-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathterms.v~
1281 lines (907 loc) · 43.4 KB
/
terms.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
(**********************************************************************)
(* Copyright 2024 Barry Jay *)
(* *)
(* Permission is hereby granted, free of charge, to any person *)
(* obtaining a copy of this software and associated documentation *)
(* files (the "Software"), to deal in the Software without *)
(* restriction, including without limitation the rights to use, copy, *)
(* modify, merge, publish, distribute, sublicense, and/or sell copies *)
(* of the Software, and to permit persons to whom the Software is *)
(* furnished to do so, subject to the following conditions: *)
(* *)
(* The above copyright notice and this permission notice shall be *)
(* included in all copies or substantial portions of the Software. *)
(* *)
(* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, *)
(* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF *)
(* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND *)
(* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT *)
(* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, *)
(* WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, *)
(* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER *)
(* DEALINGS IN THE SOFTWARE. *)
(**********************************************************************)
(**********************************************************************)
(* Tree Calculus with five reduction rules *)
(* or *)
(* Triage Calculus *)
(* *)
(* Barry Jay *)
(* *)
(**********************************************************************)
Require Import Arith Lia Bool List Nat Datatypes String.
Set Default Proof Using "Type".
Open Scope string_scope.
(* Generalities *)
Ltac refold r := unfold r; fold r.
Ltac caseEq f := generalize (refl_equal f); pattern f at -1; case f.
Ltac auto_t := eauto with TreeHintDb.
Ltac eapply2 H := eapply H; auto_t; try lia.
Ltac split_all := intros;
match goal with
| H : _ /\ _ |- _ => inversion_clear H; split_all
| H : exists _, _ |- _ => inversion H; clear H; split_all
| _ => try (split; split_all); subst; try contradiction
end; try congruence.
Ltac inv_out H := inversion H; clear H; subst.
Ltac disjunction_tac:=
match goal with
| H : _ \/ _ |- _ => inv_out H; disjunction_tac
| _ => try lia
end.
Declare Scope tree_scope.
Open Scope tree_scope.
(* Tree Calculus *)
Inductive Tree: Set :=
| Ref : string -> Tree (* variables are indexed by strings *)
| Node : Tree
| App : Tree -> Tree -> Tree
.
Hint Constructors Tree : TreeHintDb.
Notation "△" := Node : tree_scope.
Notation "x @ y" := (App x y) (at level 65, left associativity) : tree_scope.
Definition K := △ @ △.
Definition S1 x := △ @ (△@x).
Definition Sop := S1 (K @ Node) @ Node.
Definition I := S1 K @ K.
Definition KI := K@I.
Ltac unfold_op := unfold KI, I, K, S1.
(* Programs *)
Inductive program : Tree -> Prop :=
| pr_leaf : program △
| pr_stem : forall M, program M -> program (△ @ M)
| pr_fork : forall M N, program M -> program N -> program (△ @ M@ N)
.
Ltac program_tac := cbv; repeat (apply pr_stem || apply pr_fork || apply pr_leaf); auto.
(* Tree-reduction *)
(*
This differs from the reduction of untyped tree calculus, using rules suggested by Johannes Bader.
Node Node = K
Node (Node x) = S x
(Node (Node w x) y = triage {w,x,y}
*)
Inductive t_red1 : Tree -> Tree -> Prop :=
| k_red : forall y z, t_red1 (△ @ △ @ y @ z) y
| s_red: forall (x y z : Tree), t_red1 (△ @ (△ @ x) @ y @ z) (x @ z @ (y @ z))
| leaf_red: forall (w x y : Tree), t_red1 (△ @ (△ @ w @ x) @ y @ Node) w
| stem_red: forall (w x y z : Tree), t_red1 (△ @ (△ @ w @ x) @ y @ (Node @ z)) (x @ z)
| fork_red: forall (w x y z1 z2 : Tree), t_red1 (△ @ (△ @ w @ x) @ y @ (Node @ z1 @ z2)) (y @ z1 @ z2)
| appl_red : forall M M' N, t_red1 M M' -> t_red1 (M@N) (M'@N)
| appr_red : forall M N N', t_red1 N N' -> t_red1 (M@N) (M@N')
.
Hint Constructors t_red1 : TreeHintDb.
(* Multiple reduction steps *)
Inductive multi_step : (Tree -> Tree -> Prop) -> Tree -> Tree -> Prop :=
| zero_red : forall red M, multi_step red M M
| succ_red : forall (red: Tree-> Tree -> Prop) M N P,
red M N -> multi_step red N P -> multi_step red M P
.
Hint Constructors multi_step: TreeHintDb.
Definition t_red := multi_step t_red1.
Ltac tree_red :=
intros; cbv; repeat (eapply succ_red ; [ auto 10 with *; fail|]); try eapply zero_red.
(* The reduction rules for K and S are built in: here is the rule for I *)
Lemma i_red: forall x, t_red (I @ x) x. Proof. tree_red. Qed.
