list.bend

bend-mathlib@0.6.0.0 · 2,029 lines · 0 definitions

1# bend-mathlib/list.bend: List lemmas (append, reverse, length, take, drop, map, fold, filter).
2import Base
3import bend-mathlib@0.6.0.0/nat.bend as MNat
4import bend-mathlib@0.6.0.0/bool.bend as MBool
5
6# The empty list is a right identity for append: xs ++ [] = xs.
7law append_nil:
8  for -a: Quant
9  for -A: Kind(a)
10  for xs: List<a, A>
11  {List.append(a, A, xs, Nil{}) == xs : List<a, A>}
12
13def append_nil(a, A, xs):
14  match xs:
15    case Nil{}:
16      {==}
17    case h <> t:
18      %append_nil(a, A, t) : {h <> List.append(a, A, t, Nil{}) == h <> _ : List<a, A>}
19      {==}
20
21# The empty list is a left identity for append: [] ++ xs = xs.
22law nil_append:
23  for -a: Quant
24  for -A: Kind(a)
25  for -xs: List<a, A>
26  {List.append(a, A, Nil{}, xs) == xs : List<a, A>}
27
28def nil_append(a, A, xs):
29  {==}
30
31# Append is associative: (xs ++ ys) ++ zs = xs ++ (ys ++ zs).
32law append_assoc:
33  for -a: Quant
34  for -A: Kind(a)
35  for xs: List<a, A>
36  for -ys: List<a, A>
37  for -zs: List<a, A>
38  {List.append(a, A, List.append(a, A, xs, ys), zs) == List.append(a, A, xs, List.append(a, A, ys, zs)) : List<a, A>}
39
40def append_assoc(a, A, xs, ys, zs):
41  match xs:
42    case Nil{}:
43      {==}
44    case h <> t:
45      %append_assoc(a, A, t, ys, zs) : {h <> List.append(a, A, List.append(a, A, t, ys), zs) == h <> _ : List<a, A>}
46      {==}
47
48# The length of an append is the sum of the lengths.
49law length_append:
50  for -a: Quant
51  for -A: Kind(a)
52  for xs: List<a, A>
53  for -ys: List<a, A>
54  {List.length(a, A, List.append(a, A, xs, ys)) == Nat.add(List.length(a, A, xs), List.length(a, A, ys)) : Nat}
55
56def length_append(a, A, xs, ys):
57  match xs:
58    case Nil{}:
59      {==}
60    case h <> t:
61      %length_append(a, A, t, ys) : {1n+List.length(a, A, List.append(a, A, t, ys)) == 1n+_ : Nat}
62      {==}
63
64def internal_reverse_go_append(a, -A: Kind(a), xs: List<a, A>, -ys: List<a, A>, -acc: List<a, A>) -> {List.reverse.go(a, A, xs, List.append(a, A, ys, acc)) == List.append(a, A, List.reverse.go(a, A, xs, ys), acc) : List<a, A>}:
65  match xs:
66    case Nil{}:
67      {==}
68    case h <> t:
69      internal_reverse_go_append(a, A, t, h <> ys, acc)
70
71# The reverse accumulator loop appends the reversed list to the accumulator.
72law reverse_go_spec:
73  for -a: Quant
74  for -A: Kind(a)
75  for xs: List<a, A>
76  for -acc: List<a, A>
77  {List.reverse.go(a, A, xs, acc) == List.append(a, A, List.reverse(a, A, xs), acc) : List<a, A>}
78
79def reverse_go_spec(a, A, xs, acc):
80  match xs:
81    case Nil{}:
82      {==}
83    case h <> t:
84      internal_reverse_go_append(a, A, t, h <> Nil{}, acc)
85
86def internal_reverse_append_go(a, -A: Kind(a), xs: List<a, A>, -ys: List<a, A>, -acc: List<a, A>) -> {List.reverse.go(a, A, List.append(a, A, xs, ys), acc) == List.reverse.go(a, A, ys, List.reverse.go(a, A, xs, acc)) : List<a, A>}:
87  match xs:
88    case Nil{}:
89      {==}
90    case h <> t:
91      internal_reverse_append_go(a, A, t, ys, h <> acc)
92
93# Reversing an append reverses and swaps the parts: reverse (xs ++ ys) = reverse ys ++ reverse xs.
94law reverse_append:
95  for -a: Quant
96  for -A: Kind(a)
97  for xs: List<a, A>
98  for ys: List<a, A>
99  {List.reverse(a, A, List.append(a, A, xs, ys)) == List.append(a, A, List.reverse(a, A, ys), List.reverse(a, A, xs)) : List<a, A>}
100
101def reverse_append(a, A, xs, ys):
102  Equal.trans(List<a, A>, List.reverse(a, A, List.append(a, A, xs, ys)), List.reverse.go(a, A, ys, List.reverse(a, A, xs)), List.append(a, A, List.reverse(a, A, ys), List.reverse(a, A, xs)), internal_reverse_append_go(a, A, xs, ys, Nil{}), reverse_go_spec(a, A, ys, List.reverse(a, A, xs)))
103
104def internal_reverse_go_go(a, -A: Kind(a), xs: List<a, A>, -acc: List<a, A>) -> {List.reverse.go(a, A, List.reverse.go(a, A, xs, acc), Nil{}) == List.reverse.go(a, A, acc, xs) : List<a, A>}:
105  match xs:
106    case Nil{}:
107      {==}
108    case h <> t:
109      internal_reverse_go_go(a, A, t, h <> acc)
110
111# Reversing twice gives the list back.
112law reverse_reverse:
113  for -a: Quant
114  for -A: Kind(a)
115  for xs: List<a, A>
116  {List.reverse(a, A, List.reverse(a, A, xs)) == xs : List<a, A>}
117
118def reverse_reverse(a, A, xs):
119  internal_reverse_go_go(a, A, xs, Nil{})
120
121def internal_length_reverse_go(a, -A: Kind(a), xs: List<a, A>, -acc: List<a, A>, +n: Nat, e: {List.length(a, A, acc) == n : Nat}) -> {List.length(a, A, List.reverse.go(a, A, xs, acc)) == Nat.add(n, List.length(a, A, xs)) : Nat}:
122  match xs:
123    case Nil{}:
124      %Equal.sym(Nat, Nat.add(n, 0n), n, MNat.add_zero(n)) : {List.length(a, A, acc) == _ : Nat}
125      e
126    case h <> t:
127      %Equal.sym(Nat, Nat.add(n, 1n+List.length(a, A, t)), 1n+Nat.add(n, List.length(a, A, t)), MNat.add_succ(n, List.length(a, A, t))) : {List.length(a, A, List.reverse.go(a, A, t, h <> acc)) == _ : Nat}
128      internal_length_reverse_go(a, A, t, h <> acc, 1n+n, Equal.cong(Nat, Nat, k => 1n+k, List.length(a, A, acc), n, e))
129
130# Reversing preserves the length.
131law length_reverse:
132  for -a: Quant
133  for -A: Kind(a)
134  for xs: List<a, A>
135  {List.length(a, A, List.reverse(a, A, xs)) == List.length(a, A, xs) : Nat}
136
137def length_reverse(a, A, xs):
138  internal_length_reverse_go(a, A, xs, Nil{}, 0n, {==})
139
140# A right fold over an append folds the first part onto the fold of the second.
141law foldr_append:
142  for ~a: Quant
143  for ~A: Kind(a)
144  for ~B: Type
145  for ~f: A -> B -> B
146  for xs: List<a, A>
147  for -ys: List<a, A>
148  for -z: B
149  {List.foldr(~a, ~A, ~B, ~f, List.append(a, A, xs, ys), z) == List.foldr(~a, ~A, ~B, ~f, xs, List.foldr(~a, ~A, ~B, ~f, ys, z)) : B}
150
151def foldr_append(a, A, B, f, xs, ys, z):
152  match xs:
153    case Nil{}:
154      {==}
155    case h <> t:
156      %foldr_append(~a, ~A, ~B, ~f, t, ys, z) : {f(h, List.foldr(~a, ~A, ~B, ~f, List.append(a, A, t, ys), z)) == f(h, _) : B}
157      {==}
158
159# Taking n elements and appending the rest after dropping n gives the list back.
160law take_append_drop:
161  for -a: Quant
162  for -A: Kind(a)
163  for xs: List<a, A>
164  for n: Nat
165  {List.append(a, A, List.take(a, A, xs, n), List.drop(a, A, xs, n)) == xs : List<a, A>}
166
167def take_append_drop(a, A, xs, n):
168  match xs n:
169    case Nil{} _:
170      {==}
171    case h <> t 0n:
172      {==}
173    case h <> t 1n+p:
174      %take_append_drop(a, A, t, p) : {h <> List.append(a, A, List.take(a, A, t, p), List.drop(a, A, t, p)) == h <> _ : List<a, A>}
175      {==}
176
177# Mapping preserves the length.
178law length_map:
179  for ~A: Type
180  for ~B: Type
181  for ~f: A -> B
182  for xs: List<A>
183  {List.length(&1, B, List.map(~A, ~B, ~f, xs)) == List.length(&1, A, xs) : Nat}
184
185def length_map(A, B, f, xs):
186  match xs:
187    case Nil{}:
188      {==}
189    case h <> t:
190      %length_map(~A, ~B, ~f, t) : {1n+List.length(&1, B, List.map(~A, ~B, ~f, t)) == 1n+_ : Nat}
191      {==}
192
193# Mapping over an append maps each part: map f (xs ++ ys) = map f xs ++ map f ys.
194law map_append:
195  for ~A: Type
196  for ~B: Type
197  for ~f: A -> B
198  for xs: List<A>
199  for -ys: List<A>
200  {List.map(~A, ~B, ~f, List.append(&1, A, xs, ys)) == List.append(&1, B, List.map(~A, ~B, ~f, xs), List.map(~A, ~B, ~f, ys)) : List<B>}
201
202def map_append(A, B, f, xs, ys):
203  match xs:
204    case Nil{}:
205      {==}
206    case h <> t:
207      %map_append(~A, ~B, ~f, t, ys) : {f(h) <> List.map(~A, ~B, ~f, List.append(&1, A, t, ys)) == f(h) <> _ : List<B>}
208      {==}
209
210# Mapping twice is mapping the composition: map g (map f xs) = map (g . f) xs.
211law map_map:
212  for ~A: Type
213  for ~B: Type
214  for ~C: Type
215  for ~f: A -> B
216  for ~g: B -> C
217  for xs: List<A>
218  {List.map(~B, ~C, ~g, List.map(~A, ~B, ~f, xs)) == List.map(~A, ~C, ~(x => g(f(x))), xs) : List<C>}
219
220def map_map(A, B, C, f, g, xs):
221  match xs:
222    case Nil{}:
223      {==}
224    case h <> t:
225      %map_map(~A, ~B, ~C, ~f, ~g, t) : {g(f(h)) <> List.map(~B, ~C, ~g, List.map(~A, ~B, ~f, t)) == g(f(h)) <> _ : List<C>}
226      {==}
227
228# Taking zero elements gives the empty list.
229law take_zero:
230  for -a: Quant
231  for -A: Kind(a)
232  for xs: List<a, A>
233  {List.take(a, A, xs, 0n) == Nil{} : List<a, A>}
234
235def take_zero(a, A, xs):
236  match xs:
237    case Nil{}:
238      {==}
239    case h <> t:
240      {==}
241
242# Dropping zero elements gives the list back.
243law drop_zero:
244  for -a: Quant
245  for -A: Kind(a)
246  for xs: List<a, A>
247  {List.drop(a, A, xs, 0n) == xs : List<a, A>}
248
249def drop_zero(a, A, xs):
250  match xs:
251    case Nil{}:
252      {==}
253    case h <> t:
254      {==}
255
256# Taking from the empty list gives the empty list.
257law take_nil:
258  for -a: Quant
259  for -A: Kind(a)
260  for -n: Nat
261  {List.take(a, A, Nil{}, n) == Nil{} : List<a, A>}
262
263def take_nil(a, A, n):
264  {==}
265
266# Dropping from the empty list gives the empty list.
267law drop_nil:
268  for -a: Quant
269  for -A: Kind(a)
270  for -n: Nat
271  {List.drop(a, A, Nil{}, n) == Nil{} : List<a, A>}
272
273def drop_nil(a, A, n):
274  {==}
275
276# Taking n elements leaves min(n, length) of them.
277law length_take:
278  for -a: Quant
279  for -A: Kind(a)
280  for xs: List<a, A>
281  for n: Nat
282  {List.length(a, A, List.take(a, A, xs, n)) == Nat.min(n, List.length(a, A, xs)) : Nat}
283
284def length_take(a, A, xs, n):
285  match xs n:
286    case Nil{} _:
287      Equal.sym(Nat, Nat.min(n, 0n), 0n, MNat.min_zero(n))
288    case h <> t 0n:
289      {==}
290    case h <> t 1n+p:
291      %length_take(a, A, t, p) : {1n+List.length(a, A, List.take(a, A, t, p)) == 1n+_ : Nat}
292      {==}
293
294# Dropping n elements leaves length - n of them.
295law length_drop:
296  for -a: Quant
297  for -A: Kind(a)
298  for xs: List<a, A>
299  for n: Nat
300  {List.length(a, A, List.drop(a, A, xs, n)) == Nat.sub(List.length(a, A, xs), n) : Nat}
301
302def length_drop(a, A, xs, n):
303  match xs n:
304    case Nil{} _:
305      Equal.sym(Nat, Nat.sub(0n, n), 0n, MNat.zero_sub(n))
306    case h <> t 0n:
307      {==}
308    case h <> t 1n+p:
309      length_drop(a, A, t, p)
310
311# Taking as many elements as the list has gives the list back.
