT U) then (hd nodes, T')
else if not super andalso
Graph.is_edge coes_graph (nameT U, nameT T) then (T', hd nodes)
else err_bound ctxt (gen_msg err "cycle elimination produces inconsistent graph")
(fst tye_idx)
(maps find_cycle_packs cycles @ find_error_pack super T')
end;
in
build_graph G' (map (check_and_gen false) P @ map (check_and_gen true) S) (tye, idx)
end;
fun assign_bound lower G key (tye_idx as (tye, _)) =
if Type_Infer.is_paramT (Type_Infer.deref tye key) then
let
val TVar (xi, S) = Type_Infer.deref tye key;
val get_bound = if lower then get_preds else get_succs;
val raw_bound = get_bound G key;
val bound = map (Type_Infer.deref tye) raw_bound;
val not_params = filter_out Type_Infer.is_paramT bound;
fun to_fulfil T =
(case sort_of T of
NONE => NONE
| SOME S =>
SOME
(map nameT
(filter_out Type_Infer.is_paramT (map (Type_Infer.deref tye) (get_bound G T))),
S));
val styps_and_sorts = distinct (op =) (map_filter to_fulfil raw_bound);
val assignment =
if null bound orelse null not_params then NONE
else SOME (tightest lower S styps_and_sorts (map nameT not_params)
handle BOUND_ERROR msg =>
err_bound ctxt (gen_msg err msg) tye (find_error_pack lower key))
in
(case assignment of
NONE => tye_idx
| SOME T =>
if Type_Infer.is_paramT T then tye_idx
else if lower then (*upper bound check*)
let
val other_bound = map (Type_Infer.deref tye) (get_succs G key);
val s = nameT T;
in
if subset (op = o apfst nameT) (filter is_typeT other_bound, s :: styps true s)
then apfst (Vartab.update (xi, T)) tye_idx
else err_bound ctxt (gen_msg err ("assigned simple type " ^ s ^
" clashes with the upper bound of variable " ^
Syntax.string_of_typ ctxt (TVar(xi, S)))) tye (find_error_pack (not lower) key)
end
else apfst (Vartab.update (xi, T)) tye_idx)
end
else tye_idx;
val assign_lb = assign_bound true;
val assign_ub = assign_bound false;
fun assign_alternating ts' ts G tye_idx =
if ts' = ts then tye_idx
else
let
val (tye_idx' as (tye, _)) = fold (assign_lb G) ts tye_idx
|> fold (assign_ub G) ts;
in
assign_alternating ts
(filter (Type_Infer.is_paramT o Type_Infer.deref tye) ts) G tye_idx'
end;
(*Unify all weakly connected components of the constraint forest,
that contain only params. These are the only WCCs that contain
params anyway.*)
fun unify_params G (tye_idx as (tye, _)) =
let
val max_params =
filter (Type_Infer.is_paramT o Type_Infer.deref tye) (Typ_Graph.maximals G);
val to_unify = map (fn T => T :: get_preds G T) max_params;
in
fold
(fn Ts => fn tye_idx' => unify_list Ts tye_idx'
handle NO_UNIFIER (msg, tye) => err_list ctxt (gen_msg err msg) (fst tye_idx) Ts)
to_unify tye_idx
end;
fun solve_constraints G tye_idx = tye_idx
|> assign_alternating [] (Typ_Graph.keys G) G
|> unify_params G;
in
build_graph Typ_Graph.empty (map fst cs') tye_idx'
|-> solve_constraints
end;
(** coercion insertion **)
fun gen_coercion ctxt tye (T1, T2) =
(case pairself (Type_Infer.deref tye) (T1, T2) of
((Type (a, [])), (Type (b, []))) =>
if a = b
then Abs (Name.uu, Type (a, []), Bound 0)
else
(case Symreltab.lookup (coes_of ctxt) (a, b) of
NONE => raise Fail (a ^ " is not a subtype of " ^ b)
| SOME co => co)
| ((Type (a, Ts)), (Type (b, Us))) =>
if a <> b
then raise Fail ("Different constructors: " ^ a ^ " and " ^ b)
else
let
fun inst t Ts =
Term.subst_vars
(((Term.add_tvar_namesT (fastype_of t) []) ~~ rev Ts), []) t;
