(* Title: Pure/General/defs.ML
ID: $Id$
Author: Steven Obua, TU Muenchen
Checks if definitions preserve consistency of logic by enforcing that there are no cyclic definitions.
The algorithm is described in
"Cycle-free Overloading in Isabelle", Steven Obua, technical report, to be written :-)
*)
signature DEFS = sig
type graph
exception DEFS of string
exception CIRCULAR of (typ * string * string) list
exception INFINITE_CHAIN of (typ * string * string) list
exception FINAL of string * typ
exception CLASH of string * string * string
val empty : graph
val declare : graph -> string * typ -> graph (* exception DEFS *)
val define : graph -> string * typ -> string -> (string * typ) list -> graph
(* exception DEFS, CIRCULAR, INFINITE_CHAIN, CLASH, FINAL *)
val finalize : graph -> string * typ -> graph (* exception DEFS *)
val finals : graph -> (typ list) Symtab.table
(* the first argument should be the smaller graph *)
val merge : graph -> graph -> graph (* exception CIRCULAR, INFINITE_CHAIN, CLASH *)
end
structure Defs :> DEFS = struct
type tyenv = Type.tyenv
type edgelabel = (int * typ * typ * (typ * string * string) list)
type noderef = string
datatype node = Node of
string (* name of constant *)
* typ (* most general type of constant *)
* defnode Symtab.table (* a table of defnodes, each corresponding to 1 definition of the
constant for a particular type, indexed by axiom name *)
* backref Symtab.table (* a table of all back references to this node, indexed by node name *)
* typ list (* a list of all finalized types *)
and defnode = Defnode of
typ (* type of the constant in this particular definition *)
* ((noderef * (string option * edgelabel list) list) Symtab.table) (* The edges, grouped by nodes. *)
and backref = Backref of
noderef (* the name of the node that has defnodes which reference a certain node A *)
* (unit Symtab.table) (* the names of the defnodes that DIRECTLY reference A. *)
fun getnode graph noderef = the (Symtab.lookup (graph, noderef))
fun get_nodename (Node (n, _, _ ,_, _)) = n
fun get_nodedefs (Node (_, _, defs, _, _)) = defs
fun get_defnode (Node (_, _, defs, _, _)) defname = Symtab.lookup (defs, defname)
fun get_defnode' graph noderef defname = Symtab.lookup (get_nodedefs (the (Symtab.lookup (graph, noderef))), defname)
fun get_nodename (Node (n, _, _ , _, _)) = n
datatype graphaction = Declare of string * typ
| Define of string * typ * string * (string * typ) list
| Finalize of string * typ
type graph = (graphaction list) * (node Symtab.table)
val empty = ([], Symtab.empty)
exception DEFS of string;
exception CIRCULAR of (typ * string * string) list;
exception INFINITE_CHAIN of (typ * string * string) list;
exception CLASH of string * string * string;
exception FINAL of string * typ;
fun def_err s = raise (DEFS s)
fun declare (actions, g) (cty as (name, ty)) =
((Declare cty)::actions,
Symtab.update_new ((name, Node (name, Type.varifyT(Type.strip_sorts ty), Symtab.empty, Symtab.empty, [])), g))
handle Symtab.DUP _ => def_err "constant is already declared"
fun rename ty1 ty2 = incr_tvar ((maxidx_of_typ ty1)+1) ty2;
fun subst_incr_tvar inc t =
if (inc > 0) then
let
val tv = typ_tvars t
val t' = incr_tvar inc t
fun update_subst (((n,i), _), s) =
Vartab.update (((n, i), ([], TVar ((n, i+inc), []))), s)
in
(t',List.foldl update_subst Vartab.empty tv)
end
else
(t, Vartab.empty)
(* Rename tys2 so that tys2 and tys1 do not have any variables in common any more.
As a result, return the renamed tys2' and the substitution that takes tys2 to tys2'. *)
fun subst_rename max1 ty2 =
let
val max2 = (maxidx_of_typ ty2)
val (ty2', s) = subst_incr_tvar (max1 + 1) ty2
in
(ty2', s, max1 + max2 + 1)
end
fun subst s ty = Envir.norm_type s ty
fun subst_history s history = map (fn (ty, cn, dn) => (subst s ty, cn, dn)) history
fun is_instance instance_ty general_ty =
Type.typ_instance Type.empty_tsig (instance_ty, general_ty)
fun is_instance_r instance_ty general_ty =
is_instance instance_ty (rename instance_ty general_ty)
fun unify ty1 ty2 =
SOME (fst (Type.unify Type.empty_tsig (Vartab.empty, 0) (ty1, ty2)))
handle Type.TUNIFY => NONE
(*
Unifies ty1 and ty2, renaming ty1 and ty2 so that they have greater indices than max and so that they
are different. All indices in ty1 and ty2 are supposed to be less than or equal to max.