312law take_length:
313  for -A: Data
314  for +xs: List<&2, A>
315  {List.take(&2, A, xs, List.length(&2, A, xs)) == xs : List<&2, A>}
316
317def take_length(A, xs):
318  match xs:
319    case Nil{}:
320      {==}
321    case h <> t:
322      %take_length(A, t) : {h <> List.take(&2, A, t, List.length(&2, A, t)) == h <> _ : List<&2, A>}
323      {==}
324
325# Dropping as many elements as the list has gives the empty list.
326law drop_length:
327  for -A: Data
328  for +xs: List<&2, A>
329  {List.drop(&2, A, xs, List.length(&2, A, xs)) == Nil{} : List<&2, A>}
330
331def drop_length(A, xs):
332  match xs:
333    case Nil{}:
334      {==}
335    case h <> t:
336      drop_length(A, t)
337
338# Taking m from the first n is taking min(n, m).
339law take_take:
340  for -a: Quant
341  for -A: Kind(a)
342  for xs: List<a, A>
343  for n: Nat
344  for m: Nat
345  {List.take(a, A, List.take(a, A, xs, n), m) == List.take(a, A, xs, Nat.min(n, m)) : List<a, A>}
346
347def take_take(a, A, xs, n, m):
348  match xs n m:
349    case Nil{} _ _:
350      {==}
351    case h <> t 0n _:
352      {==}
353    case h <> t 1n+p 0n:
354      {==}
355    case h <> t 1n+p 1n+q:
356      %take_take(a, A, t, p, q) : {h <> List.take(a, A, List.take(a, A, t, p), q) == h <> _ : List<a, A>}
357      {==}
358
359# Dropping m after dropping n is dropping n + m.
360law drop_drop:
361  for -a: Quant
362  for -A: Kind(a)
363  for xs: List<a, A>
364  for n: Nat
365  for -m: Nat
366  {List.drop(a, A, List.drop(a, A, xs, n), m) == List.drop(a, A, xs, Nat.add(n, m)) : List<a, A>}
367
368def drop_drop(a, A, xs, n, m):
369  match xs n m:
370    case Nil{} _ _:
371      {==}
372    case h <> t 0n _:
373      {==}
374    case h <> t 1n+p _:
375      drop_drop(a, A, t, p, m)
376
377# Reversing the empty list gives the empty list.
378law reverse_nil:
379  for -a: Quant
380  for -A: Kind(a)
381  {List.reverse(a, A, Nil{}) == Nil{} : List<a, A>}
382
383def reverse_nil(a, A):
384  {==}
385
386# Reversing a one-element list gives it back.
387law reverse_singleton:
388  for -a: Quant
389  for -A: Kind(a)
390  for -x: A
391  {List.reverse(a, A, [x]) == [x] : List<a, A>}
392
393def reverse_singleton(a, A, x):
394  {==}
395
396# Replicating x n times gives a list of length n.
397law length_replicate:
398  for -A: Data
399  for n: Nat
400  for -x: A
401  {List.length(&2, A, List.replicate(A, n, x)) == n : Nat}
402
403def length_replicate(A, n, x):
404  match n:
405    case 0n:
406      {==}
407    case 1n+p:
408      %length_replicate(A, p, x) : {1n+List.length(&2, A, List.replicate(A, p, x)) == 1n+_ : Nat}
409      {==}
410
411# Range(n) has length n.
412law length_range:
413  for n: Nat
414  {List.length(&2, Nat, List.range(n)) == n : Nat}
415
416def internal_length_range_go(+n: Nat, acc: List<&2, Nat>) -> {List.length(&2, Nat, List.range.go(n, acc)) == Nat.add(n, List.length(&2, Nat, acc)) : Nat}:
417  match n:
418    case 0n:
419      {==}
420    case 1n+p:
421      %MNat.add_succ(p, List.length(&2, Nat, acc)) : {List.length(&2, Nat, List.range.go(p, p <> acc)) == _ : Nat}
422      internal_length_range_go(p, p <> acc)
423
424def length_range(n):
425  +n = n
426  %Equal.sym(Nat, List.length(&2, Nat, List.range.go(n, Nil{})), Nat.add(n, 0n), internal_length_range_go(n, Nil{})) : {_ == n : Nat}
427  MNat.add_zero(n)
428
429# Zipping two lists gives the length of the shorter one.
430law length_zip:
431  for -a: Quant
432  for -A: Kind(a)
433  for xs: List<a, A>
434  for ys: List<a, A>
435  {List.length(&1, A & A, List.zip(a, A, a, A, xs, ys)) == Nat.min(List.length(a, A, xs), List.length(a, A, ys)) : Nat}
436
437def length_zip(a, A, xs, ys):
438  match xs ys:
439    case Nil{} _:
440      {==}
441    case h <> t Nil{}:
442      {==}
443    case h <> t y <> yt:
444      %length_zip(a, A, t, yt) : {1n+List.length(&1, A & A, List.zip(a, A, a, A, t, yt)) == 1n+_ : Nat}
445      {==}
446
447# Appending after a cons: (x :: xs) ++ ys = x :: (xs ++ ys).
448law append_cons:
449  for -a: Quant
450  for -A: Kind(a)
451  for -x: A
452  for -xs: List<a, A>
453  for -ys: List<a, A>
454  {List.append(a, A, x <> xs, ys) == x <> List.append(a, A, xs, ys) : List<a, A>}
455
456def append_cons(a, A, x, xs, ys):
457  {==}
458
459# The empty list has length zero.
460law length_nil:
461  for -a: Quant
462  for -A: Kind(a)
463  {List.length(a, A, Nil{}) == 0n : Nat}
464
465def length_nil(a, A):
466  {==}
467
468# A cons is one longer than its tail.
469law length_cons:
470  for -a: Quant
471  for -A: Kind(a)
472  for -x: A
473  for -xs: List<a, A>
474  {List.length(a, A, x <> xs) == 1n+List.length(a, A, xs) : Nat}
475
476def length_cons(a, A, x, xs):
477  {==}
478
479# Concatenating an append concatenates each part: concat (xss ++ yss) = concat xss ++ concat yss.
480law concat_append:
481  for -a: Quant
482  for -A: Kind(a)
483  for xss: List<a, List<a, A>>
484  for -yss: List<a, List<a, A>>
485  {List.concat(a, A, List.append(a, List<a, A>, xss, yss)) == List.append(a, A, List.concat(a, A, xss), List.concat(a, A, yss)) : List<a, A>}
486
487def concat_append(a, A, xss, yss):
488  match xss:
489    case Nil{}:
490      {==}
491    case h <> t:
492      %Equal.sym(List<a, A>, List.concat(a, A, List.append(a, List<a, A>, t, yss)), List.append(a, A, List.concat(a, A, t), List.concat(a, A, yss)), concat_append(a, A, t, yss)) : {List.append(a, A, h, _) == List.append(a, A, List.append(a, A, h, List.concat(a, A, t)), List.concat(a, A, yss)) : List<a, A>}
493      Equal.sym(List<a, A>, List.append(a, A, List.append(a, A, h, List.concat(a, A, t)), List.concat(a, A, yss)), List.append(a, A, h, List.append(a, A, List.concat(a, A, t), List.concat(a, A, yss))), append_assoc(a, A, h, List.concat(a, A, t), List.concat(a, A, yss)))
494
495# A left fold over an append folds the second part from the fold of the first.
496law foldl_append:
497  for ~a: Quant
498  for ~A: Kind(a)
499  for ~B: Type
500  for ~f: B -> A -> B
501  for xs: List<a, A>
502  for -ys: List<a, A>
503  for -z: B
504  {List.foldl(~a, ~A, ~B, ~f, List.append(a, A, xs, ys), z) == List.foldl(~a, ~A, ~B, ~f, ys, List.foldl(~a, ~A, ~B, ~f, xs, z)) : B}
505
506def foldl_append(a, A, B, f, xs, ys, z):
507  match xs:
508    case Nil{}:
509      {==}
510    case h <> t:
511      foldl_append(~a, ~A, ~B, ~f, t, ys, f(z, h))
512
513# Mapping commutes with reversing: map f (reverse xs) = reverse (map f xs).
514law map_reverse:
515  for ~A: Type
516  for ~B: Type
517  for ~f: A -> B
518  for xs: List<A>
519  {List.map(~A, ~B, ~f, List.reverse(&1, A, xs)) == List.reverse(&1, B, List.map(~A, ~B, ~f, xs)) : List<B>}
520
521def internal_map_reverse_go(~A: Type, ~B: Type, ~f: A -> B, xs: List<A>, acc: List<A>) -> {List.map(~A, ~B, ~f, List.reverse.go(&1, A, xs, acc)) == List.reverse.go(&1, B, List.map(~A, ~B, ~f, xs), List.map(~A, ~B, ~f, acc)) : List<B>}:
522  match xs:
523    case Nil{}:
524      {==}
525    case h <> t:
526      internal_map_reverse_go(~A, ~B, ~f, t, h <> acc)
527
528def map_reverse(A, B, f, xs):
529  internal_map_reverse_go(~A, ~B, ~f, xs, Nil{})
530
531# All over an append is all over each part, joined by and.
532law all_append:
533  for ~a: Quant
534  for ~A: Kind(a)
535  for ~f: A -> Bool
536  for xs: List<a, A>
537  for -ys: List<a, A>
538  {List.all(~a, ~A, ~f, List.append(a, A, xs, ys)) == Bool.and(List.all(~a, ~A, ~f, xs), List.all(~a, ~A, ~f, ys)) : Bool}
539
540def all_append(a, A, f, xs, ys):
541  match xs:
542    case Nil{}:
543      {==}
544    case h <> t:
545      %Equal.sym(Bool, List.all(~a, ~A, ~f, List.append(a, A, t, ys)), Bool.and(List.all(~a, ~A, ~f, t), List.all(~a, ~A, ~f, ys)), all_append(~a, ~A, ~f, t, ys)) : {Bool.and(f(h), _) == Bool.and(Bool.and(f(h), List.all(~a, ~A, ~f, t)), List.all(~a, ~A, ~f, ys)) : Bool}
546      Equal.sym(Bool, Bool.and(Bool.and(f(h), List.all(~a, ~A, ~f, t)), List.all(~a, ~A, ~f, ys)), Bool.and(f(h), Bool.and(List.all(~a, ~A, ~f, t), List.all(~a, ~A, ~f, ys))), MBool.and_assoc(f(h), List.all(~a, ~A, ~f, t), List.all(~a, ~A, ~f, ys)))
547
548# Any over an append is any over each part, joined by or.
549law any_append:
550  for ~a: Quant
551  for ~A: Kind(a)
552  for ~f: A -> Bool
553  for xs: List<a, A>
554  for -ys: List<a, A>
555  {List.any(~a, ~A, ~f, List.append(a, A, xs, ys)) == Bool.or(List.any(~a, ~A, ~f, xs), List.any(~a, ~A, ~f, ys)) : Bool}
556
557def any_append(a, A, f, xs, ys):
558  match xs:
559    case Nil{}:
560      {==}
561    case h <> t:
562      %Equal.sym(Bool, List.any(~a, ~A, ~f, List.append(a, A, t, ys)), Bool.or(List.any(~a, ~A, ~f, t), List.any(~a, ~A, ~f, ys)), any_append(~a, ~A, ~f, t, ys)) : {Bool.or(f(h), _) == Bool.or(Bool.or(f(h), List.any(~a, ~A, ~f, t)), List.any(~a, ~A, ~f, ys)) : Bool}
563      Equal.sym(Bool, Bool.or(Bool.or(f(h), List.any(~a, ~A, ~f, t)), List.any(~a, ~A, ~f, ys)), Bool.or(f(h), Bool.or(List.any(~a, ~A, ~f, t), List.any(~a, ~A, ~f, ys))), MBool.or_assoc(f(h), List.any(~a, ~A, ~f, t), List.any(~a, ~A, ~f, ys)))
564
565# Filtering an append filters each part.
566law filter_append:
567  for ~A: Data
568  for ~f: A -> Bool
569  for xs: List<&2, A>
570  for ys: List<&2, A>
571  {List.filter(~A, ~f, List.append(&2, A, xs, ys)) == List.append(&2, A, List.filter(~A, ~f, xs), List.filter(~A, ~f, ys)) : List<&2, A>}
572
573def internal_filter_put_append(-A: Data, h: A, r: List<&2, A>, s: List<&2, A>, b: Bool) -> {List.append(&2, A, List.filter.put(A, h, r, b), s) == List.filter.put(A, h, List.append(&2, A, r, s), b) : List<&2, A>}:
574  match b:
575    case False{}:
576      {==}
577    case True{}:
578      {==}
579
580def filter_append(A, f, xs, ys):
581  match xs:
582    case Nil{}:
583      {==}
584    case +h <> t:
585      +ys = ys
586      +t = t
587      %Equal.sym(List<&2, A>, List.filter(~A, ~f, List.append(&2, A, t, ys)), List.append(&2, A, List.filter(~A, ~f, t), List.filter(~A, ~f, ys)), filter_append(~A, ~f, t, ys)) : {List.filter.put(A, h, _, f(h)) == List.append(&2, A, List.filter.put(A, h, List.filter(~A, ~f, t), f(h)), List.filter(~A, ~f, ys)) : List<&2, A>}
588      Equal.sym(List<&2, A>, List.append(&2, A, List.filter.put(A, h, List.filter(~A, ~f, t), f(h)), List.filter(~A, ~f, ys)), List.filter.put(A, h, List.append(&2, A, List.filter(~A, ~f, t), List.filter(~A, ~f, ys)), f(h)), internal_filter_put_append(A, h, List.filter(~A, ~f, t), List.filter(~A, ~f, ys), f(h)))
589
590# An append contains x iff either part does.