fun sub_co (COVARIANT, TU) = gen_coercion ctxt tye TU
| sub_co (CONTRAVARIANT, TU) = gen_coercion ctxt tye (swap TU);
fun ts_of [] = []
| ts_of (Type ("fun", [x1, x2]) :: xs) = x1 :: x2 :: (ts_of xs);
in
(case Symtab.lookup (tmaps_of ctxt) a of
NONE => raise Fail ("No map function for " ^ a ^ " known")
| SOME tmap =>
let
val used_coes = map sub_co ((snd tmap) ~~ (Ts ~~ Us));
in
Term.list_comb
(inst (fst tmap) (ts_of (map fastype_of used_coes)), used_coes)
end)
end
| (T, U) =>
if Type.could_unify (T, U)
then Abs (Name.uu, T, Bound 0)
else raise Fail ("Cannot generate coercion from "
^ Syntax.string_of_typ ctxt T ^ " to " ^ Syntax.string_of_typ ctxt U));
fun insert_coercions ctxt tye ts =
let
fun insert _ (Const (c, T)) =
let val T' = T;
in (Const (c, T'), T') end
| insert _ (Free (x, T)) =
let val T' = T;
in (Free (x, T'), T') end
| insert _ (Var (xi, T)) =
let val T' = T;
in (Var (xi, T'), T') end
| insert bs (Bound i) =
let val T = nth bs i handle Subscript => err_loose i;
in (Bound i, T) end
| insert bs (Abs (x, T, t)) =
let
val T' = T;
val (t', T'') = insert (T' :: bs) t;
in
(Abs (x, T', t'), T' --> T'')
end
| insert bs (t $ u) =
let
val (t', Type ("fun", [U, T])) = apsnd (Type_Infer.deref tye) (insert bs t);
val (u', U') = insert bs u;
in
if can (fn TU => strong_unify ctxt TU (tye, 0)) (U, U')
then (t' $ u', T)
else (t' $ (gen_coercion ctxt tye (U', U) $ u'), T)
end
in
map (fst o insert []) ts
end;
(** assembling the pipeline **)
fun infer_types ctxt const_type var_type raw_ts =
let
val (idx, ts) = Type_Infer.prepare ctxt const_type var_type raw_ts;
fun inf _ (t as (Const (_, T))) tye_idx = (t, T, tye_idx)
| inf _ (t as (Free (_, T))) tye_idx = (t, T, tye_idx)
| inf _ (t as (Var (_, T))) tye_idx = (t, T, tye_idx)
| inf bs (t as (Bound i)) tye_idx =
(t, snd (nth bs i handle Subscript => err_loose i), tye_idx)
| inf bs (Abs (x, T, t)) tye_idx =
let val (t', U, tye_idx') = inf ((x, T) :: bs) t tye_idx
in (Abs (x, T, t'), T --> U, tye_idx') end
| inf bs (t $ u) tye_idx =
let
val (t', T, tye_idx') = inf bs t tye_idx;
val (u', U, (tye, idx)) = inf bs u tye_idx';
val V = Type_Infer.mk_param idx [];
val (tu, tye_idx'') = (t' $ u', strong_unify ctxt (U --> V, T) (tye, idx + 1))
handle NO_UNIFIER (msg, tye') =>
raise TYPE_INFERENCE_ERROR (err_appl_msg ctxt msg tye' bs t T u U);
in (tu, V, tye_idx'') end;
fun infer_single t tye_idx =
let val (t, _, tye_idx') = inf [] t tye_idx;
in (t, tye_idx') end;
val (ts', (tye, _)) = (fold_map infer_single ts (Vartab.empty, idx)
handle TYPE_INFERENCE_ERROR err =>
let
fun gen_single t (tye_idx, constraints) =
let val (_, tye_idx', constraints') = generate_constraints ctxt err t tye_idx
in (tye_idx', constraints' @ constraints) end;
val (tye_idx, constraints) = fold gen_single ts ((Vartab.empty, idx), []);
val (tye, idx) = process_constraints ctxt err constraints tye_idx;
in
(insert_coercions ctxt tye ts, (tye, idx))
end);
val (_, ts'') = Type_Infer.finish ctxt tye ([], ts');
in ts'' end;
(** installation **)
(* term check *)
fun coercion_infer_types ctxt =
infer_types ctxt
(try (Consts.the_constraint (ProofContext.consts_of ctxt)))
(ProofContext.def_type ctxt);
val add_term_check =
Syntax.add_term_check ~100 "coercions"
(fn xs => fn ctxt =>
let val xs' = coercion_infer_types ctxt xs
in if eq_list (op aconv) (xs, xs') then NONE else SOME (xs', ctxt) end);
(* declarations *)
