1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
|
/* Lists */
// Here are some standard definitions of List and functions on Lists
datatype List<T> = Nil | Cons(T, List);
function length(l: List): nat
{
match l
case Nil => 0
case Cons(x, xs) => 1 + length(xs)
}
function concat(l: List, ys: List): List
{
match l
case Nil => ys
case Cons(x, xs) => Cons(x, concat(xs, ys))
}
function reverse(l: List): List
{
match l
case Nil => Nil
case Cons(x, xs) => concat(reverse(xs), Cons(x, Nil))
}
function revacc(l: List, acc: List): List
{
match l
case Nil => acc
case Cons(x, xs) => revacc(xs, Cons(x, acc))
}
function qreverse(l: List): List
{
revacc(l, Nil)
}
// Here are two lemmas about the List functions.
ghost method Lemma_ConcatNil()
ensures forall xs :: concat(xs, Nil) == xs;
{
}
ghost method Lemma_RevCatCommute()
ensures forall xs, ys, zs :: revacc(xs, concat(ys, zs)) == concat(revacc(xs, ys), zs);
{
}
// Here is a theorem that says "qreverse" and "reverse" calculate the same result. The proof
// is given in a calculational style. The proof is not minimal--some lines can be omitted
// and Dafny will still fill in the details.
ghost method Theorem_QReverseIsCorrect_Calc(l: List)
ensures qreverse(l) == reverse(l);
{
calc {
qreverse(l);
// def. qreverse
revacc(l, Nil);
{ Lemma_Revacc_calc(l, Nil); }
concat(reverse(l), Nil);
{ Lemma_ConcatNil(); }
reverse(l);
}
}
ghost method Lemma_Revacc_calc(xs: List, ys: List)
ensures revacc(xs, ys) == concat(reverse(xs), ys);
{
match (xs) {
case Nil =>
case Cons(x, xrest) =>
calc {
concat(reverse(xs), ys);
// def. reverse
concat(concat(reverse(xrest), Cons(x, Nil)), ys);
// induction hypothesis: Lemma_Revacc_calc(xrest, Cons(x, Nil))
concat(revacc(xrest, Cons(x, Nil)), ys);
{ Lemma_RevCatCommute(); } // forall xs,ys,zs :: revacc(xs, concat(ys, zs)) == concat(revacc(xs, ys), zs)
revacc(xrest, concat(Cons(x, Nil), ys));
{
assert forall g, gs :: concat(Cons(g, Nil), gs) == Cons(g, gs);
assert concat(Cons(x, Nil), ys) == Cons(x, ys);
}
revacc(xrest, Cons(x, ys));
// def. revacc
revacc(xs, ys);
}
}
}
// Here is a version of the same proof, as it was constructed before Dafny's "calc" construct.
ghost method Theorem_QReverseIsCorrect(l: List)
ensures qreverse(l) == reverse(l);
{
assert qreverse(l)
== // def. qreverse
revacc(l, Nil);
Lemma_Revacc(l, Nil);
assert revacc(l, Nil)
== concat(reverse(l), Nil);
Lemma_ConcatNil();
}
ghost method Lemma_Revacc(xs: List, ys: List)
ensures revacc(xs, ys) == concat(reverse(xs), ys);
{
match (xs) {
case Nil =>
case Cons(x, xrest) =>
assert revacc(xs, ys)
== // def. revacc
revacc(xrest, Cons(x, ys));
assert concat(reverse(xs), ys)
== // def. reverse
concat(concat(reverse(xrest), Cons(x, Nil)), ys)
== // induction hypothesis: Lemma3a(xrest, Cons(x, Nil))
concat(revacc(xrest, Cons(x, Nil)), ys);
Lemma_RevCatCommute(); // forall xs,ys,zs :: revacc(xs, concat(ys, zs)) == concat(revacc(xs, ys), zs)
assert concat(revacc(xrest, Cons(x, Nil)), ys)
== revacc(xrest, concat(Cons(x, Nil), ys));
assert forall g, gs :: concat(Cons(g, Nil), gs) == Cons(g, gs);
assert revacc(xrest, concat(Cons(x, Nil), ys))
== // the assert lemma just above
revacc(xrest, Cons(x, ys));
}
}
/* Fibonacci */
// To further demonstrate what the "calc" construct can do, here are some proofs about the Fibonacci function.
function Fib(n: nat): nat
{
if n < 2 then n else Fib(n - 2) + Fib(n - 1)
}
ghost method Lemma_Fib()
ensures Fib(5) < 6;
{
calc {
Fib(5);
Fib(4) + Fib(3);
calc {
Fib(2);
Fib(0) + Fib(1);
0 + 1;
1;
}
< 6;
}
}
/* List length */
// Here are some proofs that show the use of nested calculations.
ghost method Lemma_Concat_Length(xs: List, ys: List)
ensures length(concat(xs, ys)) == length(xs) + length(ys);
{}
ghost method Lemma_Reverse_Length(xs: List)
ensures length(xs) == length(reverse(xs));
{
match (xs) {
case Nil =>
case Cons(x, xrest) =>
calc {
length(reverse(xs));
// def. reverse
length(concat(reverse(xrest), Cons(x, Nil)));
{ Lemma_Concat_Length(reverse(xrest), Cons(x, Nil)); }
length(reverse(xrest)) + length(Cons(x, Nil));
// induction hypothesis
length(xrest) + length(Cons(x, Nil));
calc {
length(Cons(x, Nil));
// def. length
1 + length(Nil);
// def. length
1 + 0;
1;
}
length(xrest) + 1;
// def. length
length(xs);
}
}
}
ghost method Window(xs: List, ys: List)
ensures length(xs) == length(ys) ==> length(reverse(xs)) == length(reverse(ys));
{
calc {
length(xs) == length(ys) ==> length(reverse(xs)) == length(reverse(ys));
{ if (length(xs) == length(ys)) {
calc {
length(reverse(xs));
{ Lemma_Reverse_Length(xs); }
length(xs);
length(ys);
{ Lemma_Reverse_Length(ys); }
length(reverse(ys));
} } }
length(xs) == length(ys) ==> length(reverse(xs)) == length(reverse(xs));
true;
}
}
|