Returns SOME (max', s1, s2), so that s1(ty1) = s2(ty2) and max' is greater or equal than all
indices in s1, s2, ty1, ty2.
*)
fun unify_r max ty1 ty2 =
let
val max = Int.max(max, 0)
val max1 = max (* >= maxidx_of_typ ty1 *)
val max2 = max (* >= maxidx_of_typ ty2 *)
val max = Int.max(max, Int.max (max1, max2))
val (ty1, s1) = subst_incr_tvar (max+1) ty1
val (ty2, s2) = subst_incr_tvar (max+max1+2) ty2
val max = max+max1+max2+2
fun merge a b = Vartab.merge (fn _ => false) (a, b)
in
case unify ty1 ty2 of
NONE => NONE
| SOME s => SOME (max, merge s1 s, merge s2 s)
end
fun can_be_unified_r ty1 ty2 =
let
val ty2 = rename ty1 ty2
in
case unify ty1 ty2 of
NONE => false
| _ => true
end
fun can_be_unified ty1 ty2 =
case unify ty1 ty2 of
NONE => false
| _ => true
fun checkT (Type (a, Ts)) = Type (a, map checkT Ts)
| checkT (TVar ((a, 0), _)) = TVar ((a, 0), [])
| checkT (TVar ((a, i), _)) = def_err "type is not clean"
| checkT (TFree (a, _)) = TVar ((a, 0), [])
fun label_ord NONE NONE = EQUAL
| label_ord NONE (SOME _) = LESS
| label_ord (SOME _) NONE = GREATER
| label_ord (SOME l1) (SOME l2) = string_ord (l1,l2)
fun compare_edges (e1 as (maxidx1, u1, v1, history1)) (e2 as (maxidx2, u2, v2, history2)) =
let
val t1 = u1 --> v1
val t2 = u2 --> v2
in
if (is_instance_r t1 t2) then
(if is_instance_r t2 t1 then
SOME (int_ord (length history2, length history1))
else
SOME LESS)
else if (is_instance_r t2 t1) then
SOME GREATER
else
NONE
end
fun merge_edges_1 (x, []) = []
| merge_edges_1 (x, (y::ys)) =
(case compare_edges x y of
SOME LESS => (y::ys)
| SOME EQUAL => (y::ys)
| SOME GREATER => merge_edges_1 (x, ys)
| NONE => y::(merge_edges_1 (x, ys)))
fun merge_edges xs ys = foldl merge_edges_1 xs ys
fun pack_edges xs = merge_edges [] xs
fun merge_labelled_edges [] es = es
| merge_labelled_edges es [] = es
| merge_labelled_edges ((l1,e1)::es1) ((l2,e2)::es2) =
(case label_ord l1 l2 of
LESS => (l1, e1)::(merge_labelled_edges es1 ((l2, e2)::es2))
| GREATER => (l2, e2)::(merge_labelled_edges ((l1, e1)::es1) es2)
| EQUAL => (l1, merge_edges e1 e2)::(merge_labelled_edges es1 es2))
fun defnode_edges_foldl f a defnode =
let
val (Defnode (ty, def_edges)) = defnode
fun g (b, (_, (n, labelled_edges))) =
foldl (fn ((s, edges), b') =>
(foldl (fn (e, b'') => f ty n s e b'') b' edges))
b
labelled_edges
in
Symtab.foldl g (a, def_edges)
end
fun define (actions, graph) (name, ty) axname body =
let
val ty = checkT ty
val body = map (fn (n,t) => (n, checkT t)) body
val mainref = name
val mainnode = (case Symtab.lookup (graph, mainref) of
NONE => def_err ("constant "^mainref^" is not declared")
| SOME n => n)
val (Node (n, gty, defs, backs, finals)) = mainnode
val _ = (if is_instance_r ty gty then () else def_err "type of constant does not match declared type")
fun check_def (s, Defnode (ty', _)) =
(if can_be_unified_r ty ty' then
raise (CLASH (mainref, axname, s))
else if s = axname then
def_err "name of axiom is already used for another definition of this constant"
else false)
val _ = Symtab.exists check_def defs
fun check_final finalty =
(if can_be_unified_r finalty ty then
raise (FINAL (mainref, finalty))
else
true)
val _ = forall check_final finals
(* now we know that the only thing that can prevent acceptance of the definition is a cyclic dependency *)
(* body contains the constants that this constant definition depends on. For each element of body,
the function make_edges_to calculates a group of edges that connect this constant with
the constant that is denoted by the element of the body *)
fun make_edges_to (bodyn, bodyty) =
let
val bnode =
(case Symtab.lookup (graph, bodyn) of
NONE => def_err "body of constant definition references undeclared constant"
| SOME x => x)
val (Node (_, general_btyp, bdefs, bbacks, bfinals)) = bnode
in
case unify_r 0 bodyty general_btyp of
