perm.bend
bend-mathlib@0.5.0.0 · 347 lines · 0 definitions
1# bend-mathlib/perm.bend: step-list permutations over bendlib-kernel-list, plus sort value defs and proofs.
2# Resolves the published kernel bendlib-kernel-list@1.0.0.0 by its content hash; see packages/bendlib-kernel-list/README.md.
3import Base
4import bendlib-kernel-list@1.0.0.0/list.bend as K
5
6def internal_shift(+s: List<&2, Nat>) -> List<&2, Nat>:
7 match s:
8 case Nil{}:
9 Nil{}
10 case +i <> rest:
11 (1n+i) <> internal_shift(rest)
12
13def internal_apply_app(-A: Data, +s1: List<&2, Nat>, +s2: List<&2, Nat>, +xs: List<&2, A>) -> {K.apply(A, List.append(&2, Nat, s1, s2), xs) == K.apply(A, s2, K.apply(A, s1, xs)) : List<&2, A>}:
14 match s1:
15 case Nil{}:
16 {==}
17 case +i <> rest:
18 internal_apply_app(A, rest, s2, K.swap_at(A, i, xs))
19
20def internal_apply_shift(-A: Data, +s: List<&2, Nat>, +x: A, +xs: List<&2, A>) -> {K.apply(A, internal_shift(s), x <> xs) == x <> K.apply(A, s, xs) : List<&2, A>}:
21 match s:
22 case Nil{}:
23 {==}
24 case +i <> rest:
25 internal_apply_shift(A, rest, x, K.swap_at(A, i, xs))
26
27# Permutation is reflexive.
28def perm_refl(-A: Data, -xs: List<&2, A>) -> K.perm(A, xs, xs):
29 (Nil{}, {==})
30
31# The empty list is a permutation of itself.
32def perm_nil(-A: Data) -> K.perm(A, Nil{}, Nil{}):
33 (Nil{}, {==})
34
35# Swapping the two leading elements is a permutation.
36def perm_swap(-A: Data, -x: A, -y: A, -t: List<&2, A>) -> K.perm(A, x <> y <> t, y <> x <> t):
37 (0n <> Nil{}, {==})
38
39def internal_trans_eq(-A: Data, +s1: List<&2, Nat>, +s2: List<&2, Nat>, +xs: List<&2, A>, -ys: List<&2, A>, -zs: List<&2, A>, e1: K.perm_steps(A, s1, xs, ys), e2: K.perm_steps(A, s2, ys, zs)) -> K.perm_steps(A, List.append(&2, Nat, s1, s2), xs, zs):
40 %Equal.sym(List<&2, A>, K.apply(A, List.append(&2, Nat, s1, s2), xs), K.apply(A, s2, K.apply(A, s1, xs)), internal_apply_app(A, s1, s2, xs)) : {_ == zs : List<&2, A>}
41 %Equal.sym(List<&2, A>, K.apply(A, s1, xs), ys, e1) : {K.apply(A, s2, _) == zs : List<&2, A>}
42 e2
43
44# Permutation is transitive.
45def perm_trans(-A: Data, +xs: List<&2, A>, -ys: List<&2, A>, -zs: List<&2, A>, h1: K.perm(A, xs, ys), h2: K.perm(A, ys, zs)) -> K.perm(A, xs, zs):
46 (s1, e1) = h1
47 (s2, e2) = h2
48 +s1 = s1
49 +s2 = s2
50 (List.append(&2, Nat, s1, s2), internal_trans_eq(A, s1, s2, xs, ys, zs, e1, e2))
51
52def internal_cons_eq(-A: Data, +s: List<&2, Nat>, +x: A, +xs: List<&2, A>, -ys: List<&2, A>, e: K.perm_steps(A, s, xs, ys)) -> K.perm_steps(A, internal_shift(s), x <> xs, x <> ys):
53 %Equal.sym(List<&2, A>, K.apply(A, internal_shift(s), x <> xs), x <> K.apply(A, s, xs), internal_apply_shift(A, s, x, xs)) : {_ == x <> ys : List<&2, A>}
54 %Equal.sym(List<&2, A>, K.apply(A, s, xs), ys, e) : {x <> _ == x <> ys : List<&2, A>}
55 {==}
56
57# Prepending the same element preserves permutation.
