sort.bend
bend-mathlib@0.7.0.0 · 263 lines · 0 definitions
1# bend-mathlib/sort.bend: insertion and merge sort proved sorted over perm.bend's value defs (generic ~le).
2import Base
3import bend-mathlib@0.7.0.0/nat.bend as MNat
4import bend-mathlib@0.7.0.0/list.bend as MList
5import bend-mathlib@0.7.0.0/perm.bend as MPerm
6
7def internal_sorted_single(~A: Data, ~le: A -> A -> Bool, +x: A) -> MList.sorted_by(~A, ~le, x <> Nil{}):
8 {==}
9
10def internal_sorted_cons_cons_intro(~A: Data, ~le: A -> A -> Bool, +x: A, +y: A, +t: List<&2, A>, hxy: {le(x, y) == True{} : Bool}, hyt: MList.sorted_by(~A, ~le, y <> t)) -> MList.sorted_by(~A, ~le, x <> y <> t):
11 %Equal.sym(Bool, le(x, y), True{}, hxy) : {Bool.and(_, List.all(~&1, ~(A & A), ~(p => le(Pair.fst(A, A, p), Pair.snd(A, A, p))), List.zip(&2, A, &2, A, y <> t, t))) == True{} : Bool}
12 hyt
13
14def internal_sorted_cons_cons_elim_le(~A: Data, ~le: A -> A -> Bool, +x: A, +y: A, +t: List<&2, A>, h: MList.sorted_by(~A, ~le, x <> y <> t)) -> {le(x, y) == True{} : Bool}:
15 MList.internal_and_left(le(x, y), List.all(~&1, ~(A & A), ~(p => le(Pair.fst(A, A, p), Pair.snd(A, A, p))), List.zip(&2, A, &2, A, y <> t, t)), h)
16
17def internal_sorted_cons_cons_elim_tail(~A: Data, ~le: A -> A -> Bool, +x: A, +y: A, +t: List<&2, A>, h: MList.sorted_by(~A, ~le, x <> y <> t)) -> MList.sorted_by(~A, ~le, y <> t):
18 MList.internal_and_right(le(x, y), List.all(~&1, ~(A & A), ~(p => le(Pair.fst(A, A, p), Pair.snd(A, A, p))), List.zip(&2, A, &2, A, y <> t, t)), h)
19
20def internal_sorted_tail(~A: Data, ~le: A -> A -> Bool, +x: A, +xs: List<&2, A>, h: MList.sorted_by(~A, ~le, x <> xs)) -> MList.sorted_by(~A, ~le, xs):
21 match xs:
22 case Nil{}:
23 {==}
24 case +y <> +t:
25 internal_sorted_cons_cons_elim_tail(~A, ~le, x, y, t, h)
26
27def internal_le_flip_nat(x: Nat, y: Nat, e: {Nat.is_le(x, y) == False{} : Bool}) -> {Nat.is_le(y, x) == True{} : Bool}:
28 match x y:
29 case 0n 0n:
30 Empty.absurd({True{} == True{} : Bool}, MNat.internal_false_ne_true(Equal.sym(Bool, True{}, False{}, e)))
31 case 0n 1n+q:
32 Empty.absurd({Nat.is_le(1n+q, 0n) == True{} : Bool}, MNat.internal_false_ne_true(Equal.sym(Bool, True{}, False{}, e)))
33 case 1n+p 0n:
34 {==}
35 case 1n+p 1n+q:
36 internal_le_flip_nat(p, q, e)
37
38def internal_le_trans_nat(x: Nat, y: Nat, z: Nat, xy: {Nat.is_le(x, y) == True{} : Bool}, yz: {Nat.is_le(y, z) == True{} : Bool}) -> {Nat.is_le(x, z) == True{} : Bool}:
39 MNat.le_trans(x, y, z, xy, yz)
40
41def internal_cons_ins_step(~A: Data, ~le: A -> A -> Bool, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, b: Bool, +y: A, +x: A, +z: A, +u: List<&2, A>, e: {le(x, z) == b : Bool}, hyx: {le(y, x) == True{} : Bool}, +h: MList.sorted_by(~A, ~le, y <> z <> u), k: {le(z, x) == True{} : Bool} -> MList.sorted_by(~A, ~le, z <> MPerm.insert_by(~A, ~le, x, u))) -> MList.sorted_by(~A, ~le, y <> Bool.pick(List<&2, A>, b, x <> z <> u, z <> MPerm.insert_by(~A, ~le, x, u))):
42 match b:
43 case True{}:
44 internal_sorted_cons_cons_intro(~A, ~le, y, x, z <> u, hyx, internal_sorted_cons_cons_intro(~A, ~le, x, z, u, e, internal_sorted_cons_cons_elim_tail(~A, ~le, y, z, u, h)))
45 case False{}:
46 internal_sorted_cons_cons_intro(~A, ~le, y, z, MPerm.insert_by(~A, ~le, x, u), internal_sorted_cons_cons_elim_le(~A, ~le, y, z, u, h), k(le_total(x, z, e)))
47
48def internal_sorted_cons_ins(~A: Data, ~le: A -> A -> Bool, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, +t: List<&2, A>, +y: A, +x: A, hyx: {le(y, x) == True{} : Bool}, +h: MList.sorted_by(~A, ~le, y <> t)) -> MList.sorted_by(~A, ~le, y <> MPerm.insert_by(~A, ~le, x, t)):
49 match t:
50 case Nil{}:
51 internal_sorted_cons_cons_intro(~A, ~le, y, x, Nil{}, hyx, internal_sorted_single(~A, ~le, x))
52 case +z <> +u:
53 internal_cons_ins_step(~A, ~le, ~le_total, le(x, z), y, x, z, u, {==}, hyx, h, hzx => internal_sorted_cons_ins(~A, ~le, ~le_total, u, z, x, hzx, internal_sorted_cons_cons_elim_tail(~A, ~le, y, z, u, h)))
54
55def internal_insert_step(~A: Data, ~le: A -> A -> Bool, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, b: Bool, +x: A, +y: A, +t: List<&2, A>, e: {le(x, y) == b : Bool}, +h: MList.sorted_by(~A, ~le, y <> t)) -> MList.sorted_by(~A, ~le, Bool.pick(List<&2, A>, b, x <> y <> t, y <> MPerm.insert_by(~A, ~le, x, t))):
56 match b:
57 case True{}:
58 internal_sorted_cons_cons_intro(~A, ~le, x, y, t, e, h)
59 case False{}:
60 internal_sorted_cons_ins(~A, ~le, ~le_total, t, y, x, le_total(x, y, e), h)
61
62def insert_by_sorted(~A: Data, ~le: A -> A -> Bool, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, +x: A, +xs: List<&2, A>, +h: MList.sorted_by(~A, ~le, xs)) -> MList.sorted_by(~A, ~le, MPerm.insert_by(~A, ~le, x, xs)):
63 match xs:
64 case Nil{}:
65 internal_sorted_single(~A, ~le, x)
66 case +y <> t:
67 internal_insert_step(~A, ~le, ~le_total, le(x, y), x, y, t, {==}, h)
68
69# Insertion sort returns a sorted list.