Definition transitive red := forall (M N P: Tree), red M N -> red N P -> red M P.
Lemma transitive_red : forall red, transitive (multi_step red).
Proof. red; induction 1; split_all; eapply2 succ_red. Qed.
Definition preserves_appl (red : Tree -> Tree -> Prop) :=
forall M N M', red M M' -> red (M@N) (M'@ N).
Definition preserves_appr (red : Tree -> Tree -> Prop) :=
forall M N N', red N N' -> red (M@N) (M@N').
Definition preserves_app (red : Tree -> Tree -> Prop) :=
forall M M', red M M' -> forall N N', red N N' -> red (M@N) (M'@N').
Lemma preserves_appl_multi_step :
forall (red: Tree -> Tree -> Prop), preserves_appl red -> preserves_appl (multi_step red).
Proof. red; intros red pa M N M' r ; induction r; auto_t; eapply2 succ_red. Qed.
Lemma preserves_appr_multi_step :
forall (red: Tree -> Tree -> Prop), preserves_appr red -> preserves_appr (multi_step red).
Proof. red; intros red pa M N M' r ; induction r; auto_t; eapply2 succ_red. Qed.
Lemma preserves_app_multi_step :
forall (red: Tree -> Tree -> Prop),
preserves_appl red -> preserves_appr red ->
preserves_app (multi_step red).
Proof.
red; intros; apply transitive_red with (M'@N);
[ apply preserves_appl_multi_step |
apply preserves_appr_multi_step]
; auto.
Qed.
Lemma preserves_appl_t_red : preserves_appl t_red.
Proof. apply preserves_appl_multi_step. red; auto_t. Qed.
Lemma preserves_appr_t_red : preserves_appr t_red.
Proof. apply preserves_appr_multi_step. red; auto_t. Qed.
Lemma preserves_app_t_red : preserves_app t_red.
Proof. apply preserves_app_multi_step; red; auto_t. Qed.
Ltac zerotac := try apply zero_red; auto.
Ltac succtac :=
repeat (eapply transitive_red;
[ eapply succ_red ; auto_t;
match goal with
| |- multi_step t_red1 _ _ => idtac
| _ => fail (*gone too far ! *)
end
| ])
.
Ltac trtac := unfold_op; unfold t_red; succtac;
match goal with
| |- multi_step t_red1 △ _ => zerotac
| |- multi_step t_red1 (△ @ _ @ _ @ _) _ =>
eapply transitive_red;
[ eapply preserves_app_t_red ;
[ eapply preserves_app_t_red ;
[ eapply preserves_app_t_red ; [| trtac ] (* reduce the argument of △ *)
| ]
| ]
|] ;
zerotac; (* put the "redex" back together *)
match goal with
| |- multi_step t_red1 (△ @ ?arg @ _ @ _) _ =>
match arg with
| △ => trtac (* made progress so recurse *)
| △ @ _ => trtac (* made progress so recurse *)
| △ @ _ @ _ => trtac (* made progress so recurse *)
| _ => idtac (* no progress so stop *)
end
| _ => idtac (* for safety ? *)
end
| |- multi_step t_red1 (_ @ _) _ => (* not a ternary tree *)
eapply transitive_red; [ eapply preserves_app_t_red ; trtac |] ; (* so reduce the function *)
match goal with
| |- multi_step t_red1 (△ @ ?arg @ _ @ _) _ =>
match arg with
| △ => trtac (* made progress so recurse *)
| △ @ _ => trtac (* made progress so recurse *)
| △ @ _ @ _ => trtac (* made progress so recurse *)
| _ => idtac (* no progress so stop *)
end
| _ => idtac
end
| _ => idtac (* the head is not △ *)
end;
zerotac; succtac; zerotac.
(* 4.3: Variable Binding *)
Fixpoint substitute M x N :=
match M with
| Ref y => if eqb x y then N else Ref y
| △ => △
| M1 @ M2 => (substitute M1 x N) @ (substitute M2 x N)
end.
Lemma substitute_app:
forall M1 M2 x N, substitute (M1@ M2) x N = (substitute M1 x N) @ (substitute M2 x N).
Proof. auto. Qed.
Lemma substitute_node:
forall x N, substitute △ x N = △.
Proof. auto. Qed.
Lemma substitute_preserves_t_red:
forall M x N N', t_red N N' -> t_red (substitute M x N) (substitute M x N').
Proof.
induction M; intros; simpl; zerotac;
[ match goal with |- t_red (if ?b then _ else _) _ => caseEq b; intros; zerotac; auto end |
apply preserves_app_t_red; auto].
Qed.
(* Bracket Abstraction *)
Fixpoint bracket x M :=
match M with
| Ref y => if eqb x y then I else (K@ (Ref y))
| △ => K@ △
| App M1 M2 => S1 (bracket x M1) @ (bracket x M2)
end
.
Theorem bracket_beta: forall M x N, t_red ((bracket x M) @ N) (substitute M x N).