NONE => NONE
| SOME (maxidx, sigma1, sigma2) =>
SOME (
let
(* For each definition of the constant in the body,
check if the definition unifies with the type of the constant in the body. *)
fun make_edges ((swallowed, l),(def_name, Defnode (def_ty, _))) =
if swallowed then
(swallowed, l)
else
(case unify_r 0 bodyty def_ty of
NONE => (swallowed, l)
| SOME (maxidx, sigma1, sigma2) =>
(is_instance_r bodyty def_ty,
merge_labelled_edges l [(SOME def_name,[(maxidx, subst sigma1 ty, subst sigma2 def_ty, [])])]))
val swallowed = exists (is_instance_r bodyty) bfinals
val (swallowed, edges) = Symtab.foldl make_edges ((swallowed, []), bdefs)
in
if swallowed then
(bodyn, edges)
else
(bodyn, [(NONE, [(maxidx, subst sigma1 ty, subst sigma2 general_btyp,[])])]@edges)
end)
end
fun update_edges (b as (bodyn, bodyty), edges) =
(case make_edges_to b of
NONE => edges
| SOME m =>
(case Symtab.lookup (edges, bodyn) of
NONE => Symtab.update ((bodyn, m), edges)
| SOME (_, es') =>
let
val (_, es) = m
val es = merge_labelled_edges es es'
in
Symtab.update ((bodyn, (bodyn, es)), edges)
end
)
)
val edges = foldl update_edges Symtab.empty body
fun insert_edge edges (nodename, (defname_opt, edge)) =
let
val newlink = [(defname_opt, [edge])]
in
case Symtab.lookup (edges, nodename) of
NONE => Symtab.update ((nodename, (nodename, newlink)), edges)
| SOME (_, links) =>
let
val links' = merge_labelled_edges links newlink
in
Symtab.update ((nodename, (nodename, links')), edges)
end
end
(* We constructed all direct edges that this defnode has.
Now we have to construct the transitive hull by going a single step further. *)
val thisDefnode = Defnode (ty, edges)
fun make_trans_edges _ noderef defname_opt (max1, alpha1, beta1, history1) edges =
case defname_opt of
NONE => edges
| SOME defname =>
let
val defnode = the (get_defnode' graph noderef defname)
fun make_trans_edge _ noderef2 defname_opt2 (max2, alpha2, beta2, history2) edges =
case unify_r (Int.max (max1, max2)) beta1 alpha2 of
NONE => edges
| SOME (max, sleft, sright) =>
insert_edge edges (noderef2,
(defname_opt2,
(max, subst sleft alpha1, subst sright beta2,
(subst_history sleft history1)@
((subst sleft beta1, noderef, defname)::
(subst_history sright history2)))))
in
defnode_edges_foldl make_trans_edge edges defnode
end
val edges = defnode_edges_foldl make_trans_edges edges thisDefnode
val thisDefnode = Defnode (ty, edges)
(* We also have to add the backreferences that this new defnode induces. *)
fun hasNONElink ((NONE, _)::_) = true
| hasNONElink _ = false
fun install_backref graph noderef pointingnoderef pointingdefname =
let
val (Node (pname, _, _, _, _)) = getnode graph pointingnoderef
val (Node (name, ty, defs, backs, finals)) = getnode graph noderef
in
case Symtab.lookup (backs, pname) of
NONE =>
let
val defnames = Symtab.update ((pointingdefname, ()), Symtab.empty)
val backs = Symtab.update ((pname, Backref (pointingnoderef, defnames)), backs)
in
Symtab.update ((name, Node (name, ty, defs, backs, finals)), graph)
end
| SOME (Backref (pointingnoderef, defnames)) =>
let
val defnames = Symtab.update_new ((pointingdefname, ()), defnames)
val backs = Symtab.update ((pname, Backref (pointingnoderef, defnames)), backs)
in
Symtab.update ((name, Node (name, ty, defs, backs, finals)), graph)
end
handle Symtab.DUP _ => graph
end
fun install_backrefs (graph, (_, (noderef, labelled_edges))) =
if hasNONElink labelled_edges then
install_backref graph noderef mainref axname
else
graph
val graph = Symtab.foldl install_backrefs (graph, edges)
val (Node (_, _, _, backs, _)) = getnode graph mainref
val graph = Symtab.update ((mainref, Node (n, gty, Symtab.update_new
((axname, thisDefnode), defs), backs, finals)), graph)
(* Now we have to check all backreferences to this node and inform them about the new defnode.
