-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathNorm.v
1072 lines (914 loc) · 36.5 KB
/
Norm.v
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
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(** * Norm: Normalization of STLC *)
(* Chapter maintained by Andrew Tolmach *)
(* (Based on TAPL Ch. 12.) *)
Require Export Smallstep.
Hint Constructors multi.
(**
(This chapter is optional.)
In this chapter, we consider another fundamental theoretical property
of the simply typed lambda-calculus: the fact that the evaluation of a
well-typed program is guaranteed to halt in a finite number of
steps---i.e., every well-typed term is _normalizable_.
Unlike the type-safety properties we have considered so far, the
normalization property does not extend to full-blown programming
languages, because these languages nearly always extend the simply
typed lambda-calculus with constructs, such as general recursion
(as we discussed in the MoreStlc chapter) or recursive types, that can
be used to write nonterminating programs. However, the issue of
normalization reappears at the level of _types_ when we consider the
metatheory of polymorphic versions of the lambda calculus such as
F_omega: in this system, the language of types effectively contains a
copy of the simply typed lambda-calculus, and the termination of the
typechecking algorithm will hinge on the fact that a ``normalization''
operation on type expressions is guaranteed to terminate.
Another reason for studying normalization proofs is that they are some
of the most beautiful---and mind-blowing---mathematics to be found in
the type theory literature, often (as here) involving the fundamental
proof technique of _logical relations_.
The calculus we shall consider here is the simply typed
lambda-calculus over a single base type [bool] and with pairs. We'll
give full details of the development for the basic lambda-calculus
terms treating [bool] as an uninterpreted base type, and leave the
extension to the boolean operators and pairs to the reader. Even for
the base calculus, normalization is not entirely trivial to prove,
since each reduction of a term can duplicate redexes in subterms. *)
(** **** Exercise: 1 star *)
(** Where do we fail if we attempt to prove normalization by a
straightforward induction on the size of a well-typed term? *)
(* FILL IN HERE *)
(** [] *)
(* ###################################################################### *)
(** * Language *)
(** We begin by repeating the relevant language definition, which is
similar to those in the MoreStlc chapter, and supporting results
including type preservation and step determinism. (We won't need
progress.) You may just wish to skip down to the Normalization
section... *)
(* ###################################################################### *)
(** *** Syntax and Operational Semantics *)
Inductive ty : Type :=
| TBool : ty
| TArrow : ty -> ty -> ty
| TProd : ty -> ty -> ty
.
Inductive tm : Type :=
(* pure STLC *)
| tvar : id -> tm
| tapp : tm -> tm -> tm
| tabs : id -> ty -> tm -> tm
(* pairs *)
| tpair : tm -> tm -> tm
| tfst : tm -> tm
| tsnd : tm -> tm
(* booleans *)
| ttrue : tm
| tfalse : tm
| tif : tm -> tm -> tm -> tm.
(* i.e., [if t0 then t1 else t2] *)
(* ###################################################################### *)
(** *** Substitution *)
Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
match t with
| tvar y => if eq_id_dec x y then s else t
| tabs y T t1 => tabs y T (if eq_id_dec x y then t1 else (subst x s t1))
| tapp t1 t2 => tapp (subst x s t1) (subst x s t2)
| tpair t1 t2 => tpair (subst x s t1) (subst x s t2)
| tfst t1 => tfst (subst x s t1)
| tsnd t1 => tsnd (subst x s t1)
| ttrue => ttrue
| tfalse => tfalse
| tif t0 t1 t2 => tif (subst x s t0) (subst x s t1) (subst x s t2)
end.
Notation "'[' x ':=' s ']' t" := (subst x s t) (at level 20).
(* ###################################################################### *)
(** *** Reduction *)
Inductive value : tm -> Prop :=
| v_abs : forall x T11 t12,
value (tabs x T11 t12)
| v_pair : forall v1 v2,
value v1 ->
value v2 ->
value (tpair v1 v2)
| v_true : value ttrue
| v_false : value tfalse
.
Hint Constructors value.
Reserved Notation "t1 '==>' t2" (at level 40).