fun add_type_map raw_t context =
let
val ctxt = Context.proof_of context;
val t = singleton (Variable.polymorphic ctxt) raw_t;
fun err_str () = "\n\nthe general type signature for a map function is" ^
"\nf1 => f2 => ... => fn => C [x1, ..., xn] => C [x1, ..., xn]" ^
"\nwhere C is a constructor and fi is of type (xi => yi) or (yi => xi)";
fun gen_arg_var ([], []) = []
| gen_arg_var ((T, T') :: Ts, (U, U') :: Us) =
if T = U andalso T' = U' then COVARIANT :: gen_arg_var (Ts, Us)
else if T = U' andalso T' = U then CONTRAVARIANT :: gen_arg_var (Ts, Us)
else error ("Functions do not apply to arguments correctly:" ^ err_str ())
| gen_arg_var (_, _) =
error ("Different numbers of functions and arguments\n" ^ err_str ());
(* TODO: This function is only needed to introde the fun type map
function: "% f g h . g o h o f". There must be a better solution. *)
fun balanced (Type (_, [])) (Type (_, [])) = true
| balanced (Type (a, Ts)) (Type (b, Us)) =
a = b andalso forall I (map2 balanced Ts Us)
| balanced (TFree _) (TFree _) = true
| balanced (TVar _) (TVar _) = true
| balanced _ _ = false;
fun check_map_fun (pairs, []) (Type ("fun", [T as Type (C, Ts), U as Type (_, Us)])) =
if balanced T U
then ((pairs, Ts ~~ Us), C)
else if C = "fun"
then check_map_fun (pairs @ [(hd Ts, hd (tl Ts))], []) U
else error ("Not a proper map function:" ^ err_str ())
| check_map_fun _ _ = error ("Not a proper map function:" ^ err_str ());
val res = check_map_fun ([], []) (fastype_of t);
val res_av = gen_arg_var (fst res);
in
map_tmaps (Symtab.update (snd res, (t, res_av))) context
end;
fun add_coercion raw_t context =
let
val ctxt = Context.proof_of context;
val t = singleton (Variable.polymorphic ctxt) raw_t;
fun err_coercion () = error ("Bad type for coercion " ^
Syntax.string_of_term ctxt t ^ ":\n" ^
Syntax.string_of_typ ctxt (fastype_of t));
val (T1, T2) = Term.dest_funT (fastype_of t)
handle TYPE _ => err_coercion ();
val a =
(case T1 of
Type (x, []) => x
| _ => err_coercion ());
val b =
(case T2 of
Type (x, []) => x
| _ => err_coercion ());
fun coercion_data_update (tab, G) =
let
val G' = maybe_new_nodes [a, b] G
val G'' = Graph.add_edge_trans_acyclic (a, b) G'
handle Graph.CYCLES _ => error (a ^ " is already a subtype of " ^ b ^
"!\n\nCannot add coercion of type: " ^ a ^ " => " ^ b);
val new_edges =
flat (Graph.dest G'' |> map (fn (x, ys) => ys |> map_filter (fn y =>
if Graph.is_edge G' (x, y) then NONE else SOME (x, y))));
val G_and_new = Graph.add_edge (a, b) G';
fun complex_coercion tab G (a, b) =
let
val path = hd (Graph.irreducible_paths G (a, b))
val path' = fst (split_last path) ~~ tl path
in Abs (Name.uu, Type (a, []),
fold (fn t => fn u => t $ u) (map (the o Symreltab.lookup tab) path') (Bound 0))
end;
val tab' = fold
(fn pair => fn tab => Symreltab.update (pair, complex_coercion tab G_and_new pair) tab)
(filter (fn pair => pair <> (a, b)) new_edges)
(Symreltab.update ((a, b), t) tab);
in
(tab', G'')
end;
in
map_coes_and_graph coercion_data_update context
end;
(* theory setup *)
val setup =
Context.theory_map add_term_check #>
Attrib.setup @{binding coercion}
(Args.term >> (fn t => Thm.declaration_attribute (K (add_coercion t))))
"declaration of new coercions" #>
Attrib.setup @{binding coercion_map}
(Args.term >> (fn t => Thm.declaration_attribute (K (add_type_map t))))
"declaration of new map functions";
end;