(* Title: Pure/envir.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Environments. The type of a term variable / sort of a type variable is
part of its name. The lookup function must apply type substitutions,
since they may change the identity of a variable.
*)
signature ENVIR =
sig
type tenv
datatype env = Envir of {asol: tenv, iTs: Type.tyenv, maxidx: int}
val type_env: env -> Type.tyenv
val insert_sorts: env -> sort list -> sort list
exception SAME
val genvars: string -> env * typ list -> env * term list
val genvar: string -> env * typ -> env * term
val lookup: env * (indexname * typ) -> term option
val lookup': tenv * (indexname * typ) -> term option
val update: ((indexname * typ) * term) * env -> env
val empty: int -> env
val is_empty: env -> bool
val above: env -> int -> bool
val vupdate: ((indexname * typ) * term) * env -> env
val alist_of: env -> (indexname * (typ * term)) list
val norm_term: env -> term -> term
val norm_term_same: env -> term -> term
val norm_type: Type.tyenv -> typ -> typ
val norm_type_same: Type.tyenv -> typ -> typ
val norm_types_same: Type.tyenv -> typ list -> typ list
val beta_norm: term -> term
val head_norm: env -> term -> term
val eta_contract: term -> term
val beta_eta_contract: term -> term
val fastype: env -> typ list -> term -> typ
val typ_subst_TVars: Type.tyenv -> typ -> typ
val subst_TVars: Type.tyenv -> term -> term
val subst_Vars: tenv -> term -> term
val subst_vars: Type.tyenv * tenv -> term -> term
val expand_atom: typ -> typ * term -> term
val expand_term: (term -> (typ * term) option) -> term -> term
val expand_term_frees: ((string * typ) * term) list -> term -> term
end;
structure Envir : ENVIR =
struct
(*updating can destroy environment in 2 ways!!
(1) variables out of range (2) circular assignments
*)
type tenv = (typ * term) Vartab.table
datatype env = Envir of
{maxidx: int, (*maximum index of vars*)
asol: tenv, (*table of assignments to Vars*)
iTs: Type.tyenv} (*table of assignments to TVars*)
fun type_env (Envir {iTs, ...}) = iTs;
(*NB: type unification may invent new sorts*)
val insert_sorts = Vartab.fold (fn (_, (_, T)) => Sorts.insert_typ T) o type_env;
(*Generate a list of distinct variables.
Increments index to make them distinct from ALL present variables. *)
fun genvars name (Envir{maxidx, asol, iTs}, Ts) : env * term list =
let fun genvs (_, [] : typ list) : term list = []
| genvs (n, [T]) = [ Var((name, maxidx+1), T) ]
| genvs (n, T::Ts) =
Var((name ^ radixstring(26,"a",n), maxidx+1), T)
:: genvs(n+1,Ts)
in (Envir{maxidx=maxidx+1, asol=asol, iTs=iTs}, genvs (0,Ts)) end;
(*Generate a variable.*)
fun genvar name (env,T) : env * term =
let val (env',[v]) = genvars name (env,[T])
in (env',v) end;
fun var_clash ixn T T' = raise TYPE ("Variable " ^
quote (Term.string_of_vname ixn) ^ " has two distinct types",
[T', T], []);
fun gen_lookup f asol (xname, T) =
(case Vartab.lookup asol xname of
NONE => NONE
| SOME (U, t) => if f (T, U) then SOME t
else var_clash xname T U);
(* When dealing with environments produced by matching instead *)
(* of unification, there is no need to chase assigned TVars. *)
(* In this case, we can simply ignore the type substitution *)
(* and use = instead of eq_type. *)
fun lookup' (asol, p) = gen_lookup op = asol p;
fun lookup2 (iTs, asol) p =
if Vartab.is_empty iTs then lookup' (asol, p)
else gen_lookup (Type.eq_type iTs) asol p;
fun lookup (Envir {asol, iTs, ...}, p) = lookup2 (iTs, asol) p;
fun update (((xname, T), t), Envir {maxidx, asol, iTs}) =
Envir{maxidx=maxidx, asol=Vartab.update_new (xname, (T, t)) asol, iTs=iTs};
(*The empty environment. New variables will start with the given index+1.*)
fun empty m = Envir{maxidx=m, asol=Vartab.empty, iTs=Vartab.empty};
(*Test for empty environment*)
fun is_empty (Envir {asol, iTs, ...}) = Vartab.is_empty asol andalso Vartab.is_empty iTs;
(*Determine if the least index updated exceeds lim*)
fun above (Envir {asol, iTs, ...}) lim =
(case Vartab.min_key asol of SOME (_, i) => i > lim | NONE => true) andalso
(case Vartab.min_key iTs of SOME (_, i) => i > lim | NONE => true);
(*Update, checking Var-Var assignments: try to suppress higher indexes*)
fun vupdate ((aU as (a, U), t), env as Envir {iTs, ...}) = case t of
Var (nT as (name', T)) =>
if a = name' then env (*cycle!*)
else if TermOrd.indexname_ord (a, name') = LESS then
(case lookup (env, nT) of (*if already assigned, chase*)
NONE => update ((nT, Var (a, T)), env)
| SOME u => vupdate ((aU, u), env))
else update ((aU, t), env)
| _ => update ((aU, t), env);
(*Convert environment to alist*)
fun alist_of (Envir{asol,...}) = Vartab.dest asol;
(*** Beta normal form for terms (not eta normal form).