Inductive step : tm -> tm -> Prop :=
| ST_AppAbs : forall x T11 t12 v2,
value v2 ->
(tapp (tabs x T11 t12) v2) ==> [x:=v2]t12
| ST_App1 : forall t1 t1' t2,
t1 ==> t1' ->
(tapp t1 t2) ==> (tapp t1' t2)
| ST_App2 : forall v1 t2 t2',
value v1 ->
t2 ==> t2' ->
(tapp v1 t2) ==> (tapp v1 t2')
(* pairs *)
| ST_Pair1 : forall t1 t1' t2,
t1 ==> t1' ->
(tpair t1 t2) ==> (tpair t1' t2)
| ST_Pair2 : forall v1 t2 t2',
value v1 ->
t2 ==> t2' ->
(tpair v1 t2) ==> (tpair v1 t2')
| ST_Fst : forall t1 t1',
t1 ==> t1' ->
(tfst t1) ==> (tfst t1')
| ST_FstPair : forall v1 v2,
value v1 ->
value v2 ->
(tfst (tpair v1 v2)) ==> v1
| ST_Snd : forall t1 t1',
t1 ==> t1' ->
(tsnd t1) ==> (tsnd t1')
| ST_SndPair : forall v1 v2,
value v1 ->
value v2 ->
(tsnd (tpair v1 v2)) ==> v2
(* booleans *)
| ST_IfTrue : forall t1 t2,
(tif ttrue t1 t2) ==> t1
| ST_IfFalse : forall t1 t2,
(tif tfalse t1 t2) ==> t2
| ST_If : forall t0 t0' t1 t2,
t0 ==> t0' ->
(tif t0 t1 t2) ==> (tif t0' t1 t2)
where "t1 '==>' t2" := (step t1 t2).
Notation multistep := (multi step).
Notation "t1 '==>*' t2" := (multistep t1 t2) (at level 40).
Hint Constructors step.
Notation step_normal_form := (normal_form step).
Lemma value__normal : forall t, value t -> step_normal_form t.
Proof with eauto.
intros t H; induction H; intros [t' ST]; inversion ST...
Qed.
(* ###################################################################### *)
(** *** Typing *)
Definition context := partial_map ty.
Inductive has_type : context -> tm -> ty -> Prop :=
(* Typing rules for proper terms *)
| T_Var : forall Gamma x T,
Gamma x = Some T ->
has_type Gamma (tvar x) T
| T_Abs : forall Gamma x T11 T12 t12,
has_type (extend Gamma x T11) t12 T12 ->
has_type Gamma (tabs x T11 t12) (TArrow T11 T12)
| T_App : forall T1 T2 Gamma t1 t2,
has_type Gamma t1 (TArrow T1 T2) ->
has_type Gamma t2 T1 ->
has_type Gamma (tapp t1 t2) T2
(* pairs *)
| T_Pair : forall Gamma t1 t2 T1 T2,
has_type Gamma t1 T1 ->
has_type Gamma t2 T2 ->
has_type Gamma (tpair t1 t2) (TProd T1 T2)
| T_Fst : forall Gamma t T1 T2,
has_type Gamma t (TProd T1 T2) ->
has_type Gamma (tfst t) T1
| T_Snd : forall Gamma t T1 T2,
has_type Gamma t (TProd T1 T2) ->
has_type Gamma (tsnd t) T2
(* booleans *)
| T_True : forall Gamma,
has_type Gamma ttrue TBool
| T_False : forall Gamma,
has_type Gamma tfalse TBool
| T_If : forall Gamma t0 t1 t2 T,
has_type Gamma t0 TBool ->
has_type Gamma t1 T ->
has_type Gamma t2 T ->
has_type Gamma (tif t0 t1 t2) T
.
Hint Constructors has_type.
Hint Extern 2 (has_type _ (tapp _ _) _) => eapply T_App; auto.
Hint Extern 2 (_ = _) => compute; reflexivity.
(* ###################################################################### *)
(** *** Context Invariance *)
Inductive appears_free_in : id -> tm -> Prop :=
| afi_var : forall x,
appears_free_in x (tvar x)
| afi_app1 : forall x t1 t2,
appears_free_in x t1 -> appears_free_in x (tapp t1 t2)
| afi_app2 : forall x t1 t2,
appears_free_in x t2 -> appears_free_in x (tapp t1 t2)
| afi_abs : forall x y T11 t12,
y <> x ->
appears_free_in x t12 ->
appears_free_in x (tabs y T11 t12)
(* pairs *)
| afi_pair1 : forall x t1 t2,
appears_free_in x t1 ->
appears_free_in x (tpair t1 t2)
| afi_pair2 : forall x t1 t2,
appears_free_in x t2 ->
appears_free_in x (tpair t1 t2)
| afi_fst : forall x t,
appears_free_in x t ->
appears_free_in x (tfst t)
| afi_snd : forall x t,
appears_free_in x t ->
appears_free_in x (tsnd t)
(* booleans *)
| afi_if0 : forall x t0 t1 t2,
appears_free_in x t0 ->
appears_free_in x (tif t0 t1 t2)
| afi_if1 : forall x t0 t1 t2,
appears_free_in x t1 ->
appears_free_in x (tif t0 t1 t2)
| afi_if2 : forall x t0 t1 t2,
appears_free_in x t2 ->
appears_free_in x (tif t0 t1 t2)
.