591law contains_append:
592  for ~A: Data
593  for ~eq: A -> A -> Bool
594  for xs: List<&2, A>
595  for -ys: List<&2, A>
596  for +x: A
597  {List.contains(~A, ~eq, List.append(&2, A, xs, ys), x) == Bool.or(List.contains(~A, ~eq, xs, x), List.contains(~A, ~eq, ys, x)) : Bool}
598
599def contains_append(A, eq, xs, ys, x):
600  match xs:
601    case Nil{}:
602      {==}
603    case h <> t:
604      %Equal.sym(Bool, List.contains(~A, ~eq, List.append(&2, A, t, ys), x), Bool.or(List.contains(~A, ~eq, t, x), List.contains(~A, ~eq, ys, x)), contains_append(~A, ~eq, t, ys, x)) : {Bool.or(eq(h, x), _) == Bool.or(Bool.or(eq(h, x), List.contains(~A, ~eq, t, x)), List.contains(~A, ~eq, ys, x)) : Bool}
605      Equal.sym(Bool, Bool.or(Bool.or(eq(h, x), List.contains(~A, ~eq, t, x)), List.contains(~A, ~eq, ys, x)), Bool.or(eq(h, x), Bool.or(List.contains(~A, ~eq, t, x), List.contains(~A, ~eq, ys, x))), MBool.or_assoc(eq(h, x), List.contains(~A, ~eq, t, x), List.contains(~A, ~eq, ys, x)))
606
607# Filtering never makes a list longer.
608law length_filter_le:
609  for ~A: Data
610  for ~f: A -> Bool
611  for xs: List<&2, A>
612  {Nat.is_le(List.length(&2, A, List.filter(~A, ~f, xs)), List.length(&2, A, xs)) == True{} : Bool}
613
614def internal_length_filter_put_le(-A: Data, h: A, r: List<&2, A>, b: Bool) -> {Nat.is_le(List.length(&2, A, List.filter.put(A, h, r, b)), 1n+List.length(&2, A, r)) == True{} : Bool}:
615  match b:
616    case False{}:
617      MNat.le_succ(List.length(&2, A, r))
618    case True{}:
619      MNat.le_refl(1n+List.length(&2, A, r))
620
621def length_filter_le(A, f, xs):
622  match xs:
623    case Nil{}:
624      {==}
625    case +h <> t:
626      +t = t
627      MNat.le_trans(List.length(&2, A, List.filter.put(A, h, List.filter(~A, ~f, t), f(h))), 1n+List.length(&2, A, List.filter(~A, ~f, t)), 1n+List.length(&2, A, t), internal_length_filter_put_le(A, h, List.filter(~A, ~f, t), f(h)), MNat.succ_le_succ(List.length(&2, A, List.filter(~A, ~f, t)), List.length(&2, A, t), length_filter_le(~A, ~f, t)))
628
629# Membership in a list, as a reusable proposition.
630def mem(~A: Data, ~eq: A -> A -> Bool, +x: A, xs: List<&2, A>) -> Data:
631  {List.contains(~A, ~eq, xs, x) == True{} : Bool}
632
633# A list is sorted by a comparator when every adjacent pair is in order.
634def sorted_by(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> Data:
635  {List.all(~&1, ~(A & A), ~(p => le(Pair.fst(A, A, p), Pair.snd(A, A, p))), List.zip(&2, A, &2, A, xs, List.tail(&2, A, xs))) == True{} : Bool}
636
637def internal_or_of_right(a: Bool, -b: Bool, hb: {b == True{} : Bool}) -> {Bool.or(a, b) == True{} : Bool}:
638  match a:
639    case False{}:
640      hb
641    case True{}:
642      {==}
643
644def internal_or_of_or(a: Bool, -b: Bool, -c: Bool, h: {Bool.or(a, b) == True{} : Bool}, k: {b == True{} : Bool} -> {c == True{} : Bool}) -> {Bool.or(a, c) == True{} : Bool}:
645  match a:
646    case False{}:
647      k(h)
648    case True{}:
649      {==}
650
651def internal_and_left(a: Bool, -b: Bool, h: {Bool.and(a, b) == True{} : Bool}) -> {a == True{} : Bool}:
652  match a:
653    case False{}:
654      Empty.absurd({False{} == True{} : Bool}, MNat.internal_false_ne_true(h))
655    case True{}:
656      {==}
657
658def internal_and_right(a: Bool, -b: Bool, h: {Bool.and(a, b) == True{} : Bool}) -> {b == True{} : Bool}:
659  match a:
660    case False{}:
661      Empty.absurd({b == True{} : Bool}, MNat.internal_false_ne_true(h))
662    case True{}:
663      h
664
665def internal_mem_append_left(+xs: List<&2, Nat>, +ys: List<&2, Nat>, +x: Nat, h: {List.contains(~Nat, ~Nat.is_eq, xs, x) == True{} : Bool}) -> {List.contains(~Nat, ~Nat.is_eq, List.append(&2, Nat, xs, ys), x) == True{} : Bool}:
666  match xs:
667    case Nil{}:
668      Empty.absurd({List.contains(~Nat, ~Nat.is_eq, ys, x) == True{} : Bool}, MNat.internal_false_ne_true(h))
669    case hd <> tl:
670      internal_or_of_or(Nat.is_eq(hd, x), List.contains(~Nat, ~Nat.is_eq, tl, x), List.contains(~Nat, ~Nat.is_eq, List.append(&2, Nat, tl, ys), x), h, internal_mem_append_left(tl, ys, x))
671
672def internal_mem_append_right(+xs: List<&2, Nat>, +ys: List<&2, Nat>, +x: Nat, h: {List.contains(~Nat, ~Nat.is_eq, ys, x) == True{} : Bool}) -> {List.contains(~Nat, ~Nat.is_eq, List.append(&2, Nat, xs, ys), x) == True{} : Bool}:
673  match xs:
674    case Nil{}:
675      h
676    case hd <> tl:
677      internal_or_of_right(Nat.is_eq(hd, x), List.contains(~Nat, ~Nat.is_eq, List.append(&2, Nat, tl, ys), x), internal_mem_append_right(tl, ys, x, h))
678
679# The head of a cons is a member of it.
680law mem_cons_self:
681  for x: Nat
682  for -xs: List<&2, Nat>
683  mem(~Nat, ~Nat.is_eq, x, x <> xs)
684
685def mem_cons_self(x, xs):
686  %Equal.sym(Bool, Nat.is_eq(x, x), True{}, MNat.is_eq_refl(x)) : {Bool.or(_, List.contains(~Nat, ~Nat.is_eq, xs, x)) == True{} : Bool}
687  {==}
688
689# Membership is preserved when a new head is prepended.
690law mem_cons_of_mem:
691  for x: Nat
692  for y: Nat
693  for -xs: List<&2, Nat>
694  for h: mem(~Nat, ~Nat.is_eq, x, xs)
695  mem(~Nat, ~Nat.is_eq, x, y <> xs)
696
697def mem_cons_of_mem(x, y, xs, h):
698  +x = x
699  internal_or_of_right(Nat.is_eq(y, x), List.contains(~Nat, ~Nat.is_eq, xs, x), h)
700
701# Nothing is a member of the empty list.
702law not_mem_nil:
703  for -x: Nat
704  mem(~Nat, ~Nat.is_eq, x, Nil{}) -> Empty
705
706def not_mem_nil(x, h):
707  MNat.internal_false_ne_true(h)
708
709# Membership on the left of an append.
710law mem_append_left:
711  for xs: List<&2, Nat>
712  for ys: List<&2, Nat>
713  for x: Nat
714  for h: mem(~Nat, ~Nat.is_eq, x, xs)
715  mem(~Nat, ~Nat.is_eq, x, List.append(&2, Nat, xs, ys))
716
717def mem_append_left(xs, ys, x, h):
718  internal_mem_append_left(xs, ys, x, h)
719
720# Membership on the right of an append.
721law mem_append_right:
722  for xs: List<&2, Nat>
723  for ys: List<&2, Nat>
724  for x: Nat
725  for h: mem(~Nat, ~Nat.is_eq, x, ys)
726  mem(~Nat, ~Nat.is_eq, x, List.append(&2, Nat, xs, ys))
727
728def mem_append_right(xs, ys, x, h):
729  internal_mem_append_right(xs, ys, x, h)
730
731# The empty list is sorted by any comparator.
732law sorted_nil:
733  sorted_by(~Nat, ~Nat.is_le, Nil{})
734
735def sorted_nil():
736  {==}
737
738# A singleton list is sorted by any comparator.
739law sorted_single:
740  for -x: Nat
741  sorted_by(~Nat, ~Nat.is_le, [x])
742
743def sorted_single(x):
744  {==}
745
746# A sorted tail with an in-order head is sorted.
747law sorted_cons_cons_intro:
748  for -x: Nat
749  for -y: Nat
750  for -t: List<&2, Nat>
751  for hxy: MNat.le(x, y)
752  for hyt: sorted_by(~Nat, ~Nat.is_le, y <> t)
753  sorted_by(~Nat, ~Nat.is_le, x <> y <> t)
754
755def sorted_cons_cons_intro(x, y, t, hxy, hyt):
756  %Equal.sym(Bool, Nat.is_le(x, y), True{}, hxy) : {Bool.and(_, List.all(~&1, ~(Nat & Nat), ~(p => Nat.is_le(Pair.fst(Nat, Nat, p), Pair.snd(Nat, Nat, p))), List.zip(&2, Nat, &2, Nat, y <> t, t))) == True{} : Bool}
757  hyt
758
759# The head pair of a sorted cons-cons list is in order.
760law sorted_cons_cons_elim_le:
761  for x: Nat
762  for y: Nat
763  for -t: List<&2, Nat>
764  for h: sorted_by(~Nat, ~Nat.is_le, x <> y <> t)
765  MNat.le(x, y)
766
767def sorted_cons_cons_elim_le(x, y, t, h):
768  internal_and_left(Nat.is_le(x, y), List.all(~&1, ~(Nat & Nat), ~(p => Nat.is_le(Pair.fst(Nat, Nat, p), Pair.snd(Nat, Nat, p))), List.zip(&2, Nat, &2, Nat, y <> t, t)), h)
769
770# The tail of a sorted cons-cons list is sorted.
771law sorted_cons_cons_elim_tail:
772  for x: Nat
773  for y: Nat
774  for -t: List<&2, Nat>
775  for h: sorted_by(~Nat, ~Nat.is_le, x <> y <> t)
776  sorted_by(~Nat, ~Nat.is_le, y <> t)
777
778def sorted_cons_cons_elim_tail(x, y, t, h):
779  internal_and_right(Nat.is_le(x, y), List.all(~&1, ~(Nat & Nat), ~(p => Nat.is_le(Pair.fst(Nat, Nat, p), Pair.snd(Nat, Nat, p))), List.zip(&2, Nat, &2, Nat, y <> t, t)), h)
780
781# A sorted list has a sorted tail.
782law sorted_tail:
783  for x: Nat
784  for xs: List<&2, Nat>
785  for h: sorted_by(~Nat, ~Nat.is_le, x <> xs)
786  sorted_by(~Nat, ~Nat.is_le, xs)
787
788def sorted_tail(x, xs, h):
789  match xs:
790    case Nil{}:
791      {==}
792    case y <> t:
793      internal_and_right(Nat.is_le(x, y), List.all(~&1, ~(Nat & Nat), ~(p => Nat.is_le(Pair.fst(Nat, Nat, p), Pair.snd(Nat, Nat, p))), List.zip(&2, Nat, &2, Nat, y <> t, t)), h)
794
795# Taking the length of the first part of an append gives the first part back.
796law take_left:
797  for -a: Quant
798  for -A: Kind(a)
799  for xs: List<a, A>
800  for ys: List<a, A>
801  {List.take(a, A, List.append(a, A, xs, ys), List.length(a, A, xs)) == xs : List<a, A>}
802
803def take_left(a, A, xs, ys):
804  match xs:
805    case Nil{}:
806      take_zero(a, A, ys)
807    case h <> t:
808      %take_left(a, A, t, ys) : {h <> List.take(a, A, List.append(a, A, t, ys), List.length(a, A, t)) == h <> _ : List<a, A>}
809      {==}
810
811# Dropping the length of the first part of an append gives the second part.
812law drop_left:
813  for -a: Quant
814  for -A: Kind(a)
815  for xs: List<a, A>
816  for ys: List<a, A>
817  {List.drop(a, A, List.append(a, A, xs, ys), List.length(a, A, xs)) == ys : List<a, A>}
818
819def drop_left(a, A, xs, ys):
820  match xs:
821    case Nil{}:
822      drop_zero(a, A, ys)
823    case h <> t:
824      drop_left(a, A, t, ys)
825
826# Taking past the first part of an append keeps it and takes the rest from the second part.
827law take_append:
828  for -a: Quant
829  for -A: Kind(a)
830  for xs: List<a, A>
831  for -ys: List<a, A>
832  for -n: Nat
833  {List.take(a, A, List.append(a, A, xs, ys), Nat.add(List.length(a, A, xs), n)) == List.append(a, A, xs, List.take(a, A, ys, n)) : List<a, A>}
834
835def take_append(a, A, xs, ys, n):
836  match xs:
837    case Nil{}:
838      {==}
839    case h <> t:
840      %take_append(a, A, t, ys, n) : {h <> List.take(a, A, List.append(a, A, t, ys), Nat.add(List.length(a, A, t), n)) == h <> _ : List<a, A>}
841      {==}
842
843# Dropping past the first part of an append drops the rest from the second part.
