summaryrefslogtreecommitdiff
path: root/lib/compiler/src/v3_life.erl
blob: be3ade47ff9fe75524f1dbf71e0f16e90f841cd8 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 1999-2016. All Rights Reserved.
%%
%% Licensed under the Apache License, Version 2.0 (the "License");
%% you may not use this file except in compliance with the License.
%% You may obtain a copy of the License at
%%
%%     http://www.apache.org/licenses/LICENSE-2.0
%%
%% Unless required by applicable law or agreed to in writing, software
%% distributed under the License is distributed on an "AS IS" BASIS,
%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
%% See the License for the specific language governing permissions and
%% limitations under the License.
%%
%% %CopyrightEnd%
%%
%% Purpose : Convert annotated kernel expressions to annotated beam format.

%% This module creates beam format annotated with variable lifetime
%% information.  Each thing is given an index and for each variable we
%% store the first and last index for its occurrence.  The variable
%% database, VDB, attached to each thing is only relevant internally
%% for that thing.
%%
%% For nested things like matches the numbering continues locally and
%% the VDB for that thing refers to the variable usage within that
%% thing.  Variables which live through a such a thing are internally
%% given a very large last index.  Internally the indexes continue
%% after the index of that thing.  This creates no problems as the
%% internal variable info never escapes and externally we only see
%% variable which are alive both before or after.
%%
%% This means that variables never "escape" from a thing and the only
%% way to get values from a thing is to "return" them, with 'break' or
%% 'return'.  Externally these values become the return values of the
%% thing.  This is no real limitation as most nested things have
%% multiple threads so working out a common best variable usage is
%% difficult.

-module(v3_life).

-export([module/2]).

-export([vdb_find/2]).

-import(lists, [member/2,map/2,reverse/1,sort/1]).
-import(ordsets, [add_element/2,intersection/2,union/2]).

-include("v3_kernel.hrl").
-include("v3_life.hrl").

-type fa() :: {atom(),arity()}.

%% These are not defined in v3_kernel.hrl.
get_kanno(Kthing) -> element(2, Kthing).
%%set_kanno(Kthing, Anno) -> setelement(2, Kthing, Anno).

-spec module(#k_mdef{}, [compile:option()]) ->
                    {'ok',{module(),[fa()],[_],[_]}}.

module(#k_mdef{name=M,exports=Es,attributes=As,body=Fs0}, _Opts) ->
    Fs1 = functions(Fs0, []),
    {ok,{M,Es,As,Fs1}}.

functions([F|Fs], Acc) ->
    functions(Fs, [function(F)|Acc]);
functions([], Acc) -> reverse(Acc).

%% function(Kfunc) -> Func.

function(#k_fdef{anno=#k{a=Anno},func=F,arity=Ar,vars=Vs,body=Kb}) ->
    try
	As = var_list(Vs),
	Vdb0 = init_vars(As),
	%% Force a top-level match!
	B0 = case Kb of
		 #k_match{} -> Kb;
		 _ ->
		     Ka = get_kanno(Kb),
		     #k_match{anno=#k{us=Ka#k.us,ns=[],a=Ka#k.a},
			      vars=Vs,body=Kb,ret=[]}
	     end,
	{B1,_,Vdb1} = body(B0, 1, Vdb0),
	{function,F,Ar,As,B1,Vdb1,Anno}
    catch
	Class:Error ->
	    Stack = erlang:get_stacktrace(),
	    io:fwrite("Function: ~w/~w\n", [F,Ar]),
	    erlang:raise(Class, Error, Stack)
    end.

%% body(Kbody, I, Vdb) -> {[Expr],MaxI,Vdb}.
%%  Handle a body.

body(#k_seq{arg=Ke,body=Kb}, I, Vdb0) ->
    %%ok = io:fwrite("life ~w:~p~n", [?LINE,{Ke,I,Vdb0}]),
    A = get_kanno(Ke),
    Vdb1 = use_vars(union(A#k.us, A#k.ns), I, Vdb0),
    {Es,MaxI,Vdb2} = body(Kb, I+1, Vdb1),
    E = expr(Ke, I, Vdb2),
    {[E|Es],MaxI,Vdb2};
body(Ke, I, Vdb0) ->
    %%ok = io:fwrite("life ~w:~p~n", [?LINE,{Ke,I,Vdb0}]),
    A = get_kanno(Ke),
    Vdb1 = use_vars(union(A#k.us, A#k.ns), I, Vdb0),
    E = expr(Ke, I, Vdb1),
    {[E],I,Vdb1}.