Proof.
induction M; intros; unfold S1; simpl;
[ match goal with |- t_red ((if ?b then _ else _) @ _) _ => case b; tree_red end |
tree_red |
unfold S1; eapply succ_red; auto_t; apply preserves_app_t_red; trtac; auto]; [
eapply IHM1 |
eapply IHM2].
Qed.
(* star abstraction *)
Fixpoint occurs x M :=
match M with
| Ref y => eqb x y
| △ => false
| M1@ M2 => (occurs x M1) || (occurs x M2)
end.
Lemma substitute_occurs_false: forall M x N, occurs x M = false -> substitute M x N = M.
Proof.
induction M; simpl; intros x N e; split_all;
[ rewrite e; auto | rewrite orb_false_iff in *; rewrite IHM1; try tauto; rewrite IHM2; tauto].
Qed.
Fixpoint star x M := (* no eta-contractions because argument types must be invariant *)
match M with
| Ref y => if eqb x y then I else (K@ (Ref y))
| △ => K@ △
| App M1 M2 => if occurs x (M1 @ M2)
then S1 (star x M1) @ (star x M2)
else K@ (M1 @ M2)
end.
Notation "\" := star : tree_scope.
Theorem star_beta: forall M x N, t_red ((\x M) @ N) (substitute M x N).
Proof.
induction M as [s | | M1 ? M2]; intros x N; simpl; auto.
caseEq (x=?s); intros; tree_red .
tree_red.
unfold S1; caseEq (occurs x M1 || occurs x M2); intros; trtac. apply preserves_app_t_red; auto.
rewrite orb_false_iff in *; split_all; rewrite ! substitute_occurs_false; auto; zerotac.
Qed.
(* Recursion *)
Ltac inv1 prop :=
match goal with
| H: prop (Ref _) |- _ => inversion H; clear H; inv1 prop
| H: prop (_ @ _) |- _ => inversion H; clear H; inv1 prop
| H: prop △ |- _ => inversion H; clear H; inv1 prop
| _ => auto_t
end.
Ltac aptac := eapply transitive_red; [ eapply preserves_app_t_red |].
(* 4.2: Combinations versus Terms *)
Inductive combination : Tree -> Prop :=
| is_Node : combination △
| is_App : forall M N, combination M -> combination N -> combination (M@ N)
.
Hint Constructors combination: TreeHintDb.
Ltac combination_tac := repeat (apply is_App || apply is_Node); auto.
(* Normal Forms *)
Inductive active: Tree -> Prop :=
| active_ref: forall x, active (Ref x)
| active_app: forall M N, active M -> active (M@N)
.
Inductive normal : Tree -> Prop :=
| nf_ref: forall x, normal (Ref x)
| nf_active: forall M N, active M -> normal M -> normal N -> normal (M@N)
| nf_leaf : normal △
| nf_stem : forall M, normal M -> normal (△ @ M)
| nf_fork : forall M N, normal M -> normal N -> normal (△ @ M@ N)
.
Ltac normal_tac := cbv; repeat (apply nf_stem || apply nf_fork || apply nf_leaf || apply nf_ref); auto.
Lemma occurs_combination: forall M x, combination M -> occurs x M = false.
Proof.
induction M; simpl; split_all; inv1 combination; subst; auto; rewrite IHM1; auto; rewrite IHM2; auto.
Qed.
Lemma occurs_ref: forall x y, occurs x (Ref y) = (String.eqb x y).
Proof. auto. Qed.
Lemma occurs_node: forall x, occurs x △ = false.
Proof. auto. Qed.
Lemma occurs_app: forall x M N, occurs x (M@ N) = occurs x M || occurs x N.
Proof. auto. Qed.
Lemma star_combination: forall M x, combination M -> \x M = K@ M.
Proof.
induction M as [ | | M1 M2 M3]; intros x c; simpl;
try (split_all; inv1 combination; subst; auto; fail);
inversion c as [ | ? ? c1 c3]; inversion c3; subst; split_all; rewrite ! occurs_combination; split_all.
Qed.
Lemma star_leaf: forall x, \x △ = K@ △.
Proof. auto. Qed.
Lemma star_id: forall x, \x (Ref x) = I.
Proof. intro; unfold star, occurs; rewrite eqb_refl; auto. Qed.
Lemma star_occurs_false: forall M x, occurs x M = false -> \x M = K@ M.
Proof. induction M; simpl; intros x occ; rewrite ? occ; auto. Qed.
Lemma star_occurs_true:
forall M1 M2 x, occurs x (M1@ M2) = true -> \x (M1@ M2) = S1 (\x M1) @ (star x M2).
Proof. intros M1 M2 x occ; unfold star at 1; fold star; rewrite occ; auto. Qed.
Lemma occurs_substitute_true:
forall M x y N, x<> y -> occurs x M = true -> occurs x (substitute M y N) = true.