Hint Constructors appears_free_in.
Definition closed (t:tm) :=
forall x, ~ appears_free_in x t.
Lemma context_invariance : forall Gamma Gamma' t S,
has_type Gamma t S ->
(forall x, appears_free_in x t -> Gamma x = Gamma' x) ->
has_type Gamma' t S.
Proof with eauto.
intros. generalize dependent Gamma'.
induction H;
intros Gamma' Heqv...
- (* T_Var *)
apply T_Var... rewrite <- Heqv...
- (* T_Abs *)
apply T_Abs... apply IHhas_type. intros y Hafi.
unfold extend. destruct (eq_id_dec x y)...
- (* T_Pair *)
apply T_Pair...
- (* T_If *)
eapply T_If...
Qed.
Lemma free_in_context : forall x t T Gamma,
appears_free_in x t ->
has_type Gamma t T ->
exists T', Gamma x = Some T'.
Proof with eauto.
intros x t T Gamma Hafi Htyp.
induction Htyp; inversion Hafi; subst...
- (* T_Abs *)
destruct IHHtyp as [T' Hctx]... exists T'.
unfold extend in Hctx.
rewrite neq_id in Hctx...
Qed.
Corollary typable_empty__closed : forall t T,
has_type empty t T ->
closed t.
Proof.
intros. unfold closed. intros x H1.
destruct (free_in_context _ _ _ _ H1 H) as [T' C].
inversion C. Qed.
(* ###################################################################### *)
(** *** Preservation *)
Lemma substitution_preserves_typing : forall Gamma x U v t S,
has_type (extend Gamma x U) t S ->
has_type empty v U ->
has_type Gamma ([x:=v]t) S.
Proof with eauto.
(* Theorem: If Gamma,x:U |- t : S and empty |- v : U, then
Gamma |- ([x:=v]t) S. *)
intros Gamma x U v t S Htypt Htypv.
generalize dependent Gamma. generalize dependent S.
(* Proof: By induction on the term t. Most cases follow directly
from the IH, with the exception of tvar and tabs.
The former aren't automatic because we must reason about how the
variables interact. *)
induction t;
intros S Gamma Htypt; simpl; inversion Htypt; subst...
- (* tvar *)
simpl. rename i into y.
(* If t = y, we know that
[empty |- v : U] and
[Gamma,x:U |- y : S]
and, by inversion, [extend Gamma x U y = Some S]. We want to
show that [Gamma |- [x:=v]y : S].
There are two cases to consider: either [x=y] or [x<>y]. *)
destruct (eq_id_dec x y).
+ (* x=y *)
(* If [x = y], then we know that [U = S], and that [[x:=v]y = v].
So what we really must show is that if [empty |- v : U] then
[Gamma |- v : U]. We have already proven a more general version
of this theorem, called context invariance. *)
subst.
unfold extend in H1. rewrite eq_id in H1.
inversion H1; subst. clear H1.
eapply context_invariance...
intros x Hcontra.
destruct (free_in_context _ _ S empty Hcontra) as [T' HT']...
inversion HT'.
+ (* x<>y *)
(* If [x <> y], then [Gamma y = Some S] and the substitution has no
effect. We can show that [Gamma |- y : S] by [T_Var]. *)
apply T_Var... unfold extend in H1. rewrite neq_id in H1...
- (* tabs *)
rename i into y. rename t into T11.
(* If [t = tabs y T11 t0], then we know that
[Gamma,x:U |- tabs y T11 t0 : T11->T12]
[Gamma,x:U,y:T11 |- t0 : T12]
[empty |- v : U]
As our IH, we know that forall S Gamma,
[Gamma,x:U |- t0 : S -> Gamma |- [x:=v]t0 S].
We can calculate that
[x:=v]t = tabs y T11 (if beq_id x y then t0 else [x:=v]t0)
And we must show that [Gamma |- [x:=v]t : T11->T12]. We know
we will do so using [T_Abs], so it remains to be shown that:
[Gamma,y:T11 |- if beq_id x y then t0 else [x:=v]t0 : T12]
We consider two cases: [x = y] and [x <> y].