In this section we also check for circularity. *)
fun update_backrefs ((backs, newedges), (nodename, Backref (noderef, defnames))) =
let
val node = getnode graph noderef
fun update_defs ((defnames, newedges),(defname, _)) =
let
val (Defnode (_, defnode_edges)) = the (get_defnode node defname)
val (_, labelled_edges) = the (Symtab.lookup (defnode_edges, n))
(* the type of thisDefnode is ty *)
fun update (e as (max, alpha, beta, history), (none_edges, this_edges)) =
case unify_r max beta ty of
NONE => (e::none_edges, this_edges)
| SOME (max', s_beta, s_ty) =>
let
val alpha' = subst s_beta alpha
val ty' = subst s_ty ty
val _ =
if noderef = mainref andalso defname = axname then
(case unify alpha' ty' of
NONE =>
if (is_instance_r ty' alpha') then
raise (INFINITE_CHAIN (
(alpha', mainref, axname)::
(subst_history s_beta history)@
[(ty', mainref, axname)]))
else ()
| SOME s => raise (CIRCULAR (
(subst s alpha', mainref, axname)::
(subst_history s (subst_history s_beta history))@
[(subst s ty', mainref, axname)])))
else ()
val edge = (max', alpha', ty', subst_history s_beta history)
in
if is_instance_r beta ty then
(none_edges, edge::this_edges)
else
(e::none_edges, edge::this_edges)
end
in
case labelled_edges of
((NONE, edges)::_) =>
let
val (none_edges, this_edges) = foldl update ([], []) edges
val defnames = if none_edges = [] then defnames else Symtab.update_new ((defname, ()), defnames)
in
(defnames, (defname, none_edges, this_edges)::newedges)
end
| _ => sys_error "define: update_defs, internal error, corrupt backrefs"
end
val (defnames, newedges') = Symtab.foldl update_defs ((Symtab.empty, []), defnames)
in
if Symtab.is_empty defnames then
(backs, (noderef, newedges')::newedges)
else
let
val backs = Symtab.update_new ((nodename, Backref (noderef, defnames)), backs)
in
(backs, newedges)
end
end
val (backs, newedges) = Symtab.foldl update_backrefs ((Symtab.empty, []), backs)
(* If a Circular exception is thrown then we never reach this point. *)
(* Ok, the definition is consistent, let's update this node. *)
val graph = Symtab.update ((mainref, Node (n, gty, Symtab.update
((axname, thisDefnode), defs), backs, finals)), graph)
(* Furthermore, update all the other nodes that backreference this node. *)
fun final_update_backrefs graph noderef defname none_edges this_edges =
let
val node = getnode graph noderef
val (Node (nodename, nodety, defs, backs, finals)) = node
val (Defnode (defnode_ty, defnode_edges)) = the (get_defnode node defname)
val (_, defnode_links) = the (Symtab.lookup (defnode_edges, n))
fun update edges none_edges this_edges =
let
val u = merge_labelled_edges edges [(SOME axname, pack_edges this_edges)]
in
if none_edges = [] then
u
else
(NONE, pack_edges none_edges)::u
end
val defnode_links' =
case defnode_links of
((NONE, _) :: edges) => update edges none_edges this_edges
| edges => update edges none_edges this_edges
val defnode_edges' = Symtab.update ((n, (mainref, defnode_links')), defnode_edges)
val defs' = Symtab.update ((defname, Defnode (defnode_ty, defnode_edges')), defs)
in
Symtab.update ((nodename, Node (nodename, nodety, defs', backs, finals)), graph)
end
val graph = foldl (fn ((noderef, newedges),graph) => foldl (fn ((defname, none_edges, this_edges), graph) =>
final_update_backrefs graph noderef defname none_edges this_edges) graph newedges) graph newedges
in
((Define (name, ty, axname, body))::actions, graph)
end
fun finalize (history, graph) (c, ty) =
case Symtab.lookup (graph, c) of
NONE => def_err ("cannot finalize constant "^c^"; it is not declared")