Proof.
induction M; intros x y N d occ; simpl in *; auto;
[
match goal with H : (String.eqb x ?s) = true |- _ =>
assert(x=s) by (apply eqb_eq; auto);
assert(d2: eqb y s = false) by (apply eqb_neq; split_all);
rewrite d2; simpl; auto end
|
rewrite orb_true_iff in occ; inversion occ;
[ rewrite IHM1; auto | rewrite IHM2; auto; apply orb_true_r]
].
Qed.
Lemma star_occurs_false_app
: forall (M N : Tree) (x : string), occurs x (M@ N) = false -> \ x (M@N) = K @ (M@N).
Proof. intros; eapply star_occurs_false; cbv; auto. Qed.
Ltac occurstac :=
unfold_op; unfold occurs; fold occurs; rewrite ? orb_true_r; cbv; auto 1000 with *; fail.
Ltac startac_true x := rewrite (star_occurs_true _ _ x); [| occurstac ].
Ltac startac_false x := rewrite (star_occurs_false _ x); [ | occurstac].
Ltac startac x := unfold S1; repeat (startac_false x || startac_true x || rewrite star_id).
Ltac starstac1 xs :=
match xs with
| nil => trtac
| ?x :: ?xs1 => startac x; starstac1 xs1
end.
Ltac starstac xs := repeat (starstac1 xs).
Ltac startac2 :=
rewrite star_id
|| (rewrite star_occurs_true; [ | rewrite ? String.eqb_refl; cbv; auto; fail ])
|| (rewrite star_occurs_false; [ | cbv; auto; fail ])
|| (rewrite star_occurs_false_app; [ | cbv; auto; fail ])
; trtac.
(*** General Recursion *)
(*** Self-application and fixfunctions *)
Definition self_apply := Eval cbv in \"x" (Ref "x" @ (Ref "x")).
Definition wait M N := S1 (S1 (K@ M) @ (K@ N)) @ I.
Theorem wait_red: forall M N x, t_red (wait M N @ x) (M @ N @ x). Proof. tree_red. Qed.
Definition omega := \"w" (\"f" (Ref "f" @ (self_apply @ Ref "w" @ Ref "f"))).
Definition Yop := omega @ omega.
Theorem Yop_red: forall f, t_red (Yop @ f) (f @ (Yop @ f)).
Proof. intros; unfold Yop; unfold omega at 1; unfold star; simpl; unfold S1, I; trtac. Qed.
Definition wait2 M N x := S1 (S1 (S1 (K @ M) @ (K @ N)) @ (K @ x)) @ I.
Theorem wait2_red: forall M N x y, t_red (wait2 M N x @ y) (M @ N @ x @ y).
Proof. tree_red. Qed.
Definition omega2 := \"w" (\"f" (Ref "f" @ (wait2 (Ref "w") (Ref "w") (Ref "f")))).
Definition omega21 := K @ (△ @ (△ @ I)).
Definition omega22 := \ "w" (\ "f" (wait2 (Ref "w") (Ref "w") (Ref "f"))).
Lemma omega2_parts: omega2 = S1 omega21 @ omega22.
Proof. cbv; auto. Qed.
Definition Z f := wait2 omega2 omega2 f.
Theorem Z_red: forall f x, t_red (Z f @ x) (f @ (Z f) @ x).
Proof.
intros; unfold Z. unfold wait2 at 1. trtac.
unfold omega2. startac "f"; startac "w". trtac. do 2 (eapply preserves_app_t_red; trtac).
unfold wait2. startac "f"; startac "w". trtac.
Qed.
Theorem Z_program: forall f, program f -> program (Z f).
Proof. intros; program_tac. Qed.
Definition omega2__alt := \"w" (\"f" (Ref "f" @ (wait2 self_apply (Ref "w") (Ref "f")))).
Definition Z__alt f := wait2 self_apply omega2__alt f.
Theorem Z__alt_red: forall f x, t_red (Z__alt f @ x) (f @ (Z__alt f) @ x).
Proof.
intros; unfold Z__alt. unfold wait2 at 1. trtac. unfold self_apply at 1. trtac.
unfold omega2__alt at 1.
startac "f"; startac "w". trtac. do 2 (eapply preserves_app_t_red; trtac).
tree_red.
Qed.
Definition swap_abs := Eval cbv in \"x" (\"y" (Ref "f" @ Ref "y" @ Ref "x")).
Definition swap f := substitute swap_abs "f" f.
Theorem swap_red: forall f x y, t_red (swap f @ x @ y) (f @ y @ x). Proof. tree_red. Qed.
Definition Yop2 f := Z (swap f).
Theorem Y2_red: forall f, (program f -> program (Yop2 f)) /\ (forall x, t_red (Yop2 f @ x) (f @ x @ (Yop2 f))).
Proof. intros; split; intro; [ program_tac | eapply transitive_red; [ eapply Z_red | eapply swap_red]]. Qed.
(* generic equality *)
Definition triage f0 f1 f2 := Node @ (Node @ f0 @ f1) @ f2.