58def perm_cons(-A: Data, +x: A, +xs: List<&2, A>, -ys: List<&2, A>, h: K.perm(A, xs, ys)) -> K.perm(A, x <> xs, x <> ys):
59 (s, e) = h
60 +s = s
61 (internal_shift(s), internal_cons_eq(A, s, x, xs, ys, e))
62
63# A Perm hypothesis in existential form is reusable.
64def perm_dup(-A: Data, -xs: List<&2, A>, -ys: List<&2, A>, +h: K.perm(A, xs, ys)) -> K.perm(A, xs, ys) & K.perm(A, xs, ys):
65 (h, h)
66
67# Insertion into a list, generic comparator (structural, Base-only branching via Bool.pick).
68def insert_by(~A: Data, ~le: A -> A -> Bool, +x: A, xs: List<&2, A>) -> List<&2, A>:
69 match xs:
70 case Nil{}:
71 x <> Nil{}
72 case +y <> +t:
73 Bool.pick(List<&2, A>, le(x, y), x <> y <> t, y <> insert_by(~A, ~le, x, t))
74
75def isort_by(~A: Data, ~le: A -> A -> Bool, xs: List<&2, A>) -> List<&2, A>:
76 match xs:
77 case Nil{}:
78 Nil{}
79 case +x <> t:
80 insert_by(~A, ~le, x, isort_by(~A, ~le, t))
81
82def internal_ins_pick(-A: Data, b: Bool, +x: A, +y: A, +t: List<&2, A>, -r: List<&2, A>, ih: K.perm(A, x <> t, r)) -> K.perm(A, x <> y <> t, Bool.pick(List<&2, A>, b, x <> y <> t, y <> r)):
83 match b:
84 case True{}:
85 perm_refl(A, x <> y <> t)
86 case False{}:
87 perm_trans(A, x <> y <> t, y <> x <> t, y <> r, perm_swap(A, x, y, t), perm_cons(A, y, x <> t, r, ih))
88
89# Inserting x into xs permutes x <> xs.
90def insert_by_perm(~A: Data, ~le: A -> A -> Bool, +x: A, +xs: List<&2, A>) -> K.perm(A, x <> xs, insert_by(~A, ~le, x, xs)):
91 match xs:
92 case Nil{}:
93 perm_refl(A, x <> Nil{})
94 case +y <> t:
95 internal_ins_pick(A, le(x, y), x, y, t, insert_by(~A, ~le, x, t), insert_by_perm(~A, ~le, x, t))
96
97# Insertion sort permutes its input.
98def isort_by_perm(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> K.perm(A, xs, isort_by(~A, ~le, xs)):
99 match xs:
100 case Nil{}:
101 perm_refl(A, Nil{})
102 case +x <> t:
103 perm_trans(A, x <> t, x <> isort_by(~A, ~le, t), isort_by(~A, ~le, x <> t), perm_cons(A, x, t, isort_by(~A, ~le, t), isort_by_perm(~A, ~le, t)), insert_by_perm(~A, ~le, x, isort_by(~A, ~le, t)))
104
105# Closed instance: instantiate at Nat and at String.
106def isort_by_perm_nat(+xs: List<&2, Nat>) -> K.perm(Nat, xs, isort_by(~Nat, ~Nat.is_le, xs)):
107 isort_by_perm(~Nat, ~Nat.is_le, xs)
108
109def internal_append_nil(-A: Data, +ys: List<&2, A>) -> {List.append(&2, A, ys, Nil{}) == ys : List<&2, A>}:
110 match ys:
111 case Nil{}:
112 {==}
113 case +y <> t:
114 %Equal.sym(List<&2, A>, List.append(&2, A, t, Nil{}), t, internal_append_nil(A, t)) : {y <> _ == y <> t : List<&2, A>}
115 {==}
116
117# Appending Nil is a permutation (steps: none).
118def perm_append_nil(-A: Data, +xs: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, Nil{}), xs):
119 (Nil{}, internal_append_nil(A, xs))
120
121# Moving an element from the middle to the front is a permutation.