70def isort_by_sorted(~A: Data, ~le: A -> A -> Bool, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, +xs: List<&2, A>) -> MList.sorted_by(~A, ~le, MPerm.isort_by(~A, ~le, xs)):
71 match xs:
72 case Nil{}:
73 {==}
74 case +x <> t:
75 insert_by_sorted(~A, ~le, ~le_total, x, MPerm.isort_by(~A, ~le, t), isort_by_sorted(~A, ~le, ~le_total, t))
76
77def isort_by_sorted_nat(+xs: List<&2, Nat>) -> MList.sorted_by(~Nat, ~Nat.is_le, MPerm.isort_by(~Nat, ~Nat.is_le, xs)):
78 isort_by_sorted(~Nat, ~Nat.is_le, ~internal_le_flip_nat, xs)
79
80def internal_least(~A: Data, ~le: A -> A -> Bool, +lo: A, +xs: List<&2, A>) -> Data:
81 {List.all(~&2, ~A, ~(y => le(lo, y)), xs) == True{} : Bool}
82
83def internal_least_intro(~A: Data, ~le: A -> A -> Bool, +lo: A, +h: A, +t: List<&2, A>, hh: {le(lo, h) == True{} : Bool}, ht: internal_least(~A, ~le, lo, t)) -> internal_least(~A, ~le, lo, h <> t):
84 %Equal.sym(Bool, le(lo, h), True{}, hh) : {Bool.and(_, List.all(~&2, ~A, ~(y => le(lo, y)), t)) == True{} : Bool}
85 ht
86
87def internal_least_cons_le(~A: Data, ~le: A -> A -> Bool, +lo: A, +h: A, +t: List<&2, A>, hh: internal_least(~A, ~le, lo, h <> t)) -> {le(lo, h) == True{} : Bool}:
88 MList.internal_and_left(le(lo, h), List.all(~&2, ~A, ~(y => le(lo, y)), t), hh)
89
90def internal_least_cons_tail(~A: Data, ~le: A -> A -> Bool, +lo: A, +h: A, +t: List<&2, A>, hh: internal_least(~A, ~le, lo, h <> t)) -> internal_least(~A, ~le, lo, t):
91 MList.internal_and_right(le(lo, h), List.all(~&2, ~A, ~(y => le(lo, y)), t), hh)
92
93def internal_least_trans(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, +a: A, +b: A, +xs: List<&2, A>, +hab: {le(a, b) == True{} : Bool}, +hb: internal_least(~A, ~le, b, xs)) -> internal_least(~A, ~le, a, xs):
94 match xs:
95 case Nil{}:
96 {==}
97 case +h <> +t:
98 internal_least_intro(~A, ~le, a, h, t, le_trans(a, b, h, hab, internal_least_cons_le(~A, ~le, b, h, t, hb)), internal_least_trans(~A, ~le, ~le_trans, a, b, t, hab, internal_least_cons_tail(~A, ~le, b, h, t, hb)))
99
100def internal_sorted_least_head(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, +xs: List<&2, A>, +x: A, +h: MList.sorted_by(~A, ~le, x <> xs)) -> internal_least(~A, ~le, x, xs):
101 match xs:
102 case Nil{}:
103 {==}
104 case +y <> +t:
105 internal_least_intro(~A, ~le, x, y, t, internal_sorted_cons_cons_elim_le(~A, ~le, x, y, t, h), internal_least_trans(~A, ~le, ~le_trans, x, y, t, internal_sorted_cons_cons_elim_le(~A, ~le, x, y, t, h), internal_sorted_least_head(~A, ~le, ~le_trans, t, y, internal_sorted_cons_cons_elim_tail(~A, ~le, x, y, t, h))))
106
107def internal_sorted_cons_of_least(~A: Data, ~le: A -> A -> Bool, +lo: A, +m: List<&2, A>, hl: internal_least(~A, ~le, lo, m), hm: MList.sorted_by(~A, ~le, m)) -> MList.sorted_by(~A, ~le, lo <> m):
108 match m:
109 case Nil{}:
110 {==}
111 case +z <> +t:
112 internal_sorted_cons_cons_intro(~A, ~le, lo, z, t, internal_least_cons_le(~A, ~le, lo, z, t, hl), hm)