Definition equal :=
Yop2
(triage
(\"e" (triage K (K @ KI) (K @ (K @ KI))))
(\"x" (\"e" (triage KI (Ref "e" @ Ref "x") (K @ (K @ KI)))))
(\"x1"
(\"x2"
(\"e"
(triage KI (K @ KI)
(\"y1" (\"y2" (Ref "e" @ Ref "x1" @ Ref "y1" @ (Ref "e" @ Ref "x2" @ Ref "y2") @ KI)))
))))).
Lemma equal_leaf: t_red (equal @ Node @ Node) K.
Proof. aptac. eapply Y2_red. trtac. unfold triage. trtac. startac "e". trtac. Qed.
Lemma equal_stem: forall x y, t_red (equal @ (Node @ x) @ (Node @ y)) (equal @ x @ y).
Proof. intros. aptac. eapply Y2_red. trtac. unfold triage. trtac. startac "e"; startac "x". trtac. Qed.
Lemma equal_fork:
forall x1 x2 y1 y2, t_red (equal @ (Node @ x1 @ x2) @ (Node @ y1 @ y2))
(equal @ x1 @ y1 @ (equal @ x2 @ y2) @ KI).
Proof.
intros. aptac. eapply Y2_red. trtac.
replace ( Yop2
(triage (\ "e" (triage K (K @ KI) (K @ (K @ KI))))
(\ "x" (\ "e" (triage KI (Ref "e" @ Ref "x") (K @ (K @ KI)))))
(\ "x1"
(\ "x2"
(\ "e"
(triage KI (K @ KI)
(\ "y1" (\ "y2" (Ref "e" @ Ref "x1" @ Ref "y1" @ (Ref "e" @ Ref "x2" @ Ref "y2") @ KI))))))))) with equal by (cbv; auto).
unfold triage. trtac. startac "y2"; startac "y1"; startac "e"; startac "x2"; startac "x1"; trtac.
Qed.
Theorem equal_programs: forall M, program M -> t_red (equal @ M @ M) K.
Proof.
intros M pr; induction pr; intros.
eapply equal_leaf.
eapply transitive_red; [ eapply equal_stem | eauto].
eapply transitive_red; [ eapply equal_fork |]. aptac. aptac. eauto. eauto. trtac. trtac. trtac.
Qed.
Lemma equal_leaf_stem: forall y, t_red (equal @ Node @ (Node @ y)) KI.
Proof. intros; aptac. eapply Y2_red. trtac. unfold triage. trtac. startac "e". trtac. Qed.
Lemma equal_leaf_fork: forall y1 y2, t_red (equal @ Node @ (Node @ y1 @ y2)) KI.
Proof. intros; aptac. eapply Y2_red. trtac. unfold triage. trtac. startac "e". trtac. Qed.
Lemma equal_stem_leaf: forall x, t_red (equal @ (Node @ x) @ Node) KI.
Proof. intros; aptac. eapply Y2_red. trtac. unfold triage. trtac. startac "e"; startac "x"; trtac. Qed.
Lemma equal_stem_fork: forall x y1 y2, t_red (equal @ (Node @ x) @ (Node @ y1 @ y2)) KI.
Proof. intros; aptac. eapply Y2_red. trtac. unfold triage. trtac. startac "e"; startac "x"; trtac. Qed.
Lemma equal_fork_leaf: forall x1 x2, t_red (equal @ (Node @ x1 @ x2) @ Node) KI.
Proof. intros; aptac. eapply Y2_red. trtac. unfold triage. trtac.
startac "e"; startac "x2"; startac "x1"; trtac. Qed.
Lemma equal_fork_stem: forall x1 x2 y, t_red (equal @ (Node @ x1 @ x2) @ (Node @ y)) KI.
Proof. intros; aptac. eapply Y2_red. trtac. unfold triage. trtac.
startac "e"; startac "x2"; startac "x1"; trtac. Qed.
Theorem unequal_programs: forall M, program M -> forall N, program N -> M <> N -> t_red (equal@ M @ N) KI.
Proof.
intros M prM; induction prM; intros; inv_out H; try congruence.
eapply equal_leaf_stem.
eapply equal_leaf_fork.
eapply equal_stem_leaf.
eapply transitive_red. eapply equal_stem. eapply IHprM; eauto. intro; subst; congruence.
eapply equal_stem_fork.
eapply equal_fork_leaf.
eapply equal_fork_stem.
eapply transitive_red. eapply equal_fork.
assert(M = M0 \/ M <> M0) by (repeat decide equality). disjunction_tac.
aptac. aptac. eapply equal_programs; eauto. eapply IHprM2; eauto. intro; subst; congruence.
trtac. trtac. trtac.
aptac. aptac. eapply IHprM1; eauto. trtac. trtac. trtac. trtac.
Qed.
Definition tt := K.
Definition ff := KI.
Theorem equality_of_programs:
forall M, program M -> t_red (equal @ M @ M) tt /\ forall N, program N -> M <> N -> t_red (equal@ M @ N) ff.
Proof. intros; split; [ eapply equal_programs | eapply unequal_programs]; auto. Qed.