122def perm_move(-A: Data, +xs: List<&2, A>, +y: A, +t: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, y <> t), y <> List.append(&2, A, xs, t)):
123 match xs:
124 case Nil{}:
125 perm_refl(A, y <> t)
126 case +x <> r:
127 perm_trans(A, x <> List.append(&2, A, r, y <> t), x <> y <> List.append(&2, A, r, t), y <> x <> List.append(&2, A, r, t), perm_cons(A, x, List.append(&2, A, r, y <> t), y <> List.append(&2, A, r, t), perm_move(A, r, y, t)), perm_swap(A, x, y, List.append(&2, A, r, t)))
128
129def merge_by(~A: Data, ~le: A -> A -> Bool, xs: List<&2, A>, ys: List<&2, A>) -> List<&2, A>:
130 match xs ys:
131 case Nil{} _:
132 ys
133 case +x <> +xt Nil{}:
134 x <> xt
135 case +x <> +xt +y <> +yt:
136 Bool.pick(List<&2, A>, le(x, y), x <> merge_by(~A, ~le, xt, y <> yt), y <> merge_by(~A, ~le, x <> xt, yt))
137
138def internal_merge_pick(-A: Data, b: Bool, +x: A, +xt: List<&2, A>, +y: A, +yt: List<&2, A>, -m1: List<&2, A>, -m2: List<&2, A>, ih1: K.perm(A, List.append(&2, A, xt, y <> yt), m1), ih2: K.perm(A, x <> List.append(&2, A, xt, yt), m2)) -> K.perm(A, x <> List.append(&2, A, xt, y <> yt), Bool.pick(List<&2, A>, b, x <> m1, y <> m2)):
139 match b:
140 case True{}:
141 perm_cons(A, x, List.append(&2, A, xt, y <> yt), m1, ih1)
142 case False{}:
143 perm_trans(A, x <> List.append(&2, A, xt, y <> yt), y <> x <> List.append(&2, A, xt, yt), y <> m2, perm_move(A, x <> xt, y, yt), perm_cons(A, y, x <> List.append(&2, A, xt, yt), m2, ih2))
144
145# Merging permutes the concatenation.
146def merge_by_perm(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>, +ys: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, ys), merge_by(~A, ~le, xs, ys)):
147 match xs ys:
148 case Nil{} _:
149 perm_refl(A, ys)
150 case +x <> +xt Nil{}:
151 perm_cons(A, x, List.append(&2, A, xt, Nil{}), xt, perm_append_nil(A, xt))
152 case +x <> +xt +y <> +yt:
153 internal_merge_pick(A, le(x, y), x, xt, y, yt, merge_by(~A, ~le, xt, y <> yt), merge_by(~A, ~le, x <> xt, yt), merge_by_perm(~A, ~le, xt, y <> yt), merge_by_perm(~A, ~le, x <> xt, yt))
154
155def merge_by_perm_nat(+xs: List<&2, Nat>, +ys: List<&2, Nat>) -> K.perm(Nat, List.append(&2, Nat, xs, ys), merge_by(~Nat, ~Nat.is_le, xs, ys)):
156 merge_by_perm(~Nat, ~Nat.is_le, xs, ys)
157
158# Moving the front element into the middle is a permutation.
159def perm_move_rev(-A: Data, +xs: List<&2, A>, +y: A, +t: List<&2, A>) -> K.perm(A, y <> List.append(&2, A, xs, t), List.append(&2, A, xs, y <> t)):
160 match xs:
161 case Nil{}:
162 perm_refl(A, y <> t)
163 case +x <> r:
164 perm_trans(A, y <> x <> List.append(&2, A, r, t), x <> y <> List.append(&2, A, r, t), x <> List.append(&2, A, r, y <> t), perm_swap(A, y, x, List.append(&2, A, r, t)), perm_cons(A, x, y <> List.append(&2, A, r, t), List.append(&2, A, r, y <> t), perm_move_rev(A, r, y, t)))
165
166# Appending commutes up to permutation (derived without symmetry).