*)
apply T_Abs...
destruct (eq_id_dec x y).
+ (* x=y *)
(* If [x = y], then the substitution has no effect. Context
invariance shows that [Gamma,y:U,y:T11] and [Gamma,y:T11] are
equivalent. Since the former context shows that [t0 : T12], so
does the latter. *)
eapply context_invariance...
subst.
intros x Hafi. unfold extend.
destruct (eq_id_dec y x)...
+ (* x<>y *)
(* If [x <> y], then the IH and context invariance allow us to show that
[Gamma,x:U,y:T11 |- t0 : T12] =>
[Gamma,y:T11,x:U |- t0 : T12] =>
[Gamma,y:T11 |- [x:=v]t0 : T12] *)
apply IHt. eapply context_invariance...
intros z Hafi. unfold extend.
destruct (eq_id_dec y z)...
subst. rewrite neq_id...
Qed.
Theorem preservation : forall t t' T,
has_type empty t T ->
t ==> t' ->
has_type empty t' T.
Proof with eauto.
intros t t' T HT.
(* Theorem: If [empty |- t : T] and [t ==> t'], then [empty |- t' : T]. *)
remember (@empty ty) as Gamma. generalize dependent HeqGamma.
generalize dependent t'.
(* Proof: By induction on the given typing derivation. Many cases are
contradictory ([T_Var], [T_Abs]). We show just the interesting ones. *)
induction HT;
intros t' HeqGamma HE; subst; inversion HE; subst...
- (* T_App *)
(* If the last rule used was [T_App], then [t = t1 t2], and three rules
could have been used to show [t ==> t']: [ST_App1], [ST_App2], and
[ST_AppAbs]. In the first two cases, the result follows directly from
the IH. *)
inversion HE; subst...
+ (* ST_AppAbs *)
(* For the third case, suppose
[t1 = tabs x T11 t12]
and
[t2 = v2].
We must show that [empty |- [x:=v2]t12 : T2].
We know by assumption that
[empty |- tabs x T11 t12 : T1->T2]
and by inversion
[x:T1 |- t12 : T2]
We have already proven that substitution_preserves_typing and
[empty |- v2 : T1]
by assumption, so we are done. *)
apply substitution_preserves_typing with T1...
inversion HT1...
- (* T_Fst *)
inversion HT...
- (* T_Snd *)
inversion HT...
Qed.
(** [] *)
(* ###################################################################### *)
(** *** Determinism *)
Lemma step_deterministic :
deterministic step.
Proof with eauto.
unfold deterministic.
(* FILL IN HERE *) Admitted.
(* ###################################################################### *)
(** * Normalization *)
(** Now for the actual normalization proof.
Our goal is to prove that every well-typed term evaluates to a
normal form. In fact, it turns out to be convenient to prove
something slightly stronger, namely that every well-typed term
evaluates to a _value_. This follows from the weaker property
anyway via the Progress lemma (why?) but otherwise we don't need
Progress, and we didn't bother re-proving it above.
Here's the key definition: *)
Definition halts (t:tm) : Prop := exists t', t ==>* t' /\ value t'.
(** A trivial fact: *)
Lemma value_halts : forall v, value v -> halts v.
Proof.
intros v H. unfold halts.
exists v. split.
apply multi_refl.
assumption.
Qed.
(** The key issue in the normalization proof (as in many proofs by
induction) is finding a strong enough induction hypothesis. To this
end, we begin by defining, for each type [T], a set [R_T] of closed
terms of type [T]. We will specify these sets using a relation [R]
and write [R T t] when [t] is in [R_T]. (The sets [R_T] are sometimes
called _saturated sets_ or _reducibility candidates_.)
Here is the definition of [R] for the base language:
- [R bool t] iff [t] is a closed term of type [bool] and [t] halts in a value
- [R (T1 -> T2) t] iff [t] is a closed term of type [T1 -> T2] and [t] halts
in a value _and_ for any term [s] such that [R T1 s], we have [R
T2 (t s)]. *)
(** This definition gives us the strengthened induction hypothesis that we
need. Our primary goal is to show that all _programs_ ---i.e., all
closed terms of base type---halt. But closed terms of base type can
contain subterms of functional type, so we need to know something
about these as well. Moreover, it is not enough to know that these
subterms halt, because the application of a normalized function to a
normalized argument involves a substitution, which may enable more
evaluation steps. So we need a stronger condition for terms of
functional type: not only should they halt themselves, but, when
applied to halting arguments, they should yield halting results.