(*** Pairing and projections *)
Definition pairL x y := S1 (S1 I @ (K @ x)) @ (K @ y).
Definition fstL := S1 I @ (K@K).
Definition sndL := S1 I @ (K@KI).
Theorem fstL_red: forall x y, t_red (fstL @ (pairL x y)) x. Proof. tree_red. Qed.
Theorem sndL_red: forall x y, t_red (sndL @ (pairL x y)) y. Proof. tree_red. Qed.
(*** Function composition *)
(* given f_i : Nat ^m -> Nat and g : Nat^n -> N, define g (f_i) : Nat^m -> Nat by
g(f_i) (x_j) = g (f_i(x_j))
*)
Lemma fold_left_app_preserves_red:
forall xs f1 f2, t_red f1 f2 -> t_red (fold_left App xs f1) (fold_left App xs f2).
Proof. induction xs; intros; simpl; eauto; eapply IHxs; eapply preserves_appl_t_red; auto. Qed.
Fixpoint Kn n := (* Kn n g xs = g if length xs = n *)
match n with
| 0 => I
| S n1 => \"x" (K@ (Kn n1 @ Ref "x"))
end.
Fixpoint compose1 n := (* compose1 g f xs = g xs (f xs) if length xs = n *)
match n with
| 0 => I
| S n1 => \"g" (\"f" (\"x" (compose1 n1 @ (Ref "g" @ Ref "x") @ (Ref "f" @ Ref "x"))))
end.
Fixpoint compose0 m n := (* compose0 m n g fs xs = g xs (map fs xs) if length fs = m and length xs = n *)
match m with
| 0 => I
| S m1 => \"g" (\"f" (compose0 m1 n @ (compose1 n @ Ref "g" @ Ref "f")))
end.
Definition compose m n := S1 (K @ compose0 m n) @ (Kn n) . (* \"x" (compose0 m n @ (Kn n @ Ref "x")). *)
Lemma Kn_closed: forall n x, occurs x (Kn n) = false.
Proof.
induction n; intros; simpl; auto; rewrite IHn; simpl;
rewrite star_occurs_false; simpl; rewrite IHn; auto.
Qed.
Lemma Kn_red: forall xs g, t_red (fold_left App xs (Kn (List.length xs) @ g)) g.
Proof.
induction xs; intros; simpl; auto. repeat trtac. rewrite Kn_closed; simpl.
eapply transitive_red. eapply fold_left_app_preserves_red.
aptac. trtac. trtac. rewrite star_occurs_false; simpl. 2: eapply Kn_closed. trtac.
eapply transitive_red. eapply fold_left_app_preserves_red. trtac. eapply IHxs.
Qed.
Lemma compose1_closed: forall n x, occurs x (compose1 n) = false.
Proof.
induction n; intros; simpl; auto; rewrite ! orb_true_r;
rewrite (star_occurs_false _ "x"); eauto;
unfold S1, star; fold star; repeat (simpl; rewrite ! IHn); auto.
Qed.
Proposition compose1_red:
forall xs g f,
t_red (fold_left App xs (compose1 (List.length xs) @ g @ f)) ((fold_left App xs g) @ (fold_left App xs f)).
Proof.
induction xs; intros; simpl; auto. trtac.
rewrite compose1_closed. refold orb.
eapply transitive_red. eapply fold_left_app_preserves_red.
rewrite (star_occurs_false _ "x"); eauto. 2: eapply compose1_closed.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite (star_occurs_false _ "f"). 2: simpl; rewrite compose1_closed; auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite (star_occurs_false). 2: auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite 2 (star_occurs_false). 2, 3: auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite (star_occurs_false). 2: auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite (star_occurs_false). 2: simpl; eapply compose1_closed.
unfold S1; rewrite star_occurs_true. 2: simpl; auto.
rewrite (star_occurs_true). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_true). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_true). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_true). 2: auto.
rewrite star_id.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_true). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_true). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite (star_occurs_true). 2: auto.
rewrite (star_occurs_false). 2: auto.
rewrite star_id.
rewrite (star_occurs_false). 2: auto.
trtac.
eapply transitive_red. eapply fold_left_app_preserves_red. trtac.
eapply IHxs.
Qed.
Lemma compose0_closed: forall m n x, occurs x (compose0 m n) = false.
Proof.
induction m; intros; simpl; auto; rewrite ! orb_true_r.
rewrite ! compose1_closed; simpl. rewrite ! compose1_closed; simpl.
rewrite star_occurs_false. 2: eauto.
simpl. rewrite IHm. simpl. rewrite ! IHm.
rewrite star_occurs_false; simpl; rewrite compose1_closed; auto.
Qed.
Proposition compose0_red:
forall fs xs g,
t_red (fold_left App xs (fold_left App fs (compose0 (List.length fs) (List.length xs) @ g)))
(fold_left App (map (fun f => (fold_left App xs f)) fs) (fold_left App xs g)).
Proof.
induction fs; intros; simpl; auto.
