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
|
// RUN: %dafny /compile:0 /dprint:"%t.dprint" "%s" > "%t"
// RUN: %diff "%s.expect" "%t"
datatype List<X> = Nil | Cons(Node<X>, List<X>)
datatype Node<X> = Element(X) | Nary(List<X>)
function FlattenMain<M>(list: List<M>): List<M>
ensures IsFlat(FlattenMain(list));
{
Flatten(list, Nil)
}
function Flatten<X>(list: List<X>, ext: List<X>): List<X>
requires IsFlat(ext);
ensures IsFlat(Flatten(list, ext));
{
match list
case Nil => ext
case Cons(n, rest) =>
match n
case Element(x) => Cons(n, Flatten(rest, ext))
case Nary(nn) => Flatten(nn, Flatten(rest, ext))
}
function IsFlat<F>(list: List<F>): bool
{
match list
case Nil => true
case Cons(n, rest) =>
match n
case Element(x) => IsFlat(rest)
case Nary(nn) => false
}
function ToSeq<X>(list: List<X>): seq<X>
{
match list
case Nil => []
case Cons(n, rest) =>
match n
case Element(x) => [x] + ToSeq(rest)
case Nary(nn) => ToSeq(nn) + ToSeq(rest)
}
lemma Theorem<X>(list: List<X>)
ensures ToSeq(list) == ToSeq(FlattenMain(list));
{
Lemma(list, Nil);
}
lemma Lemma<X>(list: List<X>, ext: List<X>)
requires IsFlat(ext);
ensures ToSeq(list) + ToSeq(ext) == ToSeq(Flatten(list, ext));
{
match (list) {
case Nil =>
case Cons(n, rest) =>
match (n) {
case Element(x) =>
Lemma(rest, ext);
case Nary(nn) =>
Lemma(nn, Flatten(rest, ext));
Lemma(rest, ext);
}
}
}
// ---------------------------------------------
function NegFac(n: int): int
decreases -n;
{
if -1 <= n then -1 else - NegFac(n+1) * n
}
lemma LemmaAll()
ensures forall n :: NegFac(n) <= -1; // error: induction heuristic does not give a useful well-founded order, and thus this fails to verify
{
}
lemma LemmaOne(n: int)
ensures NegFac(n) <= -1; // error: induction heuristic does not give a useful well-founded order, and thus this fails to verify
{
}
lemma LemmaAll_Neg() //FIXME I don't understand the comment below; what trigger?
ensures forall n {:nowarn} :: NegFac(-n) <= -1; // error: fails to verify because of the minus in the trigger
{
}
lemma LemmaOne_Neg(n: int) //FIXME What trigger?
ensures NegFac(-n) <= -1; // error: fails to verify because of the minus in the trigger
{
}
lemma LemmaOneWithDecreases(n: int)
ensures NegFac(n) <= -1; // here, the programmer gives a good well-founded order, so this verifies
decreases -n;
{
}
|