The form of [R] is characteristic of the _logical relations_ proof
technique. (Since we are just dealing with unary relations here, we
could perhaps more properly say _logical predicates_.) If we want to
prove some property [P] of all closed terms of type [A], we proceed by
proving, by induction on types, that all terms of type [A] _possess_
property [P], all terms of type [A->A] _preserve_ property [P], all
terms of type [(A->A)->(A->A)] _preserve the property of preserving_
property [P], and so on. We do this by defining a family of
predicates, indexed by types. For the base type [A], the predicate is
just [P]. For functional types, it says that the function should map
values satisfying the predicate at the input type to values satisfying
the predicate at the output type.
When we come to formalize the definition of [R] in Coq, we hit a
problem. The most obvious formulation would be as a parameterized
Inductive proposition like this:
Inductive R : ty -> tm -> Prop :=
| R_bool : forall b t, has_type empty t TBool ->
halts t ->
R TBool t
| R_arrow : forall T1 T2 t, has_type empty t (TArrow T1 T2) ->
halts t ->
(forall s, R T1 s -> R T2 (tapp t s)) ->
R (TArrow T1 T2) t.
Unfortunately, Coq rejects this definition because it violates the
_strict positivity requirement_ for inductive definitions, which says
that the type being defined must not occur to the left of an arrow in
the type of a constructor argument. Here, it is the third argument to
[R_arrow], namely [(forall s, R T1 s -> R TS (tapp t s))], and
specifically the [R T1 s] part, that violates this rule. (The
outermost arrows separating the constructor arguments don't count when
applying this rule; otherwise we could never have genuinely inductive
predicates at all!) The reason for the rule is that types defined
with non-positive recursion can be used to build non-terminating
functions, which as we know would be a disaster for Coq's logical
soundness. Even though the relation we want in this case might be
perfectly innocent, Coq still rejects it because it fails the
positivity test.
Fortunately, it turns out that we _can_ define [R] using a
[Fixpoint]: *)
Fixpoint R (T:ty) (t:tm) {struct T} : Prop :=
has_type empty t T /\ halts t /\
(match T with
| TBool => True
| TArrow T1 T2 => (forall s, R T1 s -> R T2 (tapp t s))
(* FILL IN HERE *)
| TProd T1 T2 => False (* ... and delete this line *)
end).
(** As immediate consequences of this definition, we have that every
element of every set [R_T] halts in a value and is closed with type
[t] :*)
Lemma R_halts : forall {T} {t}, R T t -> halts t.
Proof.
intros. destruct T; unfold R in H; inversion H; inversion H1; assumption.
Qed.
Lemma R_typable_empty : forall {T} {t}, R T t -> has_type empty t T.
Proof.
intros. destruct T; unfold R in H; inversion H; inversion H1; assumption.
Qed.
(** Now we proceed to show the main result, which is that every
well-typed term of type [T] is an element of [R_T]. Together with
[R_halts], that will show that every well-typed term halts in a
value. *)
(* ###################################################################### *)
(** ** Membership in [R_T] is invariant under evaluation *)
(** We start with a preliminary lemma that shows a kind of strong
preservation property, namely that membership in [R_T] is _invariant_
under evaluation. We will need this property in both directions,
i.e. both to show that a term in [R_T] stays in [R_T] when it takes a
forward step, and to show that any term that ends up in [R_T] after a
step must have been in [R_T] to begin with.
First of all, an easy preliminary lemma. Note that in the forward
direction the proof depends on the fact that our language is
determinstic. This lemma might still be true for non-deterministic
languages, but the proof would be harder! *)
Lemma step_preserves_halting : forall t t', (t ==> t') -> (halts t <-> halts t').
Proof.
intros t t' ST. unfold halts.
split.
- (* -> *)
intros [t'' [STM V]].
inversion STM; subst.
apply ex_falso_quodlibet. apply value__normal in V. unfold normal_form in V. apply V. exists t'. auto.
rewrite (step_deterministic _ _ _ ST H). exists t''. split; assumption.
- (* <- *)
intros [t'0 [STM V]].
exists t'0. split; eauto.
Qed.
(** Now the main lemma, which comes in two parts, one for each
direction. Each proceeds by induction on the structure of the type
[T]. In fact, this is where we make fundamental use of the
structure of types.
One requirement for staying in [R_T] is to stay in type [T]. In the
forward direction, we get this from ordinary type Preservation. *)
Lemma step_preserves_R : forall T t t', (t ==> t') -> R T t -> R T t'.