%% protected(Kprotected, I, Vdb) -> Protected.
%%  Only used in guards.

protected(#k_protected{anno=A,arg=Ts,ret=Rs}, I, Vdb) ->
    %% Lock variables that are alive before try and used afterwards.
    %% Don't lock variables that are only used inside the protected
    %% expression.
    Pdb0 = vdb_sub(I, I+1, Vdb),
    {T,MaxI,Pdb1} = body(Ts, I+1, Pdb0),
    Pdb2 = use_vars(A#k.ns, MaxI+1, Pdb1),	%Save "return" values
    #l{ke={protected,T,var_list(Rs)},i=I,a=A#k.a,vdb=Pdb2}.

%% expr(Kexpr, I, Vdb) -> Expr.

expr(#k_test{anno=A,op=Op,args=As,inverted=Inverted}, I, _Vdb) ->
    #l{ke={test,test_op(Op),atomic_list(As),Inverted},i=I,a=A#k.a};
expr(#k_call{anno=A,op=Op,args=As,ret=Rs}, I, _Vdb) ->
    #l{ke={call,call_op(Op),atomic_list(As),var_list(Rs)},i=I,a=A#k.a};
expr(#k_enter{anno=A,op=Op,args=As}, I, _Vdb) ->
    #l{ke={enter,call_op(Op),atomic_list(As)},i=I,a=A#k.a};
expr(#k_bif{anno=A,op=Op,args=As,ret=Rs}, I, _Vdb) ->
    Bif = k_bif(A, Op, As, Rs),
    #l{ke=Bif,i=I,a=A#k.a};
expr(#k_match{anno=A,body=Kb,ret=Rs}, I, Vdb) ->
    %% Work out imported variables which need to be locked.
    Mdb = vdb_sub(I, I+1, Vdb),
    M = match(Kb, A#k.us, I+1, [], Mdb),
    #l{ke={match,M,var_list(Rs)},i=I,vdb=use_vars(A#k.us, I+1, Mdb),a=A#k.a};
expr(#k_guard_match{anno=A,body=Kb,ret=Rs}, I, Vdb) ->
    %% Work out imported variables which need to be locked.
    Mdb = vdb_sub(I, I+1, Vdb),
    M = match(Kb, A#k.us, I+1, [], Mdb),
    #l{ke={guard_match,M,var_list(Rs)},i=I,vdb=use_vars(A#k.us, I+1, Mdb),a=A#k.a};
expr(#k_try{}=Try, I, Vdb) ->
    body_try(Try, I, Vdb);
expr(#k_protected{}=Protected, I, Vdb) ->
    protected(Protected, I, Vdb);
expr(#k_try_enter{anno=A,arg=Ka,vars=Vs,body=Kb,evars=Evs,handler=Kh}, I, Vdb) ->
    %% Lock variables that are alive before the catch and used afterwards.
    %% Don't lock variables that are only used inside the try.
    Tdb0 = vdb_sub(I, I+1, Vdb),
    %% This is the tricky bit. Lock variables in Arg that are used in
    %% the body and handler. Add try tag 'variable'.
    Ab = get_kanno(Kb),
    Ah = get_kanno(Kh),
    Tdb1 = use_vars(union(Ab#k.us, Ah#k.us), I+3, Tdb0),
    Tdb2 = vdb_sub(I, I+2, Tdb1),
    Vnames = fun (Kvar) -> Kvar#k_var.name end,	%Get the variable names
    {Aes,_,Adb} = body(Ka, I+2, add_var({catch_tag,I+1}, I+1, 1000000, Tdb2)),
    {Bes,_,Bdb} = body(Kb, I+4, new_vars(sort(map(Vnames, Vs)), I+3, Tdb2)),
    {Hes,_,Hdb} = body(Kh, I+4, new_vars(sort(map(Vnames, Evs)), I+3, Tdb2)),
    #l{ke={try_enter,#l{ke={block,Aes},i=I+1,vdb=Adb,a=[]},
	   var_list(Vs),#l{ke={block,Bes},i=I+3,vdb=Bdb,a=[]},
	   var_list(Evs),#l{ke={block,Hes},i=I+3,vdb=Hdb,a=[]}},
       i=I,vdb=Tdb1,a=A#k.a};
expr(#k_catch{anno=A,body=Kb,ret=[R]}, I, Vdb) ->
    %% Lock variables that are alive before the catch and used afterwards.
    %% Don't lock variables that are only used inside the catch.
    %% Add catch tag 'variable'.
    Cdb0 = vdb_sub(I, I+1, Vdb),
    {Es,_,Cdb1} = body(Kb, I+1, add_var({catch_tag,I}, I, locked, Cdb0)),
    #l{ke={'catch',Es,variable(R)},i=I,vdb=Cdb1,a=A#k.a};
expr(#k_receive{anno=A,var=V,body=Kb,timeout=T,action=Ka,ret=Rs}, I, Vdb) ->
    %% Work out imported variables which need to be locked.
    Rdb = vdb_sub(I, I+1, Vdb),
    M = match(Kb, add_element(V#k_var.name, A#k.us), I+1, [],
	      new_vars([V#k_var.name], I, Rdb)),
    {Tes,_,Adb} = body(Ka, I+1, Rdb),
    #l{ke={receive_loop,atomic(T),variable(V),M,
	   #l{ke=Tes,i=I+1,vdb=Adb,a=[]},var_list(Rs)},
       i=I,vdb=use_vars(A#k.us, I+1, Vdb),a=A#k.a};
expr(#k_receive_accept{anno=A}, I, _Vdb) ->
    #l{ke=receive_accept,i=I,a=A#k.a};
expr(#k_receive_next{anno=A}, I, _Vdb) ->
    #l{ke=receive_next,i=I,a=A#k.a};
expr(#k_put{anno=A,arg=Arg,ret=Rs}, I, _Vdb) ->
    #l{ke={set,var_list(Rs),literal(Arg, [])},i=I,a=A#k.a};
expr(#k_break{anno=A,args=As}, I, _Vdb) ->
    #l{ke={break,atomic_list(As)},i=I,a=A#k.a};
expr(#k_guard_break{anno=A,args=As}, I, Vdb) ->
    Locked = [V || {V,_,_} <- Vdb],
    #l{ke={guard_break,atomic_list(As),Locked},i=I,a=A#k.a};
expr(#k_return{anno=A,args=As}, I, _Vdb) ->
    #l{ke={return,atomic_list(As)},i=I,a=A#k.a}.