113
114def internal_least_merge_pick(~A: Data, ~le: A -> A -> Bool, +lo: A, b: Bool, +x: A, +xt: List<&2, A>, +y: A, +yt: List<&2, A>, hx: internal_least(~A, ~le, lo, x <> xt), hy: internal_least(~A, ~le, lo, y <> yt), hm1: internal_least(~A, ~le, lo, MPerm.merge_by(~A, ~le, xt, y <> yt)), hm2: internal_least(~A, ~le, lo, MPerm.merge_by(~A, ~le, x <> xt, yt))) -> internal_least(~A, ~le, lo, Bool.pick(List<&2, A>, b, x <> MPerm.merge_by(~A, ~le, xt, y <> yt), y <> MPerm.merge_by(~A, ~le, x <> xt, yt))):
115 match b:
116 case True{}:
117 internal_least_intro(~A, ~le, lo, x, MPerm.merge_by(~A, ~le, xt, y <> yt), internal_least_cons_le(~A, ~le, lo, x, xt, hx), hm1)
118 case False{}:
119 internal_least_intro(~A, ~le, lo, y, MPerm.merge_by(~A, ~le, x <> xt, yt), internal_least_cons_le(~A, ~le, lo, y, yt, hy), hm2)
120
121def internal_least_merge(~A: Data, ~le: A -> A -> Bool, +lo: A, +xs: List<&2, A>, +ys: List<&2, A>, +hx: internal_least(~A, ~le, lo, xs), +hy: internal_least(~A, ~le, lo, ys)) -> internal_least(~A, ~le, lo, MPerm.merge_by(~A, ~le, xs, ys)):
122 match xs ys:
123 case Nil{} _:
124 hy
125 case +x <> +xt Nil{}:
126 hx
127 case +x <> +xt +y <> +yt:
128 internal_least_merge_pick(~A, ~le, lo, le(x, y), x, xt, y, yt, hx, hy, internal_least_merge(~A, ~le, lo, xt, y <> yt, internal_least_cons_tail(~A, ~le, lo, x, xt, hx), hy), internal_least_merge(~A, ~le, lo, x <> xt, yt, hx, internal_least_cons_tail(~A, ~le, lo, y, yt, hy)))
129
130def internal_merge_sorted_pick(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, b: Bool, +x: A, +xt: List<&2, A>, +y: A, +yt: List<&2, A>, +e: {le(x, y) == b : Bool}, hx: MList.sorted_by(~A, ~le, x <> xt), hy: MList.sorted_by(~A, ~le, y <> yt), hm1: MList.sorted_by(~A, ~le, MPerm.merge_by(~A, ~le, xt, y <> yt)), hm2: MList.sorted_by(~A, ~le, MPerm.merge_by(~A, ~le, x <> xt, yt))) -> MList.sorted_by(~A, ~le, Bool.pick(List<&2, A>, b, x <> MPerm.merge_by(~A, ~le, xt, y <> yt), y <> MPerm.merge_by(~A, ~le, x <> xt, yt))):
131 match b:
132 case True{}:
133 internal_sorted_cons_of_least(~A, ~le, x, MPerm.merge_by(~A, ~le, xt, y <> yt), internal_least_merge(~A, ~le, x, xt, y <> yt, internal_sorted_least_head(~A, ~le, ~le_trans, xt, x, hx), internal_least_intro(~A, ~le, x, y, yt, e, internal_least_trans(~A, ~le, ~le_trans, x, y, yt, e, internal_sorted_least_head(~A, ~le, ~le_trans, yt, y, hy)))), hm1)
134 case False{}:
135 internal_sorted_cons_of_least(~A, ~le, y, MPerm.merge_by(~A, ~le, x <> xt, yt), internal_least_merge(~A, ~le, y, x <> xt, yt, internal_least_intro(~A, ~le, y, x, xt, le_total(x, y, e), internal_least_trans(~A, ~le, ~le_trans, y, x, xt, le_total(x, y, e), internal_sorted_least_head(~A, ~le, ~le_trans, xt, x, hx))), internal_sorted_least_head(~A, ~le, ~le_trans, yt, y, hy)), hm2)
136
137# Merging two sorted lists is sorted (needs transitivity, and totality for the un-taken branch).