(* 2 *)
repeat trtac; eapply fold_left_app_preserves_red; repeat trtac.
(* 1 *)
rewrite ! compose0_closed. rewrite ! compose1_closed. unfold orb.
eapply transitive_red. eapply fold_left_app_preserves_red. eapply fold_left_app_preserves_red.
rewrite (star_occurs_false _ "f"). 2: eapply compose0_closed.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite star_occurs_false. 2: simpl; eapply compose0_closed.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite star_occurs_false. 2: auto.
unfold S1; rewrite star_occurs_true. 2: simpl; rewrite ! orb_true_r; auto.
rewrite star_occurs_false. 2: auto.
rewrite star_occurs_false. 2: eapply compose1_closed.
rewrite star_id.
rewrite star_occurs_false. 2: auto.
trtac.
eapply transitive_red. eapply IHfs. eapply fold_left_app_preserves_red.
eapply compose1_red.
Qed.
Theorem compose_red:
forall fs xs g,
t_red (fold_left App xs (fold_left App fs (compose (List.length fs) (List.length xs) @ g)))
(fold_left App (map (fun f => (fold_left App xs f)) fs) g).
Proof.
intros; unfold compose; eapply transitive_red. eapply fold_left_app_preserves_red.
eapply fold_left_app_preserves_red.
eapply transitive_red. trtac. trtac.
eapply transitive_red. eapply compose0_red.
eapply fold_left_app_preserves_red. eapply Kn_red.
Qed.
(*** Zero, successor and numerals as iterators *)
Definition zero := KI.
Definition succ1 := S1 (S1 (K @ Node) @ (S1 (K @ Node) @ K)).
Theorem succ1_red: forall n f x, t_red (succ1 @ n @ f @ x) (f @ (n @ f @ x)).
Proof. tree_red. Qed.
Definition num k := iter k (fun x => succ1 @ x) zero.
Lemma num_iterates: forall m f x, t_red (num m @ f @ x) (iter m (App f) x).
Proof.
induction m; intros; unfold num; fold num; simpl; eauto; repeat trtac. tree_red.
unfold succ1 at 1; trtac. eapply preserves_appr_t_red; eapply IHm.
Qed.
Lemma num_succ_red: forall k, t_red (num (S k)) (succ1 @ (num k)).
Proof. tree_red. Qed.
(*** The zero test and conditionals *)
Definition isZero := \"n" (Ref "n" @ (K @ ff) @ tt).
Theorem isZero_zero: t_red (isZero @ zero) tt.
Proof. tree_red. Qed.
Theorem isZero_succ: forall n, t_red (isZero @ (succ1 @ n)) ff.
Proof. tree_red. Qed.
Definition cond := I.
Theorem cond_false : forall x y, t_red (cond @ ff @ x @ y) y.
Proof. tree_red. Qed.
Theorem cond_true : forall x y, t_red (cond @ tt @ x @ y) x.
Proof. tree_red. Qed.
(*** The predecessor function - underpinning primitive recursion *)
Definition PZero := pairL zero zero. (* \ (Var 0 @ (num 0) @ (num 0)). *)
Definition PSucc := \"p" (pairL (sndL @ Ref "p") (succ1 @ (sndL @ Ref "p"))).
Definition predN := \"n" (fstL @ (Ref "n" @ PSucc @ PZero)).
Lemma PSucc_red: forall m n, t_red (PSucc @ (pairL m n)) (pairL n (succ1 @ n)).
Proof. tree_red. Qed.
Lemma pred_aux: forall k, t_red (iter k (App PSucc) PZero) (pairL (num (Nat.pred k)) (num k)).
Proof.
induction k; intros; repeat eexists; simpl; [
eapply zero_red |
aptac; [ trtac | eapply IHk | eapply PSucc_red]].
Qed.
Theorem pred_red: forall k, t_red (predN @ (num k)) (num (pred k)).
Proof.
intros; unfold predN. repeat startac2. unfold fstL; repeat startac2.
aptac. trtac. eapply transitive_red. eapply num_iterates. eapply pred_aux.
unfold pairL. trtac.
Qed.
(*** Primitive Recursion *)
Definition primrec0_abs :=
\"n"
(\"x"
(isZero @ (Ref "n")
@ Ref "g"
@ (Ref "h" @ (predN @ Ref "n") @ (Ref "x" @ (predN @ Ref "n")))
)).
Lemma primrec0_val:
primrec0_abs =
S1 (S1 (K @ △) @ (S1 (K @ △) @ (S1 (K @ K) @ (S1 (S1 (K @ isZero) @ I) @ (K @ Ref "g"))))) @
(S1 (S1 (K @ △) @ (S1 (K @ △) @ (S1 (K @ K) @ (S1 (K @ Ref "h") @ (S1 (K @ predN) @ I))))) @
(S1 (K @ (△ @ (△ @ I))) @ (S1 (K @ K) @ (S1 (K @ predN) @ I)))) .