Proof.
induction T; intros t t' E Rt; unfold R; fold R; unfold R in Rt; fold R in Rt;
destruct Rt as [typable_empty_t [halts_t RRt]].
(* TBool *)
split. eapply preservation; eauto.
split. apply (step_preserves_halting _ _ E); eauto.
auto.
(* TArrow *)
split. eapply preservation; eauto.
split. apply (step_preserves_halting _ _ E); eauto.
intros.
eapply IHT2.
apply ST_App1. apply E.
apply RRt; auto.
(* FILL IN HERE *) Admitted.
(** The generalization to multiple steps is trivial: *)
Lemma multistep_preserves_R : forall T t t',
(t ==>* t') -> R T t -> R T t'.
Proof.
intros T t t' STM; induction STM; intros.
assumption.
apply IHSTM. eapply step_preserves_R. apply H. assumption.
Qed.
(** In the reverse direction, we must add the fact that [t] has type
[T] before stepping as an additional hypothesis. *)
Lemma step_preserves_R' : forall T t t',
has_type empty t T -> (t ==> t') -> R T t' -> R T t.
Proof.
(* FILL IN HERE *) Admitted.
Lemma multistep_preserves_R' : forall T t t',
has_type empty t T -> (t ==>* t') -> R T t' -> R T t.
Proof.
intros T t t' HT STM.
induction STM; intros.
assumption.
eapply step_preserves_R'. assumption. apply H. apply IHSTM.
eapply preservation; eauto. auto.
Qed.
(* ###################################################################### *)
(** ** Closed instances of terms of type [T] belong to [R_T] *)
(** Now we proceed to show that every term of type [T] belongs to
[R_T]. Here, the induction will be on typing derivations (it would be
surprising to see a proof about well-typed terms that did not
somewhere involve induction on typing derivations!). The only
technical difficulty here is in dealing with the abstraction case.
Since we are arguing by induction, the demonstration that a term
[tabs x T1 t2] belongs to [R_(T1->T2)] should involve applying the
induction hypothesis to show that [t2] belongs to [R_(T2)]. But
[R_(T2)] is defined to be a set of _closed_ terms, while [t2] may
contain [x] free, so this does not make sense.
This problem is resolved by using a standard trick to suitably
generalize the induction hypothesis: instead of proving a statement
involving a closed term, we generalize it to cover all closed
_instances_ of an open term [t]. Informally, the statement of the
lemma will look like this:
If [x1:T1,..xn:Tn |- t : T] and [v1,...,vn] are values such that
[R T1 v1], [R T2 v2], ..., [R Tn vn], then
[R T ([x1:=v1][x2:=v2]...[xn:=vn]t)].
The proof will proceed by induction on the typing derivation
[x1:T1,..xn:Tn |- t : T]; the most interesting case will be the one
for abstraction. *)
(* ###################################################################### *)
(** *** Multisubstitutions, multi-extensions, and instantiations *)
(** However, before we can proceed to formalize the statement and
proof of the lemma, we'll need to build some (rather tedious)
machinery to deal with the fact that we are performing _multiple_
substitutions on term [t] and _multiple_ extensions of the typing
context. In particular, we must be precise about the order in which
the substitutions occur and how they act on each other. Often these
details are simply elided in informal paper proofs, but of course Coq
won't let us do that. Since here we are substituting closed terms, we
don't need to worry about how one substitution might affect the term
put in place by another. But we still do need to worry about the
_order_ of substitutions, because it is quite possible for the same
identifier to appear multiple times among the [x1,...xn] with
different associated [vi] and [Ti].
To make everything precise, we will assume that environments are
extended from left to right, and multiple substitutions are performed
from right to left. To see that this is consistent, suppose we have
an environment written as [...,y:bool,...,y:nat,...] and a
corresponding term substitution written as [...[y:=(tbool
true)]...[y:=(tnat 3)]...t]. Since environments are extended from
left to right, the binding [y:nat] hides the binding [y:bool]; since
substitutions are performed right to left, we do the substitution
[y:=(tnat 3)] first, so that the substitution [y:=(tbool true)] has
no effect. Substitution thus correctly preserves the type of the term.
With these points in mind, the following definitions should make sense.
A _multisubstitution_ is the result of applying a list of
substitutions, which we call an _environment_. *)
Definition env := list (id * tm).
Fixpoint msubst (ss:env) (t:tm) {struct ss} : tm :=
match ss with
| nil => t
| ((x,s)::ss') => msubst ss' ([x:=s]t)
end.