Chases variables in env; Does not exploit sharing of variable bindings
Does not check types, so could loop. ***)
(*raised when norm has no effect on a term, to do sharing instead of copying*)
exception SAME;
fun norm_term1 same (asol,t) : term =
let fun norm (Var wT) =
(case lookup' (asol, wT) of
SOME u => (norm u handle SAME => u)
| NONE => raise SAME)
| norm (Abs(a,T,body)) = Abs(a, T, norm body)
| norm (Abs(_,_,body) $ t) = normh(subst_bound (t, body))
| norm (f $ t) =
((case norm f of
Abs(_,_,body) => normh(subst_bound (t, body))
| nf => nf $ (norm t handle SAME => t))
handle SAME => f $ norm t)
| norm _ = raise SAME
and normh t = norm t handle SAME => t
in (if same then norm else normh) t end
fun normT iTs (Type (a, Ts)) = Type (a, normTs iTs Ts)
| normT iTs (TFree _) = raise SAME
| normT iTs (TVar vS) = (case Type.lookup iTs vS of
SOME U => normTh iTs U
| NONE => raise SAME)
and normTh iTs T = ((normT iTs T) handle SAME => T)
and normTs iTs [] = raise SAME
| normTs iTs (T :: Ts) =
((normT iTs T :: (normTs iTs Ts handle SAME => Ts))
handle SAME => T :: normTs iTs Ts);
fun norm_term2 same (asol, iTs, t) : term =
let fun norm (Const (a, T)) = Const(a, normT iTs T)
| norm (Free (a, T)) = Free(a, normT iTs T)
| norm (Var (w, T)) =
(case lookup2 (iTs, asol) (w, T) of
SOME u => normh u
| NONE => Var(w, normT iTs T))
| norm (Abs (a, T, body)) =
(Abs (a, normT iTs T, normh body) handle SAME => Abs (a, T, norm body))
| norm (Abs(_, _, body) $ t) = normh (subst_bound (t, body))
| norm (f $ t) =
((case norm f of
Abs(_, _, body) => normh (subst_bound (t, body))
| nf => nf $ normh t)
handle SAME => f $ norm t)
| norm _ = raise SAME
and normh t = (norm t) handle SAME => t
in (if same then norm else normh) t end;
fun gen_norm_term same (env as Envir{asol,iTs,...}) t : term =
if Vartab.is_empty iTs then norm_term1 same (asol, t)
else norm_term2 same (asol, iTs, t);
val norm_term = gen_norm_term false;
val norm_term_same = gen_norm_term true;
fun beta_norm t = if Term.has_abs t then norm_term (empty 0) t else t;
fun norm_type iTs = normTh iTs;
fun norm_type_same iTs =
if Vartab.is_empty iTs then raise SAME else normT iTs;
fun norm_types_same iTs =
if Vartab.is_empty iTs then raise SAME else normTs iTs;
(*Put a term into head normal form for unification.*)
fun head_norm env t =
let
fun hnorm (Var vT) = (case lookup (env, vT) of
SOME u => head_norm env u
| NONE => raise SAME)
| hnorm (Abs (a, T, body)) = Abs (a, T, hnorm body)
| hnorm (Abs (_, _, body) $ t) =
head_norm env (subst_bound (t, body))
| hnorm (f $ t) = (case hnorm f of
Abs (_, _, body) => head_norm env (subst_bound (t, body))
| nf => nf $ t)
| hnorm _ = raise SAME
in hnorm t handle SAME => t end;
(*Eta-contract a term (fully)*)
local
fun decr lev (Bound i) = if i >= lev then Bound (i - 1) else raise SAME
| decr lev (Abs (a, T, body)) = Abs (a, T, decr (lev + 1) body)
| decr lev (t $ u) = (decr lev t $ decrh lev u handle SAME => t $ decr lev u)
| decr _ _ = raise SAME
and decrh lev t = (decr lev t handle SAME => t);
fun eta (Abs (a, T, body)) =
((case eta body of