body_try(#k_try{anno=A,arg=Ka,vars=Vs,body=Kb,evars=Evs,handler=Kh,ret=Rs},
	 I, Vdb) ->
    %% Lock variables that are alive before the catch and used afterwards.
    %% Don't lock variables that are only used inside the try.
    Tdb0 = vdb_sub(I, I+1, Vdb),
    %% This is the tricky bit. Lock variables in Arg that are used in
    %% the body and handler. Add try tag 'variable'.
    Ab = get_kanno(Kb),
    Ah = get_kanno(Kh),
    Tdb1 = use_vars(union(Ab#k.us, Ah#k.us), I+3, Tdb0),
    Tdb2 = vdb_sub(I, I+2, Tdb1),
    Vnames = fun (Kvar) -> Kvar#k_var.name end,	%Get the variable names
    {Aes,_,Adb} = body(Ka, I+2, add_var({catch_tag,I+1}, I+1, locked, Tdb2)),
    {Bes,_,Bdb} = body(Kb, I+4, new_vars(sort(map(Vnames, Vs)), I+3, Tdb2)),
    {Hes,_,Hdb} = body(Kh, I+4, new_vars(sort(map(Vnames, Evs)), I+3, Tdb2)),
    #l{ke={'try',#l{ke={block,Aes},i=I+1,vdb=Adb,a=[]},
	   var_list(Vs),#l{ke={block,Bes},i=I+3,vdb=Bdb,a=[]},
	   var_list(Evs),#l{ke={block,Hes},i=I+3,vdb=Hdb,a=[]},
	   var_list(Rs)},
       i=I,vdb=Tdb1,a=A#k.a}.