167def perm_append_comm(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, ys), List.append(&2, A, ys, xs)):
168 match xs:
169 case Nil{}:
170 (Nil{}, Equal.sym(List<&2, A>, List.append(&2, A, ys, Nil{}), ys, internal_append_nil(A, ys)))
171 case +x <> r:
172 perm_trans(A, x <> List.append(&2, A, r, ys), x <> List.append(&2, A, ys, r), List.append(&2, A, ys, x <> r), perm_cons(A, x, List.append(&2, A, r, ys), List.append(&2, A, ys, r), perm_append_comm(A, r, ys)), perm_move_rev(A, ys, x, r))
173
174# Permuting the right operand of an append.
175def perm_append_right(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>, -zs: List<&2, A>, h: K.perm(A, ys, zs)) -> K.perm(A, List.append(&2, A, xs, ys), List.append(&2, A, xs, zs)):
176 match xs:
177 case Nil{}:
178 h
179 case +x <> r:
180 perm_cons(A, x, List.append(&2, A, r, ys), List.append(&2, A, r, zs), perm_append_right(A, r, ys, zs, h))
181
182# Permuting both operands of an append.
183def perm_append(-A: Data, +xs: List<&2, A>, +xs2: List<&2, A>, +ys: List<&2, A>, -ys2: List<&2, A>, p: K.perm(A, xs, xs2), q: K.perm(A, ys, ys2)) -> K.perm(A, List.append(&2, A, xs, ys), List.append(&2, A, xs2, ys2)):
184 perm_trans(A, List.append(&2, A, xs, ys), List.append(&2, A, ys, xs), List.append(&2, A, xs2, ys2), perm_append_comm(A, xs, ys), perm_trans(A, List.append(&2, A, ys, xs), List.append(&2, A, ys, xs2), List.append(&2, A, xs2, ys2), perm_append_right(A, ys, xs, xs2, p), perm_trans(A, List.append(&2, A, ys, xs2), List.append(&2, A, xs2, ys), List.append(&2, A, xs2, ys2), perm_append_comm(A, ys, xs2), perm_append_right(A, xs2, ys, ys2, q))))
185
186def evens(-A: Data, xs: List<&2, A>) -> List<&2, A>:
187 match xs:
188 case Nil{}:
189 Nil{}
190 case +x <> r:
191 match r:
192 case Nil{}:
193 x <> Nil{}
194 case +y <> t:
195 x <> evens(A, t)
196
197def odds(-A: Data, xs: List<&2, A>) -> List<&2, A>:
198 match xs:
199 case Nil{}:
200 Nil{}
201 case +x <> r:
202 match r:
203 case Nil{}:
204 Nil{}
205 case +y <> t:
206 y <> odds(A, t)
207
208# Splitting into evens and odds is a permutation.
209def split_perm(-A: Data, +xs: List<&2, A>) -> K.perm(A, xs, List.append(&2, A, evens(A, xs), odds(A, xs))):
210 match xs:
211 case Nil{}:
212 perm_refl(A, Nil{})
213 case +x <> r:
214 match r:
215 case Nil{}:
216 perm_refl(A, x <> Nil{})
217 case +y <> t:
218 perm_cons(A, x, y <> t, List.append(&2, A, evens(A, t), y <> odds(A, t)), perm_trans(A, y <> t, y <> List.append(&2, A, evens(A, t), odds(A, t)), List.append(&2, A, evens(A, t), y <> odds(A, t)), perm_cons(A, y, t, List.append(&2, A, evens(A, t), odds(A, t)), split_perm(A, t)), perm_move_rev(A, evens(A, t), y, odds(A, t))))
219
220# Merge sort with fuel (the fuel only bounds recursion; it never affects the permutation).
221def msort_by(~A: Data, ~le: A -> A -> Bool, fuel: Nat, +xs: List<&2, A>) -> List<&2, A>:
222 match fuel:
223 case 0n:
224 xs
225 case 1n++f:
226 merge_by(~A, ~le, msort_by(~A, ~le, f, evens(A, xs)), msort_by(~A, ~le, f, odds(A, xs)))
227
228# Merge sort permutes its input, for any fuel.