138def merge_by_sorted(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, +xs: List<&2, A>, +ys: List<&2, A>, +hx: MList.sorted_by(~A, ~le, xs), +hy: MList.sorted_by(~A, ~le, ys)) -> MList.sorted_by(~A, ~le, MPerm.merge_by(~A, ~le, xs, ys)):
139 match xs ys:
140 case Nil{} _:
141 hy
142 case +x <> +xt Nil{}:
143 hx
144 case +x <> +xt +y <> +yt:
145 internal_merge_sorted_pick(~A, ~le, ~le_trans, ~le_total, le(x, y), x, xt, y, yt, {==}, hx, hy, merge_by_sorted(~A, ~le, ~le_trans, ~le_total, xt, y <> yt, internal_sorted_tail(~A, ~le, x, xt, hx), hy), merge_by_sorted(~A, ~le, ~le_trans, ~le_total, x <> xt, yt, hx, internal_sorted_tail(~A, ~le, y, yt, hy)))
146
147def merge_by_sorted_nat(+xs: List<&2, Nat>, +ys: List<&2, Nat>, hx: MList.sorted_by(~Nat, ~Nat.is_le, xs), hy: MList.sorted_by(~Nat, ~Nat.is_le, ys)) -> MList.sorted_by(~Nat, ~Nat.is_le, MPerm.merge_by(~Nat, ~Nat.is_le, xs, ys)):
148 merge_by_sorted(~Nat, ~Nat.is_le, ~internal_le_trans_nat, ~internal_le_flip_nat, xs, ys, hx, hy)
149
150def internal_length_zero(~A: Data, +xs: List<&2, A>, e: {List.length(&2, A, xs) == 0n : Nat}) -> {xs == Nil{} : List<&2, A>}:
151 match xs:
152 case Nil{}:
153 {==}
154 case +x <> +t:
155 Empty.absurd({x <> t == Nil{} : List<&2, A>}, MNat.zero_ne_succ(List.length(&2, A, t), Equal.sym(Nat, List.length(&2, A, x <> t), 0n, e)))
156
157def internal_sorted_short(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>, h: MNat.le(List.length(&2, A, xs), 1n)) -> MList.sorted_by(~A, ~le, xs):
158 match xs:
159 case Nil{}:
160 {==}
161 case +x <> +t:
162 match t:
163 case Nil{}:
164 internal_sorted_single(~A, ~le, x)
165 case +y <> +u:
166 %Equal.sym(List<&2, A>, t, Nil{}, internal_length_zero(~A, t, MNat.le_zero_eq(List.length(&2, A, t), MNat.le_of_succ_le_succ(List.length(&2, A, t), 0n, h)))) : {MList.sorted_by(~A, ~le, x <> _) : Data}
167 internal_sorted_single(~A, ~le, x)
168
169def internal_length_evens_le(~A: Data, +xs: List<&2, A>) -> {Nat.is_le(List.length(&2, A, MPerm.evens(A, xs)), List.length(&2, A, xs)) == True{} : Bool}:
170 match xs:
171 case Nil{}:
172 {==}
173 case +x <> +r:
174 match r:
175 case Nil{}:
176 {==}
177 case +y <> +t:
178 MNat.succ_le_succ(List.length(&2, A, MPerm.evens(A, t)), 1n+List.length(&2, A, t), MNat.le_trans(List.length(&2, A, MPerm.evens(A, t)), List.length(&2, A, t), 1n+List.length(&2, A, t), internal_length_evens_le(~A, t), MNat.le_succ(List.length(&2, A, t))))
179
180def internal_length_odds_le(~A: Data, +xs: List<&2, A>) -> {Nat.is_le(List.length(&2, A, MPerm.odds(A, xs)), List.length(&2, A, xs)) == True{} : Bool}:
181 match xs:
182 case Nil{}:
183 {==}
184 case +x <> +r:
185 match r:
186 case Nil{}:
187 {==}
188 case +y <> +t:
189 MNat.succ_le_succ(List.length(&2, A, MPerm.odds(A, t)), 1n+List.length(&2, A, t), MNat.le_trans(List.length(&2, A, MPerm.odds(A, t)), List.length(&2, A, t), 1n+List.length(&2, A, t), internal_length_odds_le(~A, t), MNat.le_succ(List.length(&2, A, t))))
190
191def internal_least_evens(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>, +lo: A, +h: internal_least(~A, ~le, lo, xs)) -> internal_least(~A, ~le, lo, MPerm.evens(A, xs)):
192 match xs:
193 case Nil{}:
194 {==}
195 case +x <> +r:
196 match r:
197 case Nil{}:
198 h
199 case +y <> +t:
200 internal_least_intro(~A, ~le, lo, x, MPerm.evens(A, t), internal_least_cons_le(~A, ~le, lo, x, r, h), internal_least_evens(~A, ~le, t, lo, internal_least_cons_tail(~A, ~le, lo, y, t, internal_least_cons_tail(~A, ~le, lo, x, r, h))))
201
202def internal_least_odds(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>, +lo: A, +h: internal_least(~A, ~le, lo, xs)) -> internal_least(~A, ~le, lo, MPerm.odds(A, xs)):
203 match xs:
204 case Nil{}:
205 {==}
206 case +x <> +r:
207 match r:
208 case Nil{}:
209 {==}
210 case +y <> +t:
211 internal_least_intro(~A, ~le, lo, y, MPerm.odds(A, t), internal_least_cons_le(~A, ~le, lo, y, t, internal_least_cons_tail(~A, ~le, lo, x, r, h)), internal_least_odds(~A, ~le, t, lo, internal_least_cons_tail(~A, ~le, lo, y, t, internal_least_cons_tail(~A, ~le, lo, x, r, h))))
212
213def internal_evens_sorted(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, +xs: List<&2, A>, +h: MList.sorted_by(~A, ~le, xs)) -> MList.sorted_by(~A, ~le, MPerm.evens(A, xs)):
214 match xs:
215 case Nil{}:
216 {==}
217 case +x <> +r:
218 match r:
219 case Nil{}:
220 h
221 case +y <> +t:
222 internal_sorted_cons_of_least(~A, ~le, x, MPerm.evens(A, t), internal_least_evens(~A, ~le, t, x, internal_least_cons_tail(~A, ~le, x, y, t, internal_sorted_least_head(~A, ~le, ~le_trans, y <> t, x, h))), internal_evens_sorted(~A, ~le, ~le_trans, t, internal_sorted_tail(~A, ~le, y, t, internal_sorted_cons_cons_elim_tail(~A, ~le, x, y, t, h))))
223
224def internal_odds_sorted(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, +xs: List<&2, A>, +h: MList.sorted_by(~A, ~le, xs)) -> MList.sorted_by(~A, ~le, MPerm.odds(A, xs)):
225 match xs:
226 case Nil{}:
227 {==}
228 case +x <> +r:
229 match r:
230 case Nil{}:
231 {==}
232 case +y <> +t:
233 internal_sorted_cons_of_least(~A, ~le, y, MPerm.odds(A, t), internal_least_odds(~A, ~le, t, y, internal_sorted_least_head(~A, ~le, ~le_trans, t, y, internal_sorted_cons_cons_elim_tail(~A, ~le, x, y, t, h))), internal_odds_sorted(~A, ~le, ~le_trans, t, internal_sorted_tail(~A, ~le, y, t, internal_sorted_cons_cons_elim_tail(~A, ~le, x, y, t, h))))
234
235# Merge sort returns a sorted list once the fuel reaches the input length (invariant: length <= fuel+1).