%% call_op(Op) -> Op.
%% test_op(Op) -> Op.
%%  Do any necessary name translations here to munge into beam format.

call_op(#k_local{name=N}) -> N; 
call_op(#k_remote{mod=M,name=N}) -> {remote,atomic(M),atomic(N)};
call_op(Other) -> variable(Other).

test_op(#k_remote{mod=#k_atom{val=erlang},name=#k_atom{val=N}}) -> N.

%% k_bif(Anno, Op, [Arg], [Ret], Vdb) -> Expr.
%%  Build bifs.

k_bif(_A, #k_internal{name=Name}, As, Rs) ->
    {internal,Name,atomic_list(As),var_list(Rs)};
k_bif(_A, #k_remote{mod=#k_atom{val=erlang},name=#k_atom{val=Name}}, As, Rs) ->
    Ar = length(As),
    case is_gc_bif(Name, Ar) of
	false ->
	    {bif,Name,atomic_list(As),var_list(Rs)};
	true ->
	    {gc_bif,Name,atomic_list(As),var_list(Rs)}
    end.

%% match(Kexpr, [LockVar], I, Vdb) -> Expr.
%%  Convert match tree to old format.

match(#k_alt{anno=A,first=Kf,then=Kt}, Ls, I, Ctxt, Vdb0) ->
    Vdb1 = use_vars(union(A#k.us, Ls), I, Vdb0),
    F = match(Kf, Ls, I+1, Ctxt, Vdb1),
    T = match(Kt, Ls, I+1, Ctxt, Vdb1),
    #l{ke={alt,F,T},i=I,vdb=Vdb1,a=A#k.a};
match(#k_select{anno=A,var=V,types=Kts}, Ls0, I, Ctxt, Vdb0) ->
    Vanno = get_kanno(V),
    Ls1 = case member(no_usage, Vanno) of
	      false -> add_element(V#k_var.name, Ls0);
	      true -> Ls0
	  end,
    Anno = case member(reuse_for_context, Vanno) of
	       true -> [reuse_for_context|A#k.a];
	       false -> A#k.a
	   end,
    Vdb1 = use_vars(union(A#k.us, Ls1), I, Vdb0),
    Ts = [type_clause(Tc, Ls1, I+1, Ctxt, Vdb1) || Tc <- Kts],
    #l{ke={select,literal(V, Ctxt),Ts},i=I,vdb=Vdb1,a=Anno};
match(#k_guard{anno=A,clauses=Kcs}, Ls, I, Ctxt, Vdb0) ->
    Vdb1 = use_vars(union(A#k.us, Ls), I, Vdb0),
    Cs = [guard_clause(G, Ls, I+1, Ctxt, Vdb1) || G <- Kcs],
    #l{ke={guard,Cs},i=I,vdb=Vdb1,a=A#k.a};
match(Other, Ls, I, _Ctxt, Vdb0) ->
    Vdb1 = use_vars(Ls, I, Vdb0),
    {B,_,Vdb2} = body(Other, I+1, Vdb1),
    #l{ke={block,B},i=I,vdb=Vdb2,a=[]}.

type_clause(#k_type_clause{anno=A,type=T,values=Kvs}, Ls, I, Ctxt, Vdb0) ->
    %%ok = io:format("life ~w: ~p~n", [?LINE,{T,Kvs}]),
    Vdb1 = use_vars(union(A#k.us, Ls), I+1, Vdb0),
    Vs = [val_clause(Vc, Ls, I+1, Ctxt, Vdb1) || Vc <- Kvs],
    #l{ke={type_clause,type(T),Vs},i=I,vdb=Vdb1,a=A#k.a}.

val_clause(#k_val_clause{anno=A,val=V,body=Kb}, Ls0, I, Ctxt0, Vdb0) ->
    New = (get_kanno(V))#k.ns,
    Bus = (get_kanno(Kb))#k.us,
    %%ok = io:format("Ls0 = ~p, Used=~p\n  New=~p, Bus=~p\n", [Ls0,Used,New,Bus]),
    Ls1 = union(intersection(New, Bus), Ls0),	%Lock for safety
    Vdb1 = use_vars(union(A#k.us, Ls1), I+1, new_vars(New, I, Vdb0)),
    Ctxt = case V of
	       #k_binary{segs=#k_var{name=C0}} -> C0;
	       _ -> Ctxt0
	   end,
    B = match(Kb, Ls1, I+1, Ctxt, Vdb1),
    #l{ke={val_clause,literal(V, Ctxt),B},i=I,vdb=use_vars(Bus, I+1, Vdb1),a=A#k.a}.