229def msort_by_perm(~A: Data, ~le: A -> A -> Bool, fuel: Nat, +xs: List<&2, A>) -> K.perm(A, xs, msort_by(~A, ~le, fuel, xs)):
230 match fuel:
231 case 0n:
232 perm_refl(A, xs)
233 case 1n++f:
234 perm_trans(A, xs, List.append(&2, A, evens(A, xs), odds(A, xs)), msort_by(~A, ~le, 1n+f, xs), split_perm(A, xs), perm_trans(A, List.append(&2, A, evens(A, xs), odds(A, xs)), List.append(&2, A, msort_by(~A, ~le, f, evens(A, xs)), msort_by(~A, ~le, f, odds(A, xs))), msort_by(~A, ~le, 1n+f, xs), perm_append(A, evens(A, xs), msort_by(~A, ~le, f, evens(A, xs)), odds(A, xs), msort_by(~A, ~le, f, odds(A, xs)), msort_by_perm(~A, ~le, f, evens(A, xs)), msort_by_perm(~A, ~le, f, odds(A, xs))), merge_by_perm(~A, ~le, msort_by(~A, ~le, f, evens(A, xs)), msort_by(~A, ~le, f, odds(A, xs)))))
235
236def sort_by(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> List<&2, A>:
237 msort_by(~A, ~le, List.length(&2, A, xs), xs)
238
239# The sort permutes its input.
240def sort_by_perm(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> K.perm(A, xs, sort_by(~A, ~le, xs)):
241 msort_by_perm(~A, ~le, List.length(&2, A, xs), xs)
242
243def sort_by_perm_nat(+xs: List<&2, Nat>) -> K.perm(Nat, xs, sort_by(~Nat, ~Nat.is_le, xs)):
244 sort_by_perm(~Nat, ~Nat.is_le, xs)
245
246def internal_sort_nat_test() -> {sort_by(~Nat, ~Nat.is_le, [5n, 1n, 4n, 2n, 3n, 1n]) == [1n, 1n, 2n, 3n, 4n, 5n] : List<&2, Nat>}:
247 {==}
248
249def internal_swap_head_invol(-A: Data, xs: List<&2, A>) -> {K.swap_head(A, K.swap_head(A, xs)) == xs : List<&2, A>}:
250 match xs:
251 case Nil{}:
252 {==}
253 case +x <> t:
254 match t:
255 case Nil{}:
256 {==}
257 case +y <> u:
258 {==}
259
260def internal_swap_at_invol(-A: Data, +i: Nat, +xs: List<&2, A>) -> {K.swap_at(A, i, K.swap_at(A, i, xs)) == xs : List<&2, A>}:
261 match i:
262 case 0n:
263 internal_swap_head_invol(A, xs)
264 case 1n+p:
265 match xs:
266 case Nil{}:
267 {==}
268 case +x <> t:
269 %Equal.sym(List<&2, A>, K.swap_at(A, p, K.swap_at(A, p, t)), t, internal_swap_at_invol(A, p, t)) : {x <> _ == x <> t : List<&2, A>}
270 {==}
271
272def internal_invert(+s: List<&2, Nat>) -> List<&2, Nat>:
273 match s:
274 case Nil{}:
275 Nil{}
276 case +i <> rest:
277 List.append(&2, Nat, internal_invert(rest), i <> Nil{})
278
279def internal_apply_invert(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>) -> {K.apply(A, internal_invert(s), K.apply(A, s, xs)) == xs : List<&2, A>}:
280 match s:
281 case Nil{}:
282 {==}
283 case +i <> rest:
284 %Equal.sym(List<&2, A>, K.apply(A, List.append(&2, Nat, internal_invert(rest), i <> Nil{}), K.apply(A, rest, K.swap_at(A, i, xs))), K.apply(A, i <> Nil{}, K.apply(A, internal_invert(rest), K.apply(A, rest, K.swap_at(A, i, xs)))), internal_apply_app(A, internal_invert(rest), i <> Nil{}, K.apply(A, rest, K.swap_at(A, i, xs)))) : {_ == xs : List<&2, A>}
285 %Equal.sym(List<&2, A>, K.apply(A, internal_invert(rest), K.apply(A, rest, K.swap_at(A, i, xs))), K.swap_at(A, i, xs), internal_apply_invert(A, rest, K.swap_at(A, i, xs))) : {K.apply(A, i <> Nil{}, _) == xs : List<&2, A>}
286 internal_swap_at_invol(A, i, xs)
287
288def internal_sym_eq(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>, -ys: List<&2, A>, e: K.perm_steps(A, s, xs, ys)) -> K.perm_steps(A, internal_invert(s), ys, xs):
289 %e : {K.apply(A, internal_invert(s), _) == xs : List<&2, A>}
290 internal_apply_invert(A, s, xs)
291
292# Permutation is symmetric.