body' as (f $ Bound 0) =>
if loose_bvar1 (f, 0) then Abs (a, T, body')
else decrh 0 f
| body' => Abs (a, T, body')) handle SAME =>
(case body of
f $ Bound 0 =>
if loose_bvar1 (f, 0) then raise SAME
else decrh 0 f
| _ => raise SAME))
| eta (t $ u) = (eta t $ etah u handle SAME => t $ eta u)
| eta _ = raise SAME
and etah t = (eta t handle SAME => t);
in
fun eta_contract t =
if Term.has_abs t then etah t else t;
val beta_eta_contract = eta_contract o beta_norm;
end;
(*finds type of term without checking that combinations are consistent
Ts holds types of bound variables*)
fun fastype (Envir {iTs, ...}) =
let val funerr = "fastype: expected function type";
fun fast Ts (f $ u) =
(case fast Ts f of
Type ("fun", [_, T]) => T
| TVar ixnS =>
(case Type.lookup iTs ixnS of
SOME (Type ("fun", [_, T])) => T
| _ => raise TERM (funerr, [f $ u]))
| _ => raise TERM (funerr, [f $ u]))
| fast Ts (Const (_, T)) = T
| fast Ts (Free (_, T)) = T
| fast Ts (Bound i) =
(List.nth (Ts, i)
handle Subscript => raise TERM ("fastype: Bound", [Bound i]))
| fast Ts (Var (_, T)) = T
| fast Ts (Abs (_, T, u)) = T --> fast (T :: Ts) u
in fast end;
(*Substitute for type Vars in a type*)
fun typ_subst_TVars iTs T = if Vartab.is_empty iTs then T else
let fun subst(Type(a, Ts)) = Type(a, map subst Ts)
| subst(T as TFree _) = T
| subst(T as TVar ixnS) =
(case Type.lookup iTs ixnS of NONE => T | SOME(U) => U)
in subst T end;
(*Substitute for type Vars in a term*)
val subst_TVars = map_types o typ_subst_TVars;
(*Substitute for Vars in a term *)
fun subst_Vars itms t = if Vartab.is_empty itms then t else
let fun subst (v as Var ixnT) = the_default v (lookup' (itms, ixnT))
| subst (Abs (a, T, t)) = Abs (a, T, subst t)
| subst (f $ t) = subst f $ subst t
| subst t = t
in subst t end;
(*Substitute for type/term Vars in a term *)
fun subst_vars (iTs, itms) =
if Vartab.is_empty iTs then subst_Vars itms else
let fun subst (Const (a, T)) = Const(a, typ_subst_TVars iTs T)
| subst (Free (a, T)) = Free (a, typ_subst_TVars iTs T)
| subst (Var (ixn, T)) = (case lookup' (itms, (ixn, T)) of
NONE => Var (ixn, typ_subst_TVars iTs T)
| SOME t => t)
| subst (b as Bound _) = b
| subst (Abs (a, T, t)) = Abs(a, typ_subst_TVars iTs T, subst t)
| subst (f $ t) = subst f $ subst t
in subst end;
(* expand defined atoms -- with local beta reduction *)
fun expand_atom T (U, u) =
subst_TVars (Type.raw_match (U, T) Vartab.empty) u
handle Type.TYPE_MATCH => raise TYPE ("expand_atom: ill-typed replacement", [T, U], [u]);
fun expand_term get =
let
fun expand tm =
let
val (head, args) = Term.strip_comb tm;
val args' = map expand args;
fun comb head' = Term.list_comb (head', args');
in
(case head of
Abs (x, T, t) => comb (Abs (x, T, expand t))
| _ =>
(case get head of
SOME def => Term.betapplys (expand_atom (Term.fastype_of head) def, args')
| NONE => comb head)
| _ => comb head)
end;
in expand end;
fun expand_term_frees defs =
let
val eqs = map (fn ((x, U), u) => (x, (U, u))) defs;
val get = fn Free (x, _) => AList.lookup (op =) eqs x | _ => NONE;
in expand_term get end;
end;