guard_clause(#k_guard_clause{anno=A,guard=Kg,body=Kb}, Ls, I, Ctxt, Vdb0) ->
    Vdb1 = use_vars(union(A#k.us, Ls), I+2, Vdb0),
    Gdb = vdb_sub(I+1, I+2, Vdb1),
    G = protected(Kg, I+1, Gdb),
    B = match(Kb, Ls, I+2, Ctxt, Vdb1),
    #l{ke={guard_clause,G,B},
       i=I,vdb=use_vars((get_kanno(Kg))#k.us, I+2, Vdb1),
       a=A#k.a}.

%% type(Ktype) -> Type.

type(k_literal) -> literal;
type(k_int) -> integer;
%%type(k_char) -> integer;			%Hhhmmm???
type(k_float) -> float;
type(k_atom) -> atom;
type(k_nil) -> nil;
type(k_cons) -> cons;
type(k_tuple) -> tuple;
type(k_binary) -> binary;
type(k_bin_seg) -> bin_seg;
type(k_bin_int) -> bin_int;
type(k_bin_end) -> bin_end;
type(k_map) -> map.

%% variable(Klit) -> Lit.
%% var_list([Klit]) -> [Lit].

variable(#k_var{name=N}) -> {var,N}.

var_list(Ks) -> [variable(K) || K <- Ks].

%% atomic(Klit) -> Lit.
%% atomic_list([Klit]) -> [Lit].

atomic(#k_literal{val=V}) -> {literal,V};
atomic(#k_var{name=N}) -> {var,N};
atomic(#k_int{val=I}) -> {integer,I};
atomic(#k_float{val=F}) -> {float,F};
atomic(#k_atom{val=N}) -> {atom,N};
%%atomic(#k_char{val=C}) -> {char,C};
atomic(#k_nil{}) -> nil.

atomic_list(Ks) -> [atomic(K) || K <- Ks].

%% literal(Klit) -> Lit.
%% literal_list([Klit]) -> [Lit].

literal(#k_var{name=N}, _) -> {var,N};
literal(#k_literal{val=I}, _) -> {literal,I};
literal(#k_int{val=I}, _) -> {integer,I};
literal(#k_float{val=F}, _) -> {float,F};
literal(#k_atom{val=N}, _) -> {atom,N};
%%literal(#k_char{val=C}, _) -> {char,C};
literal(#k_nil{}, _) -> nil;
literal(#k_cons{hd=H,tl=T}, Ctxt) ->
    {cons,[literal(H, Ctxt),literal(T, Ctxt)]};
literal(#k_binary{segs=V}, Ctxt) ->
    {binary,literal(V, Ctxt)};
literal(#k_bin_seg{size=S,unit=U,type=T,flags=Fs,seg=Seg,next=[]}, Ctxt) ->
    %% Only occurs in patterns.
    {bin_seg,Ctxt,literal(S, Ctxt),U,T,Fs,[literal(Seg, Ctxt)]};
literal(#k_bin_seg{size=S,unit=U,type=T,flags=Fs,seg=Seg,next=N}, Ctxt) ->
    {bin_seg,Ctxt,literal(S, Ctxt),U,T,Fs,
     [literal(Seg, Ctxt),literal(N, Ctxt)]};
literal(#k_bin_int{size=S,unit=U,flags=Fs,val=Int,next=N}, Ctxt) ->
    %% Only occurs in patterns.
    {bin_int,Ctxt,literal(S, Ctxt),U,Fs,Int,
     [literal(N, Ctxt)]};
literal(#k_bin_end{}, Ctxt) ->
    {bin_end,Ctxt};
literal(#k_tuple{es=Es}, Ctxt) ->
    {tuple,literal_list(Es, Ctxt)};
literal(#k_map{op=Op,var=Var,es=Es0}, Ctxt) ->
    {map,Op,literal(Var, Ctxt),literal_list(Es0, Ctxt)};
literal(#k_map_pair{key=K,val=V}, Ctxt) ->
    {map_pair,literal(K, Ctxt),literal(V, Ctxt)}.

literal_list(Ks, Ctxt) ->
    [literal(K, Ctxt) || K <- Ks].


