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
|
// ----- Stream
codatatype Stream = Nil | Cons(head: int, tail: Stream);
function append(M: Stream, N: Stream): Stream
{
match M
case Nil => N
case Cons(t, M') => Cons(t, append(M', N))
}
function zeros(): Stream
{
Cons(0, zeros())
}
function ones(): Stream
{
Cons(1, ones())
}
copredicate atmost(a: Stream, b: Stream)
{
match a
case Nil => true
case Cons(h,t) => b.Cons? && h <= b.head && atmost(t, b.tail)
}
comethod {:induction false} Theorem0()
ensures atmost(zeros(), ones());
{
// the following shows two equivalent ways to getting essentially the
// co-inductive hypothesis
if (*) {
Theorem0#[_k-1]();
} else {
Theorem0();
}
}
ghost method Theorem0_Manual()
ensures atmost(zeros(), ones());
{
parallel (k: nat) {
Theorem0_Lemma(k);
}
}
datatype Natural = Zero | Succ(Natural);
comethod {:induction false} Theorem0_TerminationFailure_ExplicitDecreases(y: Natural)
ensures atmost(zeros(), ones());
decreases y;
{
match (y) {
case Succ(x) =>
// this is just to show that the decreases clause does kick in
Theorem0_TerminationFailure_ExplicitDecreases#[_k](x);
case Zero =>
Theorem0_TerminationFailure_ExplicitDecreases#[_k](y); // error: failure to terminate
}
Theorem0_TerminationFailure_ExplicitDecreases#[_k-1](y);
}
comethod {:induction false} Theorem0_TerminationFailure_DefaultDecreases(y: Natural)
ensures atmost(zeros(), ones());
{
match (y) {
case Succ(yy) =>
// this is just to show that the decreases clause does kick in
Theorem0_TerminationFailure_DefaultDecreases#[_k](yy);
case Zero =>
Theorem0_TerminationFailure_DefaultDecreases#[_k](y); // error: failure to terminate
}
Theorem0_TerminationFailure_DefaultDecreases#[_k-1](y);
}
ghost method {:induction true} Theorem0_Lemma(k: nat)
ensures atmost#[k](zeros(), ones());
{
}
comethod {:induction false} Theorem1()
ensures append(zeros(), ones()) == zeros();
{
Theorem1();
}
codatatype IList = ICons(head: int, tail: IList);
function UpIList(n: int): IList
{
ICons(n, UpIList(n+1))
}
copredicate Pos(s: IList)
{
s.head > 0 && Pos(s.tail)
}
comethod {:induction false} Theorem2(n: int)
requires 1 <= n;
ensures Pos(UpIList(n));
{
Theorem2(n+1);
}
comethod {:induction false} Theorem2_NotAProof(n: int)
requires 1 <= n;
ensures Pos(UpIList(n));
{ // error: this is not a proof (without automatic induction)
}
codatatype TList<T> = TCons(head: T, tail: TList);
function Next<T>(t: T): T
function FF<T>(h: T): TList<T>
{
TCons(h, FF(Next(h)))
}
function GG<T>(h: T): TList<T>
{
TCons(h, GG(Next(h)))
}
comethod {:induction false} Compare<T>(h: T)
ensures FF(h) == GG(h);
{
assert FF(h).head == GG(h).head;
Compare(Next(h));
if {
case true =>
assert FF(h).tail == GG(h).tail; // error: full equality is not known here
case true =>
assert FF(h) ==#[_k] GG(h); // yes, this is the postcondition to be proved, and it is known to hold
case true =>
assert FF(h).tail ==#[_k] GG(h).tail; // error: only _k-1 equality of the tails is known here
case true =>
assert FF(h).tail ==#[_k - 1] GG(h).tail; // yes, follows from call to Compare
case true =>
}
}
comethod {:induction false} BadTheorem(s: IList)
ensures false;
{ // error: postcondition violation
BadTheorem(s.tail);
}
|