| SOME (Node (noderef, nodety, defs, backs, finals)) =>
let
val ty = checkT ty
val _ = if (not (is_instance_r ty nodety)) then
def_err ("only type instances of the declared constant "^c^" can be finalized")
else ()
val _ = Symtab.exists (fn (def_name, Defnode (def_ty, _)) =>
if can_be_unified_r ty def_ty then
def_err ("cannot finalize constant "^c^"; clash with definition "^def_name)
else
false)
defs
fun update_finals [] = SOME [ty]
| update_finals (final_ty::finals) =
(if is_instance_r ty final_ty then NONE
else
case update_finals finals of
NONE => NONE
| (r as SOME finals) =>
if (is_instance_r final_ty ty) then
r
else
SOME (final_ty :: finals))
in
case update_finals finals of
NONE => (history, graph)
| SOME finals =>
let
val graph = Symtab.update ((noderef, Node(noderef, nodety, defs, backs, finals)), graph)
fun update_backref ((graph, backs), (backrefname, Backref (_, backdefnames))) =
let
fun update_backdef ((graph, defnames), (backdefname, _)) =
let
val (backnode as Node (_, backty, backdefs, backbacks, backfinals)) = getnode graph backrefname
val (Defnode (def_ty, all_edges)) = the (get_defnode backnode backdefname)
val (defnames', all_edges') =
case Symtab.lookup (all_edges, noderef) of
NONE => sys_error "finalize: corrupt backref"
| SOME (_, (NONE, edges)::rest) =>
let
val edges' = List.filter (fn (_, _, beta, _) => not (is_instance_r beta ty)) edges
in
if edges' = [] then
(defnames, Symtab.update ((noderef, (noderef, rest)), all_edges))
else
(Symtab.update ((backdefname, ()), defnames),
Symtab.update ((noderef, (noderef, (NONE, edges')::rest)), all_edges))
end
val defnode' = Defnode (def_ty, all_edges')
val backnode' = Node (backrefname, backty, Symtab.update ((backdefname, defnode'), backdefs),
backbacks, backfinals)
in
(Symtab.update ((backrefname, backnode'), graph), defnames')
end
val (graph', defnames') = Symtab.foldl update_backdef ((graph, Symtab.empty), backdefnames)
in
(graph', if Symtab.is_empty defnames' then backs
else Symtab.update ((backrefname, Backref (backrefname, defnames')), backs))
end
val (graph', backs') = Symtab.foldl update_backref ((graph, Symtab.empty), backs)
val Node (_, _, defs, _, _) = getnode graph' noderef
in
((Finalize (c, ty)) :: history, Symtab.update ((noderef, Node (noderef, nodety, defs, backs', finals)), graph'))
end
end
fun merge' (Declare cty, g) = (declare g cty handle _ => g)
| merge' (Define (name, ty, axname, body), g as (_, graph)) =
(case Symtab.lookup (graph, name) of
NONE => define g (name, ty) axname body
| SOME (Node (_, _, defs, _, _)) =>
(case Symtab.lookup (defs, axname) of
NONE => define g (name, ty) axname body
| SOME _ => g))
| merge' (Finalize finals, g) = finalize g finals
fun merge (actions, _) g = foldr merge' g actions
fun finals (history, graph) =
Symtab.foldl
(fn (finals, (_, Node(name, _, _, _, ftys))) => Symtab.update_new ((name, ftys), finals))
(Symtab.empty, graph)
end;
(*fun tvar name = TVar ((name, 0), [])
val bool = Type ("bool", [])
val int = Type ("int", [])
val alpha = tvar "'a"
val beta = tvar "'b"
val gamma = tvar "'c"
fun pair a b = Type ("pair", [a,b])
val _ = print "make empty"
val g = Defs.empty
val _ = print "declare"
val g = Defs.declare g "M" (alpha --> bool)
val g = Defs.declare g "N" (beta --> bool)
val _ = print "define"
val g = Defs.define g "N" (alpha --> bool) "defN" [("M", alpha --> bool)]
val g = Defs.define g "M" (alpha --> bool) "defM" [("N", int --> alpha)]
val g = Defs.declare g "0" alpha
val g = Defs.define g "0" (pair alpha beta) "zp" [("0", alpha), ("0", beta)]*)