%% is_gc_bif(Name, Arity) -> true|false
%%  Determines whether the BIF Name/Arity might do a GC.

is_gc_bif(hd, 1) -> false;
is_gc_bif(tl, 1) -> false;
is_gc_bif(self, 0) -> false;
is_gc_bif(node, 0) -> false;
is_gc_bif(node, 1) -> false;
is_gc_bif(element, 2) -> false;
is_gc_bif(get, 1) -> false;
is_gc_bif(tuple_size, 1) -> false;
is_gc_bif(Bif, Arity) ->
    not (erl_internal:bool_op(Bif, Arity) orelse
	 erl_internal:new_type_test(Bif, Arity) orelse
	 erl_internal:comp_op(Bif, Arity)).

%% Keep track of life time for variables.
%%
%% init_vars([{var,VarName}]) -> Vdb.
%% new_vars([VarName], I, Vdb) -> Vdb.
%% use_vars([VarName], I, Vdb) -> Vdb.
%% add_var(VarName, F, L, Vdb) -> Vdb.
%%
%% The list of variable names for new_vars/3 and use_vars/3
%% must be sorted.

init_vars(Vs) ->
    vdb_new(Vs).

new_vars([], _, Vdb) -> Vdb;
new_vars([V], I, Vdb) -> vdb_store_new(V, {V,I,I}, Vdb);
new_vars(Vs, I, Vdb) -> vdb_update_vars(Vs, Vdb, I).

use_vars([], _, Vdb) ->
    Vdb;
use_vars([V], I, Vdb) ->
    case vdb_find(V, Vdb) of
	{V,F,L} when I > L -> vdb_update(V, {V,F,I}, Vdb);
	{V,_,_} -> Vdb;
	error -> vdb_store_new(V, {V,I,I}, Vdb)
    end;
use_vars(Vs, I, Vdb) -> vdb_update_vars(Vs, Vdb, I).

add_var(V, F, L, Vdb) ->
    vdb_store_new(V, {V,F,L}, Vdb).

%% vdb

vdb_new(Vs) ->
    sort([{V,0,0} || {var,V} <- Vs]).

-type var() :: atom().

-spec vdb_find(var(), [vdb_entry()]) -> 'error' | vdb_entry().

vdb_find(V, Vdb) ->
    case lists:keyfind(V, 1, Vdb) of
	false -> error;
	Vd -> Vd
    end.

vdb_update(V, Update, [{V,_,_}|Vdb]) ->
    [Update|Vdb];
vdb_update(V, Update, [Vd|Vdb]) ->
    [Vd|vdb_update(V, Update, Vdb)].

vdb_store_new(V, New, [{V1,_,_}=Vd|Vdb]) when V > V1 ->
    [Vd|vdb_store_new(V, New, Vdb)];
vdb_store_new(V, New, [{V1,_,_}|_]=Vdb) when V < V1 ->
    [New|Vdb];
vdb_store_new(_, New, []) -> [New].

vdb_update_vars([V|_]=Vs, [{V1,_,_}=Vd|Vdb], I) when V > V1 ->
    [Vd|vdb_update_vars(Vs, Vdb, I)];
vdb_update_vars([V|Vs], [{V1,_,_}|_]=Vdb, I) when V < V1 ->
    %% New variable.
    [{V,I,I}|vdb_update_vars(Vs, Vdb, I)];
vdb_update_vars([V|Vs], [{_,F,L}=Vd|Vdb], I) ->
    %% Existing variable.
    if
	I > L -> [{V,F,I}|vdb_update_vars(Vs, Vdb, I)];
	true ->  [Vd|vdb_update_vars(Vs, Vdb, I)]
    end;
vdb_update_vars([V|Vs], [], I) ->
    %% New variable.
    [{V,I,I}|vdb_update_vars(Vs, [], I)];
vdb_update_vars([], Vdb, _) -> Vdb.

%% vdb_sub(Min, Max, Vdb) -> Vdb.
%%  Extract variables which are used before and after Min.  Lock
%%  variables alive after Max.

vdb_sub(Min, Max, Vdb) ->
    [ if L >= Max -> {V,F,locked};
	 true -> Vd
      end || {V,F,L}=Vd <- Vdb, F < Min, L >= Min ].