-
Notifications
You must be signed in to change notification settings - Fork 10
/
Rule8a.dfy
197 lines (177 loc) · 6.23 KB
/
Rule8a.dfy
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
// does not validate
include "Rule3.dfy"
method InternalAdd(xs: seq<NeIntRange>, range: IntRange) returns (r: seq<NeIntRange>)
requires ValidSeq(xs)
ensures ValidSeq(r)
ensures SeqToSet(r) == SeqToSet(xs) + RangeToSet(range)
{
var (start, end) := range;
if end < start {
r := xs;
assert SeqToSet(r) == SeqToSet(xs) + RangeToSet(range); // case 0 - validates
return;
}
var beforeHi := IndexAtOrBeforePlusOne(xs, start);
if beforeHi > 0 { // does not go at front
var (startBefore, endBefore) := xs[beforeHi-1];
if endBefore+1 < start {
r := InternalAdd2(xs, range);
assert SeqToSet(r) == SeqToSet(xs) + RangeToSet(range); // case 1 - validates
} else if endBefore < end {
r := xs[..beforeHi-1] + [(startBefore, end)] + xs[beforeHi..];
assume exists i: nat :: i < |r| && r[i] == (startBefore,end) && ValidSeq(r[..i+1]) && ValidSeq(r[i+1..]);
r := DeleteExtra(r, (startBefore,end));
assert SeqToSet(r) == SeqToSet(xs) + RangeToSet(range); // case 2 - fails
} else{
r := xs;
assert RangeToSet(xs[beforeHi-1]) >= RangeToSet(range);
assert xs[beforeHi-1] in xs;
SupersetOfPartsLemma(xs, xs[beforeHi-1]);
assert SeqToSet(r) == SeqToSet(xs) + RangeToSet(range); // case