236def msort_by_sorted(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, fuel: Nat, +xs: List<&2, A>, +h: {Nat.is_le(List.length(&2, A, xs), 1n+fuel) == True{} : Bool}) -> MList.sorted_by(~A, ~le, MPerm.msort_by(~A, ~le, fuel, xs)):
237 match fuel:
238 case 0n:
239 internal_sorted_short(~A, ~le, xs, h)
240 case 1n++f:
241 match xs:
242 case Nil{}:
243 merge_by_sorted(~A, ~le, ~le_trans, ~le_total, MPerm.msort_by(~A, ~le, f, MPerm.evens(A, xs)), MPerm.msort_by(~A, ~le, f, MPerm.odds(A, xs)), msort_by_sorted(~A, ~le, ~le_trans, ~le_total, f, MPerm.evens(A, xs), MNat.zero_le(1n+f)), msort_by_sorted(~A, ~le, ~le_trans, ~le_total, f, MPerm.odds(A, xs), MNat.zero_le(1n+f)))
244 case +x <> +r:
245 match r:
246 case Nil{}:
247 merge_by_sorted(~A, ~le, ~le_trans, ~le_total, MPerm.msort_by(~A, ~le, f, MPerm.evens(A, x <> r)), MPerm.msort_by(~A, ~le, f, MPerm.odds(A, x <> r)), msort_by_sorted(~A, ~le, ~le_trans, ~le_total, f, MPerm.evens(A, x <> r), MNat.le_add_right(1n, f)), msort_by_sorted(~A, ~le, ~le_trans, ~le_total, f, MPerm.odds(A, x <> r), MNat.zero_le(1n+f)))
248 case +y <> +t:
249 merge_by_sorted(~A, ~le, ~le_trans, ~le_total, MPerm.msort_by(~A, ~le, f, MPerm.evens(A, x <> y <> t)), MPerm.msort_by(~A, ~le, f, MPerm.odds(A, x <> y <> t)), msort_by_sorted(~A, ~le, ~le_trans, ~le_total, f, MPerm.evens(A, x <> y <> t), MNat.succ_le_succ(List.length(&2, A, MPerm.evens(A, t)), f, MNat.le_trans(List.length(&2, A, MPerm.evens(A, t)), List.length(&2, A, t), f, internal_length_evens_le(~A, t), MNat.le_of_succ_le_succ(List.length(&2, A, t), f, MNat.le_of_succ_le_succ(1n+List.length(&2, A, t), 1n+f, h))))), msort_by_sorted(~A, ~le, ~le_trans, ~le_total, f, MPerm.odds(A, x <> y <> t), MNat.succ_le_succ(List.length(&2, A, MPerm.odds(A, t)), f, MNat.le_trans(List.length(&2, A, MPerm.odds(A, t)), List.length(&2, A, t), f, internal_length_odds_le(~A, t), MNat.le_of_succ_le_succ(List.length(&2, A, t), f, MNat.le_of_succ_le_succ(1n+List.length(&2, A, t), 1n+f, h))))))
250
251# Merge sort of the full-length fuel sorts its input.
252def sort_by_sorted(~A: Data, ~le: A -> A -> Bool, ~le_trans: @x: A -> @y: A -> @z: A -> {le(x, y) == True{} : Bool} -> {le(y, z) == True{} : Bool} -> {le(x, z) == True{} : Bool}, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, +xs: List<&2, A>) -> MList.sorted_by(~A, ~le, MPerm.sort_by(~A, ~le, xs)):
253 match xs:
254 case Nil{}:
255 {==}
256 case +x <> +t:
257 msort_by_sorted(~A, ~le, ~le_trans, ~le_total, 1n+List.length(&2, A, t), x <> t, MNat.le_succ(1n+List.length(&2, A, t)))
258
259def msort_by_sorted_nat(+xs: List<&2, Nat>, +h: {Nat.is_le(List.length(&2, Nat, xs), 1n+List.length(&2, Nat, xs)) == True{} : Bool}) -> MList.sorted_by(~Nat, ~Nat.is_le, MPerm.msort_by(~Nat, ~Nat.is_le, List.length(&2, Nat, xs), xs)):
260 msort_by_sorted(~Nat, ~Nat.is_le, ~internal_le_trans_nat, ~internal_le_flip_nat, List.length(&2, Nat, xs), xs, h)
261
262def sort_by_sorted_nat(+xs: List<&2, Nat>) -> MList.sorted_by(~Nat, ~Nat.is_le, MPerm.sort_by(~Nat, ~Nat.is_le, xs)):
263 sort_by_sorted(~Nat, ~Nat.is_le, ~internal_le_trans_nat, ~internal_le_flip_nat, xs)