293def perm_sym(-A: Data, +xs: List<&2, A>, -ys: List<&2, A>, h: K.perm(A, xs, ys)) -> K.perm(A, ys, xs):
294 (s, e) = h
295 +s = s
296 (internal_invert(s), internal_sym_eq(A, s, xs, ys, e))
297
298# Permutation preserves length.
299def internal_swap_head_length(-A: Data, xs: List<&2, A>) -> {List.length(&2, A, K.swap_head(A, xs)) == List.length(&2, A, xs) : Nat}:
300 match xs:
301 case Nil{}:
302 {==}
303 case +x <> t:
304 match t:
305 case Nil{}:
306 {==}
307 case +y <> u:
308 {==}
309
310def internal_swap_at_length(-A: Data, +i: Nat, +xs: List<&2, A>) -> {List.length(&2, A, K.swap_at(A, i, xs)) == List.length(&2, A, xs) : Nat}:
311 match i:
312 case 0n:
313 internal_swap_head_length(A, xs)
314 case 1n+p:
315 match xs:
316 case Nil{}:
317 {==}
318 case +x <> t:
319 %Equal.sym(Nat, List.length(&2, A, K.swap_at(A, p, t)), List.length(&2, A, t), internal_swap_at_length(A, p, t)) : {1n+_ == 1n+List.length(&2, A, t) : Nat}
320 {==}
321
322def internal_apply_length(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>) -> {List.length(&2, A, K.apply(A, s, xs)) == List.length(&2, A, xs) : Nat}:
323 match s:
324 case Nil{}:
325 {==}
326 case +i <> rest:
327 Equal.trans(Nat, List.length(&2, A, K.apply(A, rest, K.swap_at(A, i, xs))), List.length(&2, A, K.swap_at(A, i, xs)), List.length(&2, A, xs), internal_apply_length(A, rest, K.swap_at(A, i, xs)), internal_swap_at_length(A, i, xs))
328
329def internal_length_eq(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>, -ys: List<&2, A>, e: K.perm_steps(A, s, xs, ys)) -> {List.length(&2, A, ys) == List.length(&2, A, xs) : Nat}:
330 %e : {List.length(&2, A, _) == List.length(&2, A, xs) : Nat}
331 internal_apply_length(A, s, xs)
332
333def perm_length(-A: Data, +xs: List<&2, A>, -ys: List<&2, A>, h: K.perm(A, xs, ys)) -> {List.length(&2, A, ys) == List.length(&2, A, xs) : Nat}:
334 (s, e) = h
335 internal_length_eq(A, s, xs, ys, e)
336
337def internal_perm_nil_nat() -> K.perm(Nat, Nil{}, Nil{}):
338 perm_nil(Nat)
339
340def internal_perm_dup_nat(+xs: List<&2, Nat>, -ys: List<&2, Nat>, h: K.perm(Nat, xs, ys)) -> K.perm(Nat, xs, ys) & K.perm(Nat, xs, ys):
341 perm_dup(Nat, xs, ys, h)
342
343def internal_perm_sym_nat(+xs: List<&2, Nat>, -ys: List<&2, Nat>, h: K.perm(Nat, xs, ys)) -> K.perm(Nat, ys, xs):
344 perm_sym(Nat, xs, ys, h)
345
346def internal_perm_length_nat(+xs: List<&2, Nat>, -ys: List<&2, Nat>, h: K.perm(Nat, xs, ys)) -> {List.length(&2, Nat, ys) == List.length(&2, Nat, xs) : Nat}:
347 perm_length(Nat, xs, ys, h)