(** We need similar machinery to talk about repeated extension of a
typing context using a list of (identifier, type) pairs, which we
call a _type assignment_. *)
Definition tass := list (id * ty).
Fixpoint mextend (Gamma : context) (xts : tass) :=
match xts with
| nil => Gamma
| ((x,v)::xts') => extend (mextend Gamma xts') x v
end.
(** We will need some simple operations that work uniformly on
environments and type assigments *)
Fixpoint lookup {X:Set} (k : id) (l : list (id * X)) {struct l} : option X :=
match l with
| nil => None
| (j,x) :: l' =>
if eq_id_dec j k then Some x else lookup k l'
end.
Fixpoint drop {X:Set} (n:id) (nxs:list (id * X)) {struct nxs} : list (id * X) :=
match nxs with
| nil => nil
| ((n',x)::nxs') => if eq_id_dec n' n then drop n nxs' else (n',x)::(drop n nxs')
end.
(** An _instantiation_ combines a type assignment and a value
environment with the same domains, where corresponding elements are
in R *)
Inductive instantiation : tass -> env -> Prop :=
| V_nil : instantiation nil nil
| V_cons : forall x T v c e, value v -> R T v -> instantiation c e -> instantiation ((x,T)::c) ((x,v)::e).
(** We now proceed to prove various properties of these definitions. *)
(* ###################################################################### *)
(** *** More Substitution Facts *)
(** First we need some additional lemmas on (ordinary) substitution. *)
Lemma vacuous_substitution : forall t x,
~ appears_free_in x t ->
forall t', [x:=t']t = t.
Proof with eauto.
(* FILL IN HERE *) Admitted.
Lemma subst_closed: forall t,
closed t ->
forall x t', [x:=t']t = t.
Proof.
intros. apply vacuous_substitution. apply H. Qed.
Lemma subst_not_afi : forall t x v, closed v -> ~ appears_free_in x ([x:=v]t).
Proof with eauto. (* rather slow this way *)
unfold closed, not.
induction t; intros x v P A; simpl in A.
- (* tvar *)
destruct (eq_id_dec x i)...
inversion A; subst. auto.
- (* tapp *)
inversion A; subst...
- (* tabs *)
destruct (eq_id_dec x i)...
inversion A; subst...
inversion A; subst...
- (* tpair *)
inversion A; subst...
- (* tfst *)
inversion A; subst...
- (* tsnd *)
inversion A; subst...
- (* ttrue *)
inversion A.
- (* tfalse *)
inversion A.
- (* tif *)
inversion A; subst...
Qed.
Lemma duplicate_subst : forall t' x t v,
closed v -> [x:=t]([x:=v]t') = [x:=v]t'.
Proof.
intros. eapply vacuous_substitution. apply subst_not_afi. auto.
Qed.
Lemma swap_subst : forall t x x1 v v1, x <> x1 -> closed v -> closed v1 ->
[x1:=v1]([x:=v]t) = [x:=v]([x1:=v1]t).
Proof with eauto.
induction t; intros; simpl.
- (* tvar *)
destruct (eq_id_dec x i); destruct (eq_id_dec x1 i).
subst. apply ex_falso_quodlibet...
subst. simpl. rewrite eq_id. apply subst_closed...
subst. simpl. rewrite eq_id. rewrite subst_closed...
simpl. rewrite neq_id... rewrite neq_id...
(* FILL IN HERE *) Admitted.
(* ###################################################################### *)
(** *** Properties of multi-substitutions *)
Lemma msubst_closed: forall t, closed t -> forall ss, msubst ss t = t.
Proof.
induction ss.
reflexivity.
destruct a. simpl. rewrite subst_closed; assumption.
Qed.
(** Closed environments are those that contain only closed terms. *)
Fixpoint closed_env (env:env) {struct env} :=
match env with
| nil => True
| (x,t)::env' => closed t /\ closed_env env'
end.
(** Next come a series of lemmas charcterizing how [msubst] of closed terms
distributes over [subst] and over each term form *)
Lemma subst_msubst: forall env x v t, closed v -> closed_env env ->
msubst env ([x:=v]t) = [x:=v](msubst (drop x env) t).
Proof.
induction env0; intros.
auto.
destruct a. simpl.
inversion H0. fold closed_env in H2.
destruct (eq_id_dec i x).
subst. rewrite duplicate_subst; auto.
simpl. rewrite swap_subst; eauto.
Qed.