Proof. unfold primrec0_abs; startac "x"; startac "n"; auto. Qed.
Definition primrec0 g h :=
Yop2 (S1 (S1 (K @ Node) @ (S1 (K @ Node) @ (S1 (K @ K) @ (S1 (S1 (K @ isZero) @ I) @ (K @ g))))) @
(S1 (S1 (K @ △) @ (S1 (K @ Node) @ (S1 (K @ K) @ (S1 (K @ h) @ (S1 (K @ predN) @ I))))) @
(S1 (K @ (Node @ (△ @ I))) @ (S1 (K @ K) @ (S1 (K @ predN) @ I))))) .
Lemma primrec0_red_zero :
forall g h, t_red (primrec0 g h @ zero) g.
Proof.
intros; unfold primrec0; startac "y"; startac "x". eapply transitive_red.
eapply Y2_red. trtac. aptac. aptac. aptac. trtac.
trtac. eapply isZero_zero. trtac. trtac. trtac. trtac.
Qed.
Lemma primrec0_red_succ :
forall k g h, t_red (primrec0 g h @ (num (S k))) (h @ (num k) @ (primrec0 g h @ (num k))).
Proof.
intros; unfold primrec0; startac "y"; startac "x". eapply transitive_red.
eapply Y2_red. trtac.
aptac. aptac. eapply isZero_succ. trtac. trtac. trtac. unfold ff. trtac.
eapply preserves_app_t_red. eapply preserves_app_t_red. trtac.
eapply pred_red.
eapply preserves_app_t_red. trtac.
eapply pred_red.
Qed.
Definition primrec g h xs := primrec0 (fold_left App xs g) (fold_left App xs h).
Theorem primrec_red_zero:
forall xs g h, t_red (primrec g h xs @ zero) (fold_left App xs g).
Proof. intros; eapply primrec0_red_zero. Qed.
Theorem primrec_red_succ:
forall xs g h k,
t_red (primrec g h xs @ (num (S k))) (fold_left App xs h @ (num k) @ (primrec g h xs @ (num k))).
Proof. intros; simpl; auto; eapply primrec0_red_succ. Qed.
Definition prim_plus0 n := primrec I (K @ (K @ (\"z" (succ1 @ (Ref "z"))))) (n::nil).
Definition prim_plus := \"n" (prim_plus0 (Ref "n")).
Theorem prim_plus_zero: forall n, t_red (prim_plus @ n @ zero) n.
Proof.
intros. aptac. eapply star_beta. trtac.
replace (substitute (prim_plus0 (Ref "n")) "n" n) with (prim_plus0 n) by (cbv; auto).
eapply transitive_red; [ eapply primrec_red_zero | tree_red].
Qed.
Theorem prim_plus0_succ1: forall m n, t_red (prim_plus0 m @ (num (S n))) (succ1 @ (prim_plus0 m @ (num n))).
Proof. intros; eapply transitive_red; [ eapply primrec_red_succ | simpl; trtac]. Qed.
(*** Minimisation *)
Definition minrec_abs := \"r1" (\"r0" (Ref "f" @ Ref "r1" @ Ref "r1" @ (Ref "r0" @ (succ1 @ (Ref "r1"))))).
Lemma min_rec_abs_val :
minrec_abs = △ @
(△ @
(△ @ (△ @ (△ @ △ @ △)) @
(△ @ (△ @ (△ @ △ @ △)) @
(△ @ (△ @ (△ @ △ @ (△ @ △))) @
(△ @ (△ @ (△ @ (△ @ (△ @ △ @ Ref "f")) @ (△ @ (△ @ (△ @ △)) @ (△ @ △)))) @
(△ @ (△ @ (△ @ △)) @ (△ @ △))))))) @
(△ @ (△ @ (△ @ △ @ (△ @ (△ @ (△ @ (△ @ (△ @ △)) @ (△ @ △)))))) @ \ "r1" (K @ (succ1 @ (Ref "r1")))).
Proof. unfold minrec_abs; repeat startac2. Qed.
Definition minrec0 f := Yop2 ( △ @
(△ @
(△ @ (△ @ (△ @ △ @ △)) @
(△ @ (△ @ (△ @ △ @ △)) @
(△ @ (△ @ (△ @ △ @ (△ @ △))) @
(△ @ (△ @ (△ @ (△ @ (△ @ △ @ f)) @ (△ @ (△ @ (△ @ △)) @ (△ @ △)))) @
(△ @ (△ @ (△ @ △)) @ (△ @ △))))))) @
(△ @ (△ @ (△ @ △ @ (△ @ (△ @ (△ @ (△ @ (△ @ △)) @ (△ @ △)))))) @ \ "r1" (K @ (succ1 @ (Ref "r1"))))).
Lemma minrec0_found: forall f n, t_red (f @ n) tt -> t_red (minrec0 f @ n) n.
Proof.
intros; eapply transitive_red; [ eapply Y2_red | trtac].
aptac. aptac. eauto. trtac. trtac. trtac. trtac.
Qed.