844law drop_append:
845  for -a: Quant
846  for -A: Kind(a)
847  for xs: List<a, A>
848  for -ys: List<a, A>
849  for -n: Nat
850  {List.drop(a, A, List.append(a, A, xs, ys), Nat.add(List.length(a, A, xs), n)) == List.drop(a, A, ys, n) : List<a, A>}
851
852def drop_append(a, A, xs, ys, n):
853  match xs:
854    case Nil{}:
855      {==}
856    case h <> t:
857      drop_append(a, A, t, ys, n)
858
859# Taking at most the length of the first part of an append only sees the first part.
860law take_append_of_le_length:
861  for -a: Quant
862  for -A: Kind(a)
863  for xs: List<a, A>
864  for ys: List<a, A>
865  for n: Nat
866  for h: MNat.le(n, List.length(a, A, xs))
867  {List.take(a, A, List.append(a, A, xs, ys), n) == List.take(a, A, xs, n) : List<a, A>}
868
869def take_append_of_le_length(a, A, xs, ys, n, h):
870  match xs n:
871    case Nil{} 0n:
872      take_zero(a, A, ys)
873    case Nil{} 1n+p:
874      Empty.absurd({List.take(a, A, ys, 1n+p) == Nil{} : List<a, A>}, MNat.internal_false_ne_true(h))
875    case x <> t 0n:
876      {==}
877    case x <> t 1n+p:
878      %take_append_of_le_length(a, A, t, ys, p, h) : {x <> List.take(a, A, List.append(a, A, t, ys), p) == x <> _ : List<a, A>}
879      {==}
880
881# Dropping at most the length of the first part of an append drops only from the first part.
882law drop_append_of_le_length:
883  for -a: Quant
884  for -A: Kind(a)
885  for xs: List<a, A>
886  for ys: List<a, A>
887  for n: Nat
888  for h: MNat.le(n, List.length(a, A, xs))
889  {List.drop(a, A, List.append(a, A, xs, ys), n) == List.append(a, A, List.drop(a, A, xs, n), ys) : List<a, A>}
890
891def drop_append_of_le_length(a, A, xs, ys, n, h):
892  match xs n:
893    case Nil{} 0n:
894      drop_zero(a, A, ys)
895    case Nil{} 1n+p:
896      Empty.absurd({List.drop(a, A, ys, 1n+p) == ys : List<a, A>}, MNat.internal_false_ne_true(h))
897    case x <> t 0n:
898      {==}
899    case x <> t 1n+p:
900      drop_append_of_le_length(a, A, t, ys, p, h)
901
902# Mapping commutes with taking: map f (take n xs) = take n (map f xs).
903law map_take:
904  for ~A: Type
905  for ~B: Type
906  for ~f: A -> B
907  for xs: List<A>
908  for n: Nat
909  {List.map(~A, ~B, ~f, List.take(&1, A, xs, n)) == List.take(&1, B, List.map(~A, ~B, ~f, xs), n) : List<B>}
910
911def map_take(A, B, f, xs, n):
912  match xs n:
913    case Nil{} _:
914      {==}
915    case h <> t 0n:
916      {==}
917    case h <> t 1n+p:
918      %map_take(~A, ~B, ~f, t, p) : {f(h) <> List.map(~A, ~B, ~f, List.take(&1, A, t, p)) == f(h) <> _ : List<B>}
919      {==}
920
921# Mapping commutes with dropping: map f (drop n xs) = drop n (map f xs).
922law map_drop:
923  for ~A: Type
924  for ~B: Type
925  for ~f: A -> B
926  for xs: List<A>
927  for n: Nat
928  {List.map(~A, ~B, ~f, List.drop(&1, A, xs, n)) == List.drop(&1, B, List.map(~A, ~B, ~f, xs), n) : List<B>}
929
930def map_drop(A, B, f, xs, n):
931  match xs n:
932    case Nil{} _:
933      {==}
934    case h <> t 0n:
935      {==}
936    case h <> t 1n+p:
937      map_drop(~A, ~B, ~f, t, p)
938
939# Taking m from n copies of x gives min(m, n) copies of x.
940law take_replicate:
941  for -A: Data
942  for n: Nat
943  for m: Nat
944  for -x: A
945  {List.take(&2, A, List.replicate(A, n, x), m) == List.replicate(A, Nat.min(m, n), x) : List<&2, A>}
946
947def take_replicate(A, n, m, x):
948  match n m:
949    case 0n _:
950      %MNat.min_zero_sym(m) : {Nil{} == List.replicate(A, _, x) : List<&2, A>}
951      {==}
952    case 1n+p 0n:
953      {==}
954    case 1n+p 1n+q:
955      %take_replicate(A, p, q, x) : {x <> List.take(&2, A, List.replicate(A, p, x), q) == x <> _ : List<&2, A>}
956      {==}
957
958# Dropping m from n copies of x gives n - m copies of x.
959law drop_replicate:
960  for -A: Data
961  for n: Nat
962  for m: Nat
963  for -x: A
964  {List.drop(&2, A, List.replicate(A, n, x), m) == List.replicate(A, Nat.sub(n, m), x) : List<&2, A>}
965
966def drop_replicate(A, n, m, x):
967  match n m:
968    case 0n _:
969      {==}
970    case 1n+p 0n:
971      {==}
972    case 1n+p 1n+q:
973      drop_replicate(A, p, q, x)
974
975# Taking n elements gives at most n of them.
976law length_take_le:
977  for -a: Quant
978  for -A: Kind(a)
979  for xs: List<a, A>
980  for n: Nat
981  MNat.le(List.length(a, A, List.take(a, A, xs, n)), n)
982
983def length_take_le(a, A, xs, n):
984  match xs n:
985    case Nil{} _:
986      MNat.zero_le(n)
987    case h <> t 0n:
988      {==}
989    case h <> t 1n+p:
990      MNat.succ_le_succ(List.length(a, A, List.take(a, A, t, p)), p, length_take_le(a, A, t, p))
991
992# The tail is one shorter than the list, or empty: length (tail xs) = length xs - 1.
993law length_tail:
994  for -a: Quant
995  for -A: Kind(a)
996  for xs: List<a, A>
997  {List.length(a, A, List.tail(a, A, xs)) == Nat.sub(List.length(a, A, xs), 1n) : Nat}
998
999def length_tail(a, A, xs):
1000  match xs:
1001    case Nil{}:
1002      {==}
1003    case h <> t:
1004      MNat.sub_zero_sym(List.length(a, A, t))
1005
1006# Mapping the identity gives the list back.
1007law map_id:
1008  for ~A: Type
1009  for xs: List<A>
1010  {List.map(~A, ~A, ~(x => x), xs) == xs : List<A>}
1011
1012def map_id(A, xs):
1013  match xs:
1014    case Nil{}:
1015      {==}
1016    case h <> t:
1017      %map_id(~A, t) : {h <> List.map(~A, ~A, ~(x => x), t) == h <> _ : List<A>}
1018      {==}
1019
1020# A right fold over a map folds with the mapped function: foldr g z (map f xs) = foldr (g . f) z xs.
1021law foldr_map:
1022  for ~A: Type
1023  for ~B: Type
1024  for ~C: Type
1025  for ~f: A -> B
1026  for ~g: B -> C -> C
1027  for xs: List<A>
1028  for -z: C
1029  {List.foldr(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, xs), z) == List.foldr(~&1, ~A, ~C, ~(x => y => g(f(x), y)), xs, z) : C}
1030
1031def foldr_map(A, B, C, f, g, xs, z):
1032  match xs:
1033    case Nil{}:
1034      {==}
1035    case h <> t:
1036      %foldr_map(~A, ~B, ~C, ~f, ~g, t, z) : {g(f(h), List.foldr(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, t), z)) == g(f(h), _) : C}
1037      {==}
1038
1039# A left fold over a map folds with the mapped function: foldl g z (map f xs) = foldl (fun acc x => g acc (f x)) z xs.
1040law foldl_map:
1041  for ~A: Type
1042  for ~B: Type
1043  for ~C: Type
1044  for ~f: A -> B
1045  for ~g: C -> B -> C
1046  for xs: List<A>
1047  for -z: C
1048  {List.foldl(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, xs), z) == List.foldl(~&1, ~A, ~C, ~(x => y => g(x, f(y))), xs, z) : C}
1049
1050def foldl_map(A, B, C, f, g, xs, z):
1051  match xs:
1052    case Nil{}:
1053      {==}
1054    case h <> t:
1055      foldl_map(~A, ~B, ~C, ~f, ~g, t, g(z, f(h)))
1056
1057# Replicating m + n times appends m copies to n copies.
1058law replicate_add:
1059  for -A: Data
1060  for m: Nat
1061  for -n: Nat
1062  for -x: A
1063  {List.replicate(A, Nat.add(m, n), x) == List.append(&2, A, List.replicate(A, m, x), List.replicate(A, n, x)) : List<&2, A>}
1064
1065def replicate_add(A, m, n, x):
1066  match m:
1067    case 0n:
1068      {==}
1069    case 1n+p:
1070      %replicate_add(A, p, n, x) : {x <> List.replicate(A, Nat.add(p, n), x) == x <> _ : List<&2, A>}
1071      {==}
1072
1073def internal_replicate_append_nil(-A: Data, n: Nat, -x: A) -> {List.append(&2, A, List.replicate(A, n, x), Nil{}) == List.replicate(A, n, x) : List<&2, A>}:
1074  match n:
1075    case 0n:
1076      {==}
1077    case 1n+p:
1078      %internal_replicate_append_nil(A, p, x) : {x <> List.append(&2, A, List.replicate(A, p, x), Nil{}) == x <> _ : List<&2, A>}
1079      {==}
1080
1081def internal_replicate_append_cons(-A: Data, n: Nat, -x: A, -acc: List<&2, A>) -> {List.append(&2, A, List.replicate(A, n, x), x <> acc) == x <> List.append(&2, A, List.replicate(A, n, x), acc) : List<&2, A>}:
1082  match n:
1083    case 0n:
1084      {==}
1085    case 1n+p:
1086      %internal_replicate_append_cons(A, p, x, acc) : {x <> List.append(&2, A, List.replicate(A, p, x), x <> acc) == x <> _ : List<&2, A>}
1087      {==}
1088
1089def internal_reverse_go_replicate(-A: Data, n: Nat, -x: A, -acc: List<&2, A>) -> {List.reverse.go(&2, A, List.replicate(A, n, x), acc) == List.append(&2, A, List.replicate(A, n, x), acc) : List<&2, A>}:
1090  match n:
1091    case 0n:
1092      {==}
1093    case 1n+p:
1094      +p = p
1095      Equal.trans(List<&2, A>, List.reverse.go(&2, A, List.replicate(A, p, x), x <> acc), List.append(&2, A, List.replicate(A, p, x), x <> acc), x <> List.append(&2, A, List.replicate(A, p, x), acc), internal_reverse_go_replicate(A, p, x, x <> acc), internal_replicate_append_cons(A, p, x, acc))
1096
1097# Reversing n copies of x gives them back.
1098law reverse_replicate:
1099  for -A: Data
1100  for n: Nat
1101  for -x: A
1102  {List.reverse(&2, A, List.replicate(A, n, x)) == List.replicate(A, n, x) : List<&2, A>}
1103
1104def reverse_replicate(A, n, x):
1105  +n = n
1106  Equal.trans(List<&2, A>, List.reverse(&2, A, List.replicate(A, n, x)), List.append(&2, A, List.replicate(A, n, x), Nil{}), List.replicate(A, n, x), internal_reverse_go_replicate(A, n, x, Nil{}), internal_replicate_append_nil(A, n, x))
1107
1108# Filtering with an always-true predicate gives the list back.
1109law filter_true:
1110  for ~A: Data
1111  for xs: List<&2, A>
1112  {List.filter(~A, ~(x => True{}), xs) == xs : List<&2, A>}
1113
1114def filter_true(A, xs):
1115  match xs:
1116    case Nil{}:
1117      {==}
1118    case h <> t:
1119      %filter_true(~A, t) : {h <> List.filter(~A, ~(x => True{}), t) == h <> _ : List<&2, A>}
1120      {==}
1121
1122# Filtering with an always-false predicate gives the empty list.
1123law filter_false:
1124  for ~A: Data
1125  for xs: List<&2, A>
1126  {List.filter(~A, ~(x => False{}), xs) == Nil{} : List<&2, A>}
1127
1128def filter_false(A, xs):
1129  match xs:
1130    case Nil{}:
1131      {==}
1132    case h <> t:
1133      filter_false(~A, t)
1134
1135def internal_filter_put_filter(~A: Data, ~p: A -> Bool, +h: A, r: List<&2, A>, b: Bool) -> {List.filter(~A, ~p, List.filter.put(A, h, r, b)) == List.filter.put(A, h, List.filter(~A, ~p, r), Bool.and(p(h), b)) : List<&2, A>}:
1136  match b:
1137    case False{}:
1138      %MBool.and_false_sym(p(h)) : {List.filter(~A, ~p, r) == List.filter.put(A, h, List.filter(~A, ~p, r), _) : List<&2, A>}
1139      {==}
1140    case True{}:
1141      %MBool.and_true_sym(p(h)) : {List.filter(~A, ~p, h <> r) == List.filter.put(A, h, List.filter(~A, ~p, r), _) : List<&2, A>}
1142      {==}
1143
1144# Filtering twice is filtering by both predicates: filter p (filter q xs) = filter (p and q) xs.
1145law filter_filter:
1146  for ~A: Data
1147  for ~p: A -> Bool
1148  for ~q: A -> Bool
1149  for xs: List<&2, A>
1150  {List.filter(~A, ~p, List.filter(~A, ~q, xs)) == List.filter(~A, ~(+x => Bool.and(p(x), q(x))), xs) : List<&2, A>}
1151
1152def filter_filter(A, p, q, xs):
1153  match xs:
1154    case Nil{}:
1155      {==}
1156    case +h <> t:
1157      +t = t
1158      %filter_filter(~A, ~p, ~q, t) : {List.filter(~A, ~p, List.filter.put(A, h, List.filter(~A, ~q, t), q(h))) == List.filter.put(A, h, _, Bool.and(p(h), q(h))) : List<&2, A>}
1159      internal_filter_put_filter(~A, ~p, h, List.filter(~A, ~q, t), q(h))
1160
1161def internal_all_filter_put(~A: Data, ~q: A -> Bool, -h: A, r: List<&2, A>, b: Bool) -> {List.all(~&2, ~A, ~q, List.filter.put(A, h, r, b)) == Bool.and(Bool.or(Bool.not(b), q(h)), List.all(~&2, ~A, ~q, r)) : Bool}:
1162  match b:
1163    case False{}:
1164      {==}
1165    case True{}:
1166      {==}
1167
1168# All over a filter checks q only where p holds: all q (filter p xs) = all (not p or q) xs.
1169law all_filter:
1170  for ~A: Data
1171  for ~p: A -> Bool
1172  for ~q: A -> Bool
1173  for xs: List<&2, A>
1174  {List.all(~&2, ~A, ~q, List.filter(~A, ~p, xs)) == List.all(~&2, ~A, ~(+x => Bool.or(Bool.not(p(x)), q(x))), xs) : Bool}
1175
1176def all_filter(A, p, q, xs):
1177  match xs:
1178    case Nil{}:
1179      {==}
1180    case +h <> t:
1181      +t = t
1182      %all_filter(~A, ~p, ~q, t) : {List.all(~&2, ~A, ~q, List.filter.put(A, h, List.filter(~A, ~p, t), p(h))) == Bool.and(Bool.or(Bool.not(p(h)), q(h)), _) : Bool}
1183      internal_all_filter_put(~A, ~q, h, List.filter(~A, ~p, t), p(h))
1184
1185def internal_any_filter_put(~A: Data, ~q: A -> Bool, -h: A, r: List<&2, A>, b: Bool) -> {List.any(~&2, ~A, ~q, List.filter.put(A, h, r, b)) == Bool.or(Bool.and(b, q(h)), List.any(~&2, ~A, ~q, r)) : Bool}:
1186  match b:
1187    case False{}:
1188      {==}
1189    case True{}:
1190      {==}
1191
1192# Any over a filter looks for q only where p holds: any q (filter p xs) = any (p and q) xs.
1193law any_filter:
1194  for ~A: Data
1195  for ~p: A -> Bool
1196  for ~q: A -> Bool
1197  for xs: List<&2, A>
1198  {List.any(~&2, ~A, ~q, List.filter(~A, ~p, xs)) == List.any(~&2, ~A, ~(+x => Bool.and(p(x), q(x))), xs) : Bool}
1199
1200def any_filter(A, p, q, xs):
1201  match xs:
1202    case Nil{}:
1203      {==}
1204    case +h <> t:
1205      +t = t
1206      %any_filter(~A, ~p, ~q, t) : {List.any(~&2, ~A, ~q, List.filter.put(A, h, List.filter(~A, ~p, t), p(h))) == Bool.or(Bool.and(p(h), q(h)), _) : Bool}
1207      internal_any_filter_put(~A, ~q, h, List.filter(~A, ~p, t), p(h))
1208
1209# All over a map checks the composed predicate: all p (map f xs) = all (p . f) xs.
1210law all_map:
1211  for ~A: Type
1212  for ~B: Type
1213  for ~f: A -> B
1214  for ~p: B -> Bool
1215  for xs: List<A>
1216  {List.all(~&1, ~B, ~p, List.map(~A, ~B, ~f, xs)) == List.all(~&1, ~A, ~(x => p(f(x))), xs) : Bool}
1217
1218def all_map(A, B, f, p, xs):
1219  match xs:
1220    case Nil{}:
1221      {==}
1222    case h <> t:
1223      %all_map(~A, ~B, ~f, ~p, t) : {Bool.and(p(f(h)), List.all(~&1, ~B, ~p, List.map(~A, ~B, ~f, t))) == Bool.and(p(f(h)), _) : Bool}
1224      {==}
1225
1226# Any over a map checks the composed predicate: any p (map f xs) = any (p . f) xs.
1227law any_map:
1228  for ~A: Type
1229  for ~B: Type
1230  for ~f: A -> B
1231  for ~p: B -> Bool
1232  for xs: List<A>
1233  {List.any(~&1, ~B, ~p, List.map(~A, ~B, ~f, xs)) == List.any(~&1, ~A, ~(x => p(f(x))), xs) : Bool}
1234
1235def any_map(A, B, f, p, xs):
1236  match xs:
1237    case Nil{}:
1238      {==}
1239    case h <> t:
1240      %any_map(~A, ~B, ~f, ~p, t) : {Bool.or(p(f(h)), List.any(~&1, ~B, ~p, List.map(~A, ~B, ~f, t))) == Bool.or(p(f(h)), _) : Bool}
1241      {==}
1242
1243def internal_all_reverse_go(~a: Quant, ~A: Kind(a), ~f: A -> Bool, xs: List<a, A>, -acc: List<a, A>, c: Bool, e: {List.all(~a, ~A, ~f, acc) == c : Bool}) -> {List.all(~a, ~A, ~f, List.reverse.go(a, A, xs, acc)) == Bool.and(c, List.all(~a, ~A, ~f, xs)) : Bool}:
1244  match xs c:
1245    case Nil{} True{}:
1246      e
1247    case Nil{} False{}:
1248      e
1249    case h <> t True{}:
1250      +v = f(h)
1251      internal_all_reverse_go(~a, ~A, ~f, t, h <> acc, v, Equal.trans(Bool, Bool.and(v, List.all(~a, ~A, ~f, acc)), Bool.and(v, True{}), v, Equal.cong(Bool, Bool, k => Bool.and(v, k), List.all(~a, ~A, ~f, acc), True{}, e), MBool.and_true(v)))
1252    case h <> t False{}:
1253      internal_all_reverse_go(~a, ~A, ~f, t, h <> acc, False{}, Equal.trans(Bool, Bool.and(f(h), List.all(~a, ~A, ~f, acc)), Bool.and(f(h), False{}), False{}, Equal.cong(Bool, Bool, k => Bool.and(f(h), k), List.all(~a, ~A, ~f, acc), False{}, e), MBool.and_false(f(h))))
1254
1255# Reversing does not change whether all elements satisfy a predicate.
1256law all_reverse:
1257  for ~a: Quant
1258  for ~A: Kind(a)
1259  for ~f: A -> Bool
1260  for xs: List<a, A>
1261  {List.all(~a, ~A, ~f, List.reverse(a, A, xs)) == List.all(~a, ~A, ~f, xs) : Bool}
1262
1263def all_reverse(a, A, f, xs):
1264  internal_all_reverse_go(~a, ~A, ~f, xs, Nil{}, True{}, {==})
1265
1266def internal_any_reverse_go(~a: Quant, ~A: Kind(a), ~f: A -> Bool, xs: List<a, A>, -acc: List<a, A>, c: Bool, e: {List.any(~a, ~A, ~f, acc) == c : Bool}) -> {List.any(~a, ~A, ~f, List.reverse.go(a, A, xs, acc)) == Bool.or(c, List.any(~a, ~A, ~f, xs)) : Bool}:
1267  match xs c:
1268    case Nil{} True{}:
1269      e
1270    case Nil{} False{}:
1271      e
1272    case h <> t True{}:
1273      internal_any_reverse_go(~a, ~A, ~f, t, h <> acc, True{}, Equal.trans(Bool, Bool.or(f(h), List.any(~a, ~A, ~f, acc)), Bool.or(f(h), True{}), True{}, Equal.cong(Bool, Bool, k => Bool.or(f(h), k), List.any(~a, ~A, ~f, acc), True{}, e), MBool.or_true(f(h))))
1274    case h <> t False{}:
1275      +v = f(h)
1276      internal_any_reverse_go(~a, ~A, ~f, t, h <> acc, v, Equal.trans(Bool, Bool.or(v, List.any(~a, ~A, ~f, acc)), Bool.or(v, False{}), v, Equal.cong(Bool, Bool, k => Bool.or(v, k), List.any(~a, ~A, ~f, acc), False{}, e), MBool.or_false(v)))
1277
1278# Reversing does not change whether some element satisfies a predicate.
1279law any_reverse:
1280  for ~a: Quant
1281  for ~A: Kind(a)
1282  for ~f: A -> Bool
1283  for xs: List<a, A>
1284  {List.any(~a, ~A, ~f, List.reverse(a, A, xs)) == List.any(~a, ~A, ~f, xs) : Bool}
1285
1286def any_reverse(a, A, f, xs):
1287  internal_any_reverse_go(~a, ~A, ~f, xs, Nil{}, False{}, {==})
1288
1289def internal_contains_reverse_go(~A: Data, ~eq: A -> A -> Bool, xs: List<&2, A>, -acc: List<&2, A>, +x: A, c: Bool, e: {List.contains(~A, ~eq, acc, x) == c : Bool}) -> {List.contains(~A, ~eq, List.reverse.go(&2, A, xs, acc), x) == Bool.or(c, List.contains(~A, ~eq, xs, x)) : Bool}:
1290  match xs c:
1291    case Nil{} True{}:
1292      e
1293    case Nil{} False{}:
1294      e
1295    case h <> t True{}:
1296      internal_contains_reverse_go(~A, ~eq, t, h <> acc, x, True{}, Equal.trans(Bool, Bool.or(eq(h, x), List.contains(~A, ~eq, acc, x)), Bool.or(eq(h, x), True{}), True{}, Equal.cong(Bool, Bool, k => Bool.or(eq(h, x), k), List.contains(~A, ~eq, acc, x), True{}, e), MBool.or_true(eq(h, x))))
1297    case h <> t False{}:
1298      +v = eq(h, x)
1299      internal_contains_reverse_go(~A, ~eq, t, h <> acc, x, v, Equal.trans(Bool, Bool.or(v, List.contains(~A, ~eq, acc, x)), Bool.or(v, False{}), v, Equal.cong(Bool, Bool, k => Bool.or(v, k), List.contains(~A, ~eq, acc, x), False{}, e), MBool.or_false(v)))
1300
1301# Reversing does not change whether a list contains x.
1302law contains_reverse:
1303  for ~A: Data
1304  for ~eq: A -> A -> Bool
1305  for xs: List<&2, A>
1306  for +x: A
1307  {List.contains(~A, ~eq, List.reverse(&2, A, xs), x) == List.contains(~A, ~eq, xs, x) : Bool}
1308
1309def contains_reverse(A, eq, xs, x):
1310  internal_contains_reverse_go(~A, ~eq, xs, Nil{}, x, False{}, {==})
1311
1312# --- generated: _sym twins (tools/mathlib/twins.ts), do not edit ---
1313
1314# The empty list is a right identity for append: xs ++ [] = xs, reversed to rewrite toward the simple side.
1315law append_nil_sym:
1316  for -a: Quant
1317  for -A: Kind(a)
1318  for xs: List<a, A>
1319  {xs == List.append(a, A, xs, Nil{}) : List<a, A>}
1320
1321def append_nil_sym(a, A, xs):
1322  Equal.sym(List<a, A>, List.append(a, A, xs, Nil{}), xs, append_nil(a, A, xs))
1323
1324# The empty list is a left identity for append: [] ++ xs = xs, reversed to rewrite toward the simple side.
1325law nil_append_sym:
1326  for -a: Quant
1327  for -A: Kind(a)
1328  for -xs: List<a, A>
1329  {xs == List.append(a, A, Nil{}, xs) : List<a, A>}
1330
1331def nil_append_sym(a, A, xs):
1332  Equal.sym(List<a, A>, List.append(a, A, Nil{}, xs), xs, nil_append(a, A, xs))
1333
1334# Append is associative: (xs ++ ys) ++ zs = xs ++ (ys ++ zs), reversed to rewrite toward the simple side.
1335law append_assoc_sym:
1336  for -a: Quant
1337  for -A: Kind(a)
1338  for xs: List<a, A>
1339  for -ys: List<a, A>
1340  for -zs: List<a, A>
1341  {List.append(a, A, xs, List.append(a, A, ys, zs)) == List.append(a, A, List.append(a, A, xs, ys), zs) : List<a, A>}
1342
1343def append_assoc_sym(a, A, xs, ys, zs):
1344  Equal.sym(List<a, A>, List.append(a, A, List.append(a, A, xs, ys), zs), List.append(a, A, xs, List.append(a, A, ys, zs)), append_assoc(a, A, xs, ys, zs))
1345
1346# The length of an append is the sum of the lengths, reversed to rewrite toward the simple side.
1347law length_append_sym:
1348  for -a: Quant
1349  for -A: Kind(a)
1350  for xs: List<a, A>
1351  for -ys: List<a, A>
1352  {Nat.add(List.length(a, A, xs), List.length(a, A, ys)) == List.length(a, A, List.append(a, A, xs, ys)) : Nat}
1353
1354def length_append_sym(a, A, xs, ys):
1355  Equal.sym(Nat, List.length(a, A, List.append(a, A, xs, ys)), Nat.add(List.length(a, A, xs), List.length(a, A, ys)), length_append(a, A, xs, ys))
1356
1357# The reverse accumulator loop appends the reversed list to the accumulator, reversed to rewrite toward the simple side.
1358law reverse_go_spec_sym:
1359  for -a: Quant
1360  for -A: Kind(a)
1361  for xs: List<a, A>
1362  for -acc: List<a, A>
1363  {List.append(a, A, List.reverse(a, A, xs), acc) == List.reverse.go(a, A, xs, acc) : List<a, A>}
1364
1365def reverse_go_spec_sym(a, A, xs, acc):
1366  Equal.sym(List<a, A>, List.reverse.go(a, A, xs, acc), List.append(a, A, List.reverse(a, A, xs), acc), reverse_go_spec(a, A, xs, acc))
1367
1368# Reversing an append reverses and swaps the parts: reverse (xs ++ ys) = reverse ys ++ reverse xs, reversed to rewrite toward the simple side.
1369law reverse_append_sym:
1370  for -a: Quant
1371  for -A: Kind(a)
1372  for xs: List<a, A>
1373  for ys: List<a, A>
1374  {List.append(a, A, List.reverse(a, A, ys), List.reverse(a, A, xs)) == List.reverse(a, A, List.append(a, A, xs, ys)) : List<a, A>}
1375
1376def reverse_append_sym(a, A, xs, ys):
1377  Equal.sym(List<a, A>, List.reverse(a, A, List.append(a, A, xs, ys)), List.append(a, A, List.reverse(a, A, ys), List.reverse(a, A, xs)), reverse_append(a, A, xs, ys))
1378
1379# Reversing twice gives the list back, reversed to rewrite toward the simple side.
1380law reverse_reverse_sym:
1381  for -a: Quant
1382  for -A: Kind(a)
1383  for xs: List<a, A>
1384  {xs == List.reverse(a, A, List.reverse(a, A, xs)) : List<a, A>}
1385
1386def reverse_reverse_sym(a, A, xs):
1387  Equal.sym(List<a, A>, List.reverse(a, A, List.reverse(a, A, xs)), xs, reverse_reverse(a, A, xs))
1388
1389# Reversing preserves the length, reversed to rewrite toward the simple side.
1390law length_reverse_sym:
1391  for -a: Quant
1392  for -A: Kind(a)
1393  for xs: List<a, A>
1394  {List.length(a, A, xs) == List.length(a, A, List.reverse(a, A, xs)) : Nat}
1395
1396def length_reverse_sym(a, A, xs):
1397  Equal.sym(Nat, List.length(a, A, List.reverse(a, A, xs)), List.length(a, A, xs), length_reverse(a, A, xs))
1398
1399# A right fold over an append folds the first part onto the fold of the second, reversed to rewrite toward the simple side.
1400law foldr_append_sym:
1401  for ~a: Quant
1402  for ~A: Kind(a)
1403  for ~B: Type
1404  for ~f: A -> B -> B
1405  for xs: List<a, A>
1406  for -ys: List<a, A>
1407  for -z: B
1408  {List.foldr(~a, ~A, ~B, ~f, xs, List.foldr(~a, ~A, ~B, ~f, ys, z)) == List.foldr(~a, ~A, ~B, ~f, List.append(a, A, xs, ys), z) : B}
1409
1410def foldr_append_sym(a, A, B, f, xs, ys, z):
1411  Equal.sym(B, List.foldr(~a, ~A, ~B, ~f, List.append(a, A, xs, ys), z), List.foldr(~a, ~A, ~B, ~f, xs, List.foldr(~a, ~A, ~B, ~f, ys, z)), foldr_append(~a, ~A, ~B, ~f, xs, ys, z))
1412
1413# Taking n elements and appending the rest after dropping n gives the list back, reversed to rewrite toward the simple side.
1414law take_append_drop_sym:
1415  for -a: Quant
1416  for -A: Kind(a)
1417  for xs: List<a, A>
1418  for n: Nat
1419  {xs == List.append(a, A, List.take(a, A, xs, n), List.drop(a, A, xs, n)) : List<a, A>}
1420
1421def take_append_drop_sym(a, A, xs, n):
1422  Equal.sym(List<a, A>, List.append(a, A, List.take(a, A, xs, n), List.drop(a, A, xs, n)), xs, take_append_drop(a, A, xs, n))
1423
1424# Mapping preserves the length, reversed to rewrite toward the simple side.
1425law length_map_sym:
1426  for ~A: Type
1427  for ~B: Type
1428  for ~f: A -> B
1429  for xs: List<A>
1430  {List.length(&1, A, xs) == List.length(&1, B, List.map(~A, ~B, ~f, xs)) : Nat}
1431
1432def length_map_sym(A, B, f, xs):
1433  Equal.sym(Nat, List.length(&1, B, List.map(~A, ~B, ~f, xs)), List.length(&1, A, xs), length_map(~A, ~B, ~f, xs))
1434
1435# Mapping over an append maps each part: map f (xs ++ ys) = map f xs ++ map f ys, reversed to rewrite toward the simple side.
1436law map_append_sym:
1437  for ~A: Type
1438  for ~B: Type
1439  for ~f: A -> B
1440  for xs: List<A>
1441  for -ys: List<A>
1442  {List.append(&1, B, List.map(~A, ~B, ~f, xs), List.map(~A, ~B, ~f, ys)) == List.map(~A, ~B, ~f, List.append(&1, A, xs, ys)) : List<B>}
1443
1444def map_append_sym(A, B, f, xs, ys):
1445  Equal.sym(List<B>, List.map(~A, ~B, ~f, List.append(&1, A, xs, ys)), List.append(&1, B, List.map(~A, ~B, ~f, xs), List.map(~A, ~B, ~f, ys)), map_append(~A, ~B, ~f, xs, ys))
1446
1447# Mapping twice is mapping the composition: map g (map f xs) = map (g . f) xs, reversed to rewrite toward the simple side.
1448law map_map_sym:
1449  for ~A: Type
1450  for ~B: Type
1451  for ~C: Type
1452  for ~f: A -> B
1453  for ~g: B -> C
1454  for xs: List<A>
1455  {List.map(~A, ~C, ~(x => g(f(x))), xs) == List.map(~B, ~C, ~g, List.map(~A, ~B, ~f, xs)) : List<C>}
1456
1457def map_map_sym(A, B, C, f, g, xs):
1458  Equal.sym(List<C>, List.map(~B, ~C, ~g, List.map(~A, ~B, ~f, xs)), List.map(~A, ~C, ~(x => g(f(x))), xs), map_map(~A, ~B, ~C, ~f, ~g, xs))
1459
1460# Taking zero elements gives the empty list, reversed to rewrite toward the simple side.
1461law take_zero_sym:
1462  for -a: Quant
1463  for -A: Kind(a)
1464  for xs: List<a, A>
1465  {Nil{} == List.take(a, A, xs, 0n) : List<a, A>}
1466
1467def take_zero_sym(a, A, xs):
1468  Equal.sym(List<a, A>, List.take(a, A, xs, 0n), Nil{}, take_zero(a, A, xs))
1469
1470# Dropping zero elements gives the list back, reversed to rewrite toward the simple side.
1471law drop_zero_sym:
1472  for -a: Quant
1473  for -A: Kind(a)
1474  for xs: List<a, A>
1475  {xs == List.drop(a, A, xs, 0n) : List<a, A>}
1476
1477def drop_zero_sym(a, A, xs):
1478  Equal.sym(List<a, A>, List.drop(a, A, xs, 0n), xs, drop_zero(a, A, xs))
1479
1480# Taking from the empty list gives the empty list, reversed to rewrite toward the simple side.
1481law take_nil_sym:
1482  for -a: Quant
1483  for -A: Kind(a)
1484  for -n: Nat
1485  {Nil{} == List.take(a, A, Nil{}, n) : List<a, A>}
1486
1487def take_nil_sym(a, A, n):
1488  Equal.sym(List<a, A>, List.take(a, A, Nil{}, n), Nil{}, take_nil(a, A, n))
1489
1490# Dropping from the empty list gives the empty list, reversed to rewrite toward the simple side.
1491law drop_nil_sym:
1492  for -a: Quant
1493  for -A: Kind(a)
1494  for -n: Nat
1495  {Nil{} == List.drop(a, A, Nil{}, n) : List<a, A>}
1496
1497def drop_nil_sym(a, A, n):
1498  Equal.sym(List<a, A>, List.drop(a, A, Nil{}, n), Nil{}, drop_nil(a, A, n))
1499
1500# Taking n elements leaves min(n, length) of them, reversed to rewrite toward the simple side.
1501law length_take_sym:
1502  for -a: Quant
1503  for -A: Kind(a)
1504  for xs: List<a, A>
1505  for n: Nat
1506  {Nat.min(n, List.length(a, A, xs)) == List.length(a, A, List.take(a, A, xs, n)) : Nat}
1507
1508def length_take_sym(a, A, xs, n):
1509  Equal.sym(Nat, List.length(a, A, List.take(a, A, xs, n)), Nat.min(n, List.length(a, A, xs)), length_take(a, A, xs, n))
1510
1511# Dropping n elements leaves length - n of them, reversed to rewrite toward the simple side.
1512law length_drop_sym:
1513  for -a: Quant
1514  for -A: Kind(a)
1515  for xs: List<a, A>
1516  for n: Nat
1517  {Nat.sub(List.length(a, A, xs), n) == List.length(a, A, List.drop(a, A, xs, n)) : Nat}
1518
1519def length_drop_sym(a, A, xs, n):
1520  Equal.sym(Nat, List.length(a, A, List.drop(a, A, xs, n)), Nat.sub(List.length(a, A, xs), n), length_drop(a, A, xs, n))
1521
1522# Taking as many elements as the list has gives the list back, reversed to rewrite toward the simple side.
1523law take_length_sym:
1524  for -A: Data
1525  for +xs: List<&2, A>
1526  {xs == List.take(&2, A, xs, List.length(&2, A, xs)) : List<&2, A>}
1527
1528def take_length_sym(A, xs):
1529  Equal.sym(List<&2, A>, List.take(&2, A, xs, List.length(&2, A, xs)), xs, take_length(A, xs))
1530
1531# Dropping as many elements as the list has gives the empty list, reversed to rewrite toward the simple side.
1532law drop_length_sym:
1533  for -A: Data
1534  for +xs: List<&2, A>
1535  {Nil{} == List.drop(&2, A, xs, List.length(&2, A, xs)) : List<&2, A>}
1536
1537def drop_length_sym(A, xs):
1538  Equal.sym(List<&2, A>, List.drop(&2, A, xs, List.length(&2, A, xs)), Nil{}, drop_length(A, xs))
1539
1540# Taking m from the first n is taking min(n, m), reversed to rewrite toward the simple side.
1541law take_take_sym:
1542  for -a: Quant
1543  for -A: Kind(a)
1544  for xs: List<a, A>
1545  for n: Nat
1546  for m: Nat
1547  {List.take(a, A, xs, Nat.min(n, m)) == List.take(a, A, List.take(a, A, xs, n), m) : List<a, A>}
1548
1549def take_take_sym(a, A, xs, n, m):
1550  Equal.sym(List<a, A>, List.take(a, A, List.take(a, A, xs, n), m), List.take(a, A, xs, Nat.min(n, m)), take_take(a, A, xs, n, m))
1551
1552# Dropping m after dropping n is dropping n + m, reversed to rewrite toward the simple side.
1553law drop_drop_sym:
1554  for -a: Quant
1555  for -A: Kind(a)
1556  for xs: List<a, A>
1557  for n: Nat
1558  for -m: Nat
1559  {List.drop(a, A, xs, Nat.add(n, m)) == List.drop(a, A, List.drop(a, A, xs, n), m) : List<a, A>}
1560
1561def drop_drop_sym(a, A, xs, n, m):
1562  Equal.sym(List<a, A>, List.drop(a, A, List.drop(a, A, xs, n), m), List.drop(a, A, xs, Nat.add(n, m)), drop_drop(a, A, xs, n, m))
1563
1564# Reversing the empty list gives the empty list, reversed to rewrite toward the simple side.
1565law reverse_nil_sym:
1566  for -a: Quant
1567  for -A: Kind(a)
1568  {Nil{} == List.reverse(a, A, Nil{}) : List<a, A>}
1569
1570def reverse_nil_sym(a, A):
1571  Equal.sym(List<a, A>, List.reverse(a, A, Nil{}), Nil{}, reverse_nil(a, A))
1572
1573# Reversing a one-element list gives it back, reversed to rewrite toward the simple side.
1574law reverse_singleton_sym:
1575  for -a: Quant
1576  for -A: Kind(a)
1577  for -x: A
1578  {[x] == List.reverse(a, A, [x]) : List<a, A>}
1579
1580def reverse_singleton_sym(a, A, x):
1581  Equal.sym(List<a, A>, List.reverse(a, A, [x]), [x], reverse_singleton(a, A, x))
1582
1583# Replicating x n times gives a list of length n, reversed to rewrite toward the simple side.
1584law length_replicate_sym:
1585  for -A: Data
1586  for n: Nat
1587  for -x: A
1588  {n == List.length(&2, A, List.replicate(A, n, x)) : Nat}
1589
1590def length_replicate_sym(A, n, x):
1591  Equal.sym(Nat, List.length(&2, A, List.replicate(A, n, x)), n, length_replicate(A, n, x))
1592
1593# Range(n) has length n, reversed to rewrite toward the simple side.
1594law length_range_sym:
1595  for n: Nat
1596  {n == List.length(&2, Nat, List.range(n)) : Nat}
1597
1598def length_range_sym(n):
1599  Equal.sym(Nat, List.length(&2, Nat, List.range(n)), n, length_range(n))
1600
1601# Zipping two lists gives the length of the shorter one, reversed to rewrite toward the simple side.
1602law length_zip_sym:
1603  for -a: Quant
1604  for -A: Kind(a)
1605  for xs: List<a, A>
1606  for ys: List<a, A>
1607  {Nat.min(List.length(a, A, xs), List.length(a, A, ys)) == List.length(&1, A & A, List.zip(a, A, a, A, xs, ys)) : Nat}
1608
1609def length_zip_sym(a, A, xs, ys):
1610  Equal.sym(Nat, List.length(&1, A & A, List.zip(a, A, a, A, xs, ys)), Nat.min(List.length(a, A, xs), List.length(a, A, ys)), length_zip(a, A, xs, ys))
1611
1612# Appending after a cons: (x :: xs) ++ ys = x :: (xs ++ ys), reversed to rewrite toward the simple side.
1613law append_cons_sym:
1614  for -a: Quant
1615  for -A: Kind(a)
1616  for -x: A
1617  for -xs: List<a, A>
1618  for -ys: List<a, A>
1619  {x <> List.append(a, A, xs, ys) == List.append(a, A, x <> xs, ys) : List<a, A>}
1620
1621def append_cons_sym(a, A, x, xs, ys):
1622  Equal.sym(List<a, A>, List.append(a, A, x <> xs, ys), x <> List.append(a, A, xs, ys), append_cons(a, A, x, xs, ys))
1623
1624# The empty list has length zero, reversed to rewrite toward the simple side.
1625law length_nil_sym:
1626  for -a: Quant
1627  for -A: Kind(a)
1628  {0n == List.length(a, A, Nil{}) : Nat}
1629
1630def length_nil_sym(a, A):
1631  Equal.sym(Nat, List.length(a, A, Nil{}), 0n, length_nil(a, A))
1632
1633# A cons is one longer than its tail, reversed to rewrite toward the simple side.
1634law length_cons_sym:
1635  for -a: Quant
1636  for -A: Kind(a)
1637  for -x: A
1638  for -xs: List<a, A>
1639  {1n+List.length(a, A, xs) == List.length(a, A, x <> xs) : Nat}
1640
1641def length_cons_sym(a, A, x, xs):
1642  Equal.sym(Nat, List.length(a, A, x <> xs), 1n+List.length(a, A, xs), length_cons(a, A, x, xs))
1643
1644# Concatenating an append concatenates each part: concat (xss ++ yss) = concat xss ++ concat yss, reversed to rewrite toward the simple side.
1645law concat_append_sym:
1646  for -a: Quant
1647  for -A: Kind(a)
1648  for xss: List<a, List<a, A>>
1649  for -yss: List<a, List<a, A>>
1650  {List.append(a, A, List.concat(a, A, xss), List.concat(a, A, yss)) == List.concat(a, A, List.append(a, List<a, A>, xss, yss)) : List<a, A>}
1651
1652def concat_append_sym(a, A, xss, yss):
1653  Equal.sym(List<a, A>, List.concat(a, A, List.append(a, List<a, A>, xss, yss)), List.append(a, A, List.concat(a, A, xss), List.concat(a, A, yss)), concat_append(a, A, xss, yss))
1654
1655# A left fold over an append folds the second part from the fold of the first, reversed to rewrite toward the simple side.
1656law foldl_append_sym:
1657  for ~a: Quant
1658  for ~A: Kind(a)
1659  for ~B: Type
1660  for ~f: B -> A -> B
1661  for xs: List<a, A>
1662  for -ys: List<a, A>
1663  for -z: B
1664  {List.foldl(~a, ~A, ~B, ~f, ys, List.foldl(~a, ~A, ~B, ~f, xs, z)) == List.foldl(~a, ~A, ~B, ~f, List.append(a, A, xs, ys), z) : B}
1665
1666def foldl_append_sym(a, A, B, f, xs, ys, z):
1667  Equal.sym(B, List.foldl(~a, ~A, ~B, ~f, List.append(a, A, xs, ys), z), List.foldl(~a, ~A, ~B, ~f, ys, List.foldl(~a, ~A, ~B, ~f, xs, z)), foldl_append(~a, ~A, ~B, ~f, xs, ys, z))
1668
1669# Mapping commutes with reversing: map f (reverse xs) = reverse (map f xs), reversed to rewrite toward the simple side.
1670law map_reverse_sym:
1671  for ~A: Type
1672  for ~B: Type
1673  for ~f: A -> B
1674  for xs: List<A>
1675  {List.reverse(&1, B, List.map(~A, ~B, ~f, xs)) == List.map(~A, ~B, ~f, List.reverse(&1, A, xs)) : List<B>}
1676
1677def map_reverse_sym(A, B, f, xs):
1678  Equal.sym(List<B>, List.map(~A, ~B, ~f, List.reverse(&1, A, xs)), List.reverse(&1, B, List.map(~A, ~B, ~f, xs)), map_reverse(~A, ~B, ~f, xs))
1679
1680# All over an append is all over each part, joined by and, reversed to rewrite toward the simple side.
1681law all_append_sym:
1682  for ~a: Quant
1683  for ~A: Kind(a)
1684  for ~f: A -> Bool
1685  for xs: List<a, A>
1686  for -ys: List<a, A>
1687  {Bool.and(List.all(~a, ~A, ~f, xs), List.all(~a, ~A, ~f, ys)) == List.all(~a, ~A, ~f, List.append(a, A, xs, ys)) : Bool}
1688
1689def all_append_sym(a, A, f, xs, ys):
1690  Equal.sym(Bool, List.all(~a, ~A, ~f, List.append(a, A, xs, ys)), Bool.and(List.all(~a, ~A, ~f, xs), List.all(~a, ~A, ~f, ys)), all_append(~a, ~A, ~f, xs, ys))
1691
1692# Any over an append is any over each part, joined by or, reversed to rewrite toward the simple side.
1693law any_append_sym:
1694  for ~a: Quant
1695  for ~A: Kind(a)
1696  for ~f: A -> Bool
1697  for xs: List<a, A>
1698  for -ys: List<a, A>
1699  {Bool.or(List.any(~a, ~A, ~f, xs), List.any(~a, ~A, ~f, ys)) == List.any(~a, ~A, ~f, List.append(a, A, xs, ys)) : Bool}
1700
1701def any_append_sym(a, A, f, xs, ys):
1702  Equal.sym(Bool, List.any(~a, ~A, ~f, List.append(a, A, xs, ys)), Bool.or(List.any(~a, ~A, ~f, xs), List.any(~a, ~A, ~f, ys)), any_append(~a, ~A, ~f, xs, ys))
1703
1704# Filtering an append filters each part, reversed to rewrite toward the simple side.
1705law filter_append_sym:
1706  for ~A: Data
1707  for ~f: A -> Bool
1708  for xs: List<&2, A>
1709  for ys: List<&2, A>
1710  {List.append(&2, A, List.filter(~A, ~f, xs), List.filter(~A, ~f, ys)) == List.filter(~A, ~f, List.append(&2, A, xs, ys)) : List<&2, A>}
1711
1712def filter_append_sym(A, f, xs, ys):
1713  Equal.sym(List<&2, A>, List.filter(~A, ~f, List.append(&2, A, xs, ys)), List.append(&2, A, List.filter(~A, ~f, xs), List.filter(~A, ~f, ys)), filter_append(~A, ~f, xs, ys))
1714
1715# An append contains x iff either part does, reversed to rewrite toward the simple side.
1716law contains_append_sym:
1717  for ~A: Data
1718  for ~eq: A -> A -> Bool
1719  for xs: List<&2, A>
1720  for -ys: List<&2, A>
1721  for +x: A
1722  {Bool.or(List.contains(~A, ~eq, xs, x), List.contains(~A, ~eq, ys, x)) == List.contains(~A, ~eq, List.append(&2, A, xs, ys), x) : Bool}
1723
1724def contains_append_sym(A, eq, xs, ys, x):
1725  Equal.sym(Bool, List.contains(~A, ~eq, List.append(&2, A, xs, ys), x), Bool.or(List.contains(~A, ~eq, xs, x), List.contains(~A, ~eq, ys, x)), contains_append(~A, ~eq, xs, ys, x))
1726
1727# Filtering never makes a list longer, reversed to rewrite toward the simple side.
1728law length_filter_le_sym:
1729  for ~A: Data
1730  for ~f: A -> Bool
1731  for xs: List<&2, A>
1732  {True{} == Nat.is_le(List.length(&2, A, List.filter(~A, ~f, xs)), List.length(&2, A, xs)) : Bool}
1733
1734def length_filter_le_sym(A, f, xs):
1735  Equal.sym(Bool, Nat.is_le(List.length(&2, A, List.filter(~A, ~f, xs)), List.length(&2, A, xs)), True{}, length_filter_le(~A, ~f, xs))
1736
1737# Taking the length of the first part of an append gives the first part back, reversed to rewrite toward the simple side.
1738law take_left_sym:
1739  for -a: Quant
1740  for -A: Kind(a)
1741  for xs: List<a, A>
1742  for ys: List<a, A>
1743  {xs == List.take(a, A, List.append(a, A, xs, ys), List.length(a, A, xs)) : List<a, A>}
1744
1745def take_left_sym(a, A, xs, ys):
1746  Equal.sym(List<a, A>, List.take(a, A, List.append(a, A, xs, ys), List.length(a, A, xs)), xs, take_left(a, A, xs, ys))
1747
1748# Dropping the length of the first part of an append gives the second part, reversed to rewrite toward the simple side.
1749law drop_left_sym:
1750  for -a: Quant
1751  for -A: Kind(a)
1752  for xs: List<a, A>
1753  for ys: List<a, A>
1754  {ys == List.drop(a, A, List.append(a, A, xs, ys), List.length(a, A, xs)) : List<a, A>}
1755
1756def drop_left_sym(a, A, xs, ys):
1757  Equal.sym(List<a, A>, List.drop(a, A, List.append(a, A, xs, ys), List.length(a, A, xs)), ys, drop_left(a, A, xs, ys))
1758
1759# Taking past the first part of an append keeps it and takes the rest from the second part, reversed to rewrite toward the simple side.
1760law take_append_sym:
1761  for -a: Quant
1762  for -A: Kind(a)
1763  for xs: List<a, A>
1764  for -ys: List<a, A>
1765  for -n: Nat
1766  {List.append(a, A, xs, List.take(a, A, ys, n)) == List.take(a, A, List.append(a, A, xs, ys), Nat.add(List.length(a, A, xs), n)) : List<a, A>}
1767
1768def take_append_sym(a, A, xs, ys, n):
1769  Equal.sym(List<a, A>, List.take(a, A, List.append(a, A, xs, ys), Nat.add(List.length(a, A, xs), n)), List.append(a, A, xs, List.take(a, A, ys, n)), take_append(a, A, xs, ys, n))
1770
1771# Dropping past the first part of an append drops the rest from the second part, reversed to rewrite toward the simple side.
1772law drop_append_sym:
1773  for -a: Quant
1774  for -A: Kind(a)
1775  for xs: List<a, A>
1776  for -ys: List<a, A>
1777  for -n: Nat
1778  {List.drop(a, A, ys, n) == List.drop(a, A, List.append(a, A, xs, ys), Nat.add(List.length(a, A, xs), n)) : List<a, A>}
1779
1780def drop_append_sym(a, A, xs, ys, n):
1781  Equal.sym(List<a, A>, List.drop(a, A, List.append(a, A, xs, ys), Nat.add(List.length(a, A, xs), n)), List.drop(a, A, ys, n), drop_append(a, A, xs, ys, n))
1782
1783# Taking at most the length of the first part of an append only sees the first part, reversed to rewrite toward the simple side.
1784law take_append_of_le_length_sym:
1785  for -a: Quant
1786  for -A: Kind(a)
1787  for xs: List<a, A>
1788  for ys: List<a, A>
1789  for n: Nat
1790  for h: MNat.le(n, List.length(a, A, xs))
1791  {List.take(a, A, xs, n) == List.take(a, A, List.append(a, A, xs, ys), n) : List<a, A>}
1792
1793def take_append_of_le_length_sym(a, A, xs, ys, n, h):
1794  Equal.sym(List<a, A>, List.take(a, A, List.append(a, A, xs, ys), n), List.take(a, A, xs, n), take_append_of_le_length(a, A, xs, ys, n, h))
1795
1796# Dropping at most the length of the first part of an append drops only from the first part, reversed to rewrite toward the simple side.
1797law drop_append_of_le_length_sym:
1798  for -a: Quant
1799  for -A: Kind(a)
1800  for xs: List<a, A>
1801  for ys: List<a, A>
1802  for n: Nat
1803  for h: MNat.le(n, List.length(a, A, xs))
1804  {List.append(a, A, List.drop(a, A, xs, n), ys) == List.drop(a, A, List.append(a, A, xs, ys), n) : List<a, A>}
1805
1806def drop_append_of_le_length_sym(a, A, xs, ys, n, h):
1807  Equal.sym(List<a, A>, List.drop(a, A, List.append(a, A, xs, ys), n), List.append(a, A, List.drop(a, A, xs, n), ys), drop_append_of_le_length(a, A, xs, ys, n, h))
1808
1809# Mapping commutes with taking: map f (take n xs) = take n (map f xs), reversed to rewrite toward the simple side.
1810law map_take_sym:
1811  for ~A: Type
1812  for ~B: Type
1813  for ~f: A -> B
1814  for xs: List<A>
1815  for n: Nat
1816  {List.take(&1, B, List.map(~A, ~B, ~f, xs), n) == List.map(~A, ~B, ~f, List.take(&1, A, xs, n)) : List<B>}
1817
1818def map_take_sym(A, B, f, xs, n):
1819  Equal.sym(List<B>, List.map(~A, ~B, ~f, List.take(&1, A, xs, n)), List.take(&1, B, List.map(~A, ~B, ~f, xs), n), map_take(~A, ~B, ~f, xs, n))
1820
1821# Mapping commutes with dropping: map f (drop n xs) = drop n (map f xs), reversed to rewrite toward the simple side.
1822law map_drop_sym:
1823  for ~A: Type
1824  for ~B: Type
1825  for ~f: A -> B
1826  for xs: List<A>
1827  for n: Nat
1828  {List.drop(&1, B, List.map(~A, ~B, ~f, xs), n) == List.map(~A, ~B, ~f, List.drop(&1, A, xs, n)) : List<B>}
1829
1830def map_drop_sym(A, B, f, xs, n):
1831  Equal.sym(List<B>, List.map(~A, ~B, ~f, List.drop(&1, A, xs, n)), List.drop(&1, B, List.map(~A, ~B, ~f, xs), n), map_drop(~A, ~B, ~f, xs, n))
1832
1833# Taking m from n copies of x gives min(m, n) copies of x, reversed to rewrite toward the simple side.
1834law take_replicate_sym:
1835  for -A: Data
1836  for n: Nat
1837  for m: Nat
1838  for -x: A
1839  {List.replicate(A, Nat.min(m, n), x) == List.take(&2, A, List.replicate(A, n, x), m) : List<&2, A>}
1840
1841def take_replicate_sym(A, n, m, x):
1842  Equal.sym(List<&2, A>, List.take(&2, A, List.replicate(A, n, x), m), List.replicate(A, Nat.min(m, n), x), take_replicate(A, n, m, x))
1843
1844# Dropping m from n copies of x gives n - m copies of x, reversed to rewrite toward the simple side.
1845law drop_replicate_sym:
1846  for -A: Data
1847  for n: Nat
1848  for m: Nat
1849  for -x: A
1850  {List.replicate(A, Nat.sub(n, m), x) == List.drop(&2, A, List.replicate(A, n, x), m) : List<&2, A>}
1851
1852def drop_replicate_sym(A, n, m, x):
1853  Equal.sym(List<&2, A>, List.drop(&2, A, List.replicate(A, n, x), m), List.replicate(A, Nat.sub(n, m), x), drop_replicate(A, n, m, x))
1854
1855# The tail is one shorter than the list, or empty: length (tail xs) = length xs - 1, reversed to rewrite toward the simple side.
1856law length_tail_sym:
1857  for -a: Quant
1858  for -A: Kind(a)
1859  for xs: List<a, A>
1860  {Nat.sub(List.length(a, A, xs), 1n) == List.length(a, A, List.tail(a, A, xs)) : Nat}
1861
1862def length_tail_sym(a, A, xs):
1863  Equal.sym(Nat, List.length(a, A, List.tail(a, A, xs)), Nat.sub(List.length(a, A, xs), 1n), length_tail(a, A, xs))
1864
1865# Mapping the identity gives the list back, reversed to rewrite toward the simple side.
1866law map_id_sym:
1867  for ~A: Type
1868  for xs: List<A>
1869  {xs == List.map(~A, ~A, ~(x => x), xs) : List<A>}
1870
1871def map_id_sym(A, xs):
1872  Equal.sym(List<A>, List.map(~A, ~A, ~(x => x), xs), xs, map_id(~A, xs))
1873
1874# A right fold over a map folds with the mapped function: foldr g z (map f xs) = foldr (g . f) z xs, reversed to rewrite toward the simple side.
1875law foldr_map_sym:
1876  for ~A: Type
1877  for ~B: Type
1878  for ~C: Type
1879  for ~f: A -> B
1880  for ~g: B -> C -> C
1881  for xs: List<A>
1882  for -z: C
1883  {List.foldr(~&1, ~A, ~C, ~(x => y => g(f(x), y)), xs, z) == List.foldr(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, xs), z) : C}
1884
1885def foldr_map_sym(A, B, C, f, g, xs, z):
1886  Equal.sym(C, List.foldr(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, xs), z), List.foldr(~&1, ~A, ~C, ~(x => y => g(f(x), y)), xs, z), foldr_map(~A, ~B, ~C, ~f, ~g, xs, z))
1887
1888# A left fold over a map folds with the mapped function: foldl g z (map f xs) = foldl (fun acc x => g acc (f x)) z xs, reversed to rewrite toward the simple side.
1889law foldl_map_sym:
1890  for ~A: Type
1891  for ~B: Type
1892  for ~C: Type
1893  for ~f: A -> B
1894  for ~g: C -> B -> C
1895  for xs: List<A>
1896  for -z: C
1897  {List.foldl(~&1, ~A, ~C, ~(x => y => g(x, f(y))), xs, z) == List.foldl(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, xs), z) : C}
1898
1899def foldl_map_sym(A, B, C, f, g, xs, z):
1900  Equal.sym(C, List.foldl(~&1, ~B, ~C, ~g, List.map(~A, ~B, ~f, xs), z), List.foldl(~&1, ~A, ~C, ~(x => y => g(x, f(y))), xs, z), foldl_map(~A, ~B, ~C, ~f, ~g, xs, z))
1901
1902# Replicating m + n times appends m copies to n copies, reversed to rewrite toward the simple side.
1903law replicate_add_sym:
1904  for -A: Data
1905  for m: Nat
1906  for -n: Nat
1907  for -x: A
1908  {List.append(&2, A, List.replicate(A, m, x), List.replicate(A, n, x)) == List.replicate(A, Nat.add(m, n), x) : List<&2, A>}
1909
1910def replicate_add_sym(A, m, n, x):
1911  Equal.sym(List<&2, A>, List.replicate(A, Nat.add(m, n), x), List.append(&2, A, List.replicate(A, m, x), List.replicate(A, n, x)), replicate_add(A, m, n, x))
1912
1913# Reversing n copies of x gives them back, reversed to rewrite toward the simple side.
1914law reverse_replicate_sym:
1915  for -A: Data
1916  for n: Nat
1917  for -x: A
1918  {List.replicate(A, n, x) == List.reverse(&2, A, List.replicate(A, n, x)) : List<&2, A>}
1919
1920def reverse_replicate_sym(A, n, x):
1921  Equal.sym(List<&2, A>, List.reverse(&2, A, List.replicate(A, n, x)), List.replicate(A, n, x), reverse_replicate(A, n, x))
1922
1923# Filtering with an always-true predicate gives the list back, reversed to rewrite toward the simple side.
1924law filter_true_sym:
1925  for ~A: Data
1926  for xs: List<&2, A>
1927  {xs == List.filter(~A, ~(x => True{}), xs) : List<&2, A>}
1928
1929def filter_true_sym(A, xs):
1930  Equal.sym(List<&2, A>, List.filter(~A, ~(x => True{}), xs), xs, filter_true(~A, xs))
1931
1932# Filtering with an always-false predicate gives the empty list, reversed to rewrite toward the simple side.
1933law filter_false_sym:
1934  for ~A: Data
1935  for xs: List<&2, A>
1936  {Nil{} == List.filter(~A, ~(x => False{}), xs) : List<&2, A>}
1937
1938def filter_false_sym(A, xs):
1939  Equal.sym(List<&2, A>, List.filter(~A, ~(x => False{}), xs), Nil{}, filter_false(~A, xs))
1940
1941# Filtering twice is filtering by both predicates: filter p (filter q xs) = filter (p and q) xs, reversed to rewrite toward the simple side.
1942law filter_filter_sym:
1943  for ~A: Data
1944  for ~p: A -> Bool
1945  for ~q: A -> Bool
1946  for xs: List<&2, A>
1947  {List.filter(~A, ~(+x => Bool.and(p(x), q(x))), xs) == List.filter(~A, ~p, List.filter(~A, ~q, xs)) : List<&2, A>}
1948
1949def filter_filter_sym(A, p, q, xs):
1950  Equal.sym(List<&2, A>, List.filter(~A, ~p, List.filter(~A, ~q, xs)), List.filter(~A, ~(+x => Bool.and(p(x), q(x))), xs), filter_filter(~A, ~p, ~q, xs))
1951
1952# All over a filter checks q only where p holds: all q (filter p xs) = all (not p or q) xs, reversed to rewrite toward the simple side.
1953law all_filter_sym:
1954  for ~A: Data
1955  for ~p: A -> Bool
1956  for ~q: A -> Bool
1957  for xs: List<&2, A>
1958  {List.all(~&2, ~A, ~(+x => Bool.or(Bool.not(p(x)), q(x))), xs) == List.all(~&2, ~A, ~q, List.filter(~A, ~p, xs)) : Bool}
1959
1960def all_filter_sym(A, p, q, xs):
1961  Equal.sym(Bool, List.all(~&2, ~A, ~q, List.filter(~A, ~p, xs)), List.all(~&2, ~A, ~(+x => Bool.or(Bool.not(p(x)), q(x))), xs), all_filter(~A, ~p, ~q, xs))
1962
1963# Any over a filter looks for q only where p holds: any q (filter p xs) = any (p and q) xs, reversed to rewrite toward the simple side.
1964law any_filter_sym:
1965  for ~A: Data
1966  for ~p: A -> Bool
1967  for ~q: A -> Bool
1968  for xs: List<&2, A>
1969  {List.any(~&2, ~A, ~(+x => Bool.and(p(x), q(x))), xs) == List.any(~&2, ~A, ~q, List.filter(~A, ~p, xs)) : Bool}
1970
1971def any_filter_sym(A, p, q, xs):
1972  Equal.sym(Bool, List.any(~&2, ~A, ~q, List.filter(~A, ~p, xs)), List.any(~&2, ~A, ~(+x => Bool.and(p(x), q(x))), xs), any_filter(~A, ~p, ~q, xs))
1973
1974# All over a map checks the composed predicate: all p (map f xs) = all (p . f) xs, reversed to rewrite toward the simple side.
1975law all_map_sym:
1976  for ~A: Type
1977  for ~B: Type
1978  for ~f: A -> B
1979  for ~p: B -> Bool
1980  for xs: List<A>
1981  {List.all(~&1, ~A, ~(x => p(f(x))), xs) == List.all(~&1, ~B, ~p, List.map(~A, ~B, ~f, xs)) : Bool}
1982
1983def all_map_sym(A, B, f, p, xs):
1984  Equal.sym(Bool, List.all(~&1, ~B, ~p, List.map(~A, ~B, ~f, xs)), List.all(~&1, ~A, ~(x => p(f(x))), xs), all_map(~A, ~B, ~f, ~p, xs))
1985
1986# Any over a map checks the composed predicate: any p (map f xs) = any (p . f) xs, reversed to rewrite toward the simple side.
1987law any_map_sym:
1988  for ~A: Type
1989  for ~B: Type
1990  for ~f: A -> B
1991  for ~p: B -> Bool
1992  for xs: List<A>
1993  {List.any(~&1, ~A, ~(x => p(f(x))), xs) == List.any(~&1, ~B, ~p, List.map(~A, ~B, ~f, xs)) : Bool}
1994
1995def any_map_sym(A, B, f, p, xs):
1996  Equal.sym(Bool, List.any(~&1, ~B, ~p, List.map(~A, ~B, ~f, xs)), List.any(~&1, ~A, ~(x => p(f(x))), xs), any_map(~A, ~B, ~f, ~p, xs))
1997
1998# Reversing does not change whether all elements satisfy a predicate, reversed to rewrite toward the simple side.
1999law all_reverse_sym:
2000  for ~a: Quant
2001  for ~A: Kind(a)
2002  for ~f: A -> Bool
2003  for xs: List<a, A>
2004  {List.all(~a, ~A, ~f, xs) == List.all(~a, ~A, ~f, List.reverse(a, A, xs)) : Bool}
2005
2006def all_reverse_sym(a, A, f, xs):
2007  Equal.sym(Bool, List.all(~a, ~A, ~f, List.reverse(a, A, xs)), List.all(~a, ~A, ~f, xs), all_reverse(~a, ~A, ~f, xs))
2008
2009# Reversing does not change whether some element satisfies a predicate, reversed to rewrite toward the simple side.
2010law any_reverse_sym:
2011  for ~a: Quant
2012  for ~A: Kind(a)
2013  for ~f: A -> Bool
2014  for xs: List<a, A>
2015  {List.any(~a, ~A, ~f, xs) == List.any(~a, ~A, ~f, List.reverse(a, A, xs)) : Bool}
2016
2017def any_reverse_sym(a, A, f, xs):
2018  Equal.sym(Bool, List.any(~a, ~A, ~f, List.reverse(a, A, xs)), List.any(~a, ~A, ~f, xs), any_reverse(~a, ~A, ~f, xs))
2019
2020# Reversing does not change whether a list contains x, reversed to rewrite toward the simple side.
2021law contains_reverse_sym:
2022  for ~A: Data
2023  for ~eq: A -> A -> Bool
2024  for xs: List<&2, A>
2025  for +x: A
2026  {List.contains(~A, ~eq, xs, x) == List.contains(~A, ~eq, List.reverse(&2, A, xs), x) : Bool}
2027
2028def contains_reverse_sym(A, eq, xs, x):
2029  Equal.sym(Bool, List.contains(~A, ~eq, List.reverse(&2, A, xs), x), List.contains(~A, ~eq, xs, x), contains_reverse(~A, ~eq, xs, x))