Lemma msubst_var: forall ss x, closed_env ss ->
msubst ss (tvar x) =
match lookup x ss with
| Some t => t
| None => tvar x
end.
Proof.
induction ss; intros.
reflexivity.
destruct a.
simpl. destruct (eq_id_dec i x).
apply msubst_closed. inversion H; auto.
apply IHss. inversion H; auto.
Qed.
Lemma msubst_abs: forall ss x T t,
msubst ss (tabs x T t) = tabs x T (msubst (drop x ss) t).
Proof.
induction ss; intros.
reflexivity.
destruct a.
simpl. destruct (eq_id_dec i x); simpl; auto.
Qed.
Lemma msubst_app : forall ss t1 t2, msubst ss (tapp t1 t2) = tapp (msubst ss t1) (msubst ss t2).
Proof.
induction ss; intros.
reflexivity.
destruct a.
simpl. rewrite <- IHss. auto.
Qed.
(** You'll need similar functions for the other term constructors. *)
(* FILL IN HERE *)
(* ###################################################################### *)
(** *** Properties of multi-extensions *)
(** We need to connect the behavior of type assignments with that of their
corresponding contexts. *)
Lemma mextend_lookup : forall (c : tass) (x:id), lookup x c = (mextend empty c) x.
Proof.
induction c; intros.
auto.
destruct a. unfold lookup, mextend, extend. destruct (eq_id_dec i x); auto.
Qed.
Lemma mextend_drop : forall (c: tass) Gamma x x',
mextend Gamma (drop x c) x' = if eq_id_dec x x' then Gamma x' else mextend Gamma c x'.
induction c; intros.
destruct (eq_id_dec x x'); auto.
destruct a. simpl.
destruct (eq_id_dec i x).
subst. rewrite IHc.
destruct (eq_id_dec x x'). auto. unfold extend. rewrite neq_id; auto.
simpl. unfold extend. destruct (eq_id_dec i x').
subst.
destruct (eq_id_dec x x').
subst. exfalso. auto.
auto.
auto.
Qed.
(* ###################################################################### *)
(** *** Properties of Instantiations *)
(** These are strightforward. *)
Lemma instantiation_domains_match: forall {c} {e},
instantiation c e -> forall {x} {T}, lookup x c = Some T -> exists t, lookup x e = Some t.
Proof.
intros c e V. induction V; intros x0 T0 C.
solve by inversion .
simpl in *.
destruct (eq_id_dec x x0); eauto.
Qed.
Lemma instantiation_env_closed : forall c e, instantiation c e -> closed_env e.
Proof.
intros c e V; induction V; intros.
econstructor.
unfold closed_env. fold closed_env.
split. eapply typable_empty__closed. eapply R_typable_empty. eauto.
auto.
Qed.
Lemma instantiation_R : forall c e, instantiation c e ->
forall x t T, lookup x c = Some T ->
lookup x e = Some t -> R T t.
Proof.
intros c e V. induction V; intros x' t' T' G E.
solve by inversion.
unfold lookup in *. destruct (eq_id_dec x x').
inversion G; inversion E; subst. auto.
eauto.
Qed.
Lemma instantiation_drop : forall c env,
instantiation c env -> forall x, instantiation (drop x c) (drop x env).
Proof.
intros c e V. induction V.
intros. simpl. constructor.
intros. unfold drop. destruct (eq_id_dec x x0); auto. constructor; eauto.
Qed.
(* ###################################################################### *)
(** *** Congruence lemmas on multistep *)
(** We'll need just a few of these; add them as the demand arises. *)
Lemma multistep_App2 : forall v t t',
value v -> (t ==>* t') -> (tapp v t) ==>* (tapp v t').
Proof.
intros v t t' V STM. induction STM.
apply multi_refl.
eapply multi_step.
apply ST_App2; eauto. auto.
Qed.
(* FILL IN HERE *)
(* ###################################################################### *)
(** *** The R Lemma. *)
(** We finally put everything together.
The key lemma about preservation of typing under substitution can
be lifted to multi-substitutions: *)
Lemma msubst_preserves_typing : forall c e,
instantiation c e ->
forall Gamma t S, has_type (mextend Gamma c) t S ->
has_type Gamma (msubst e t) S.
Proof.
induction 1; intros.
simpl in H. simpl. auto.
simpl in H2. simpl.
apply IHinstantiation.
eapply substitution_preserves_typing; eauto.
apply (R_typable_empty H0).
Qed.
(** And at long last, the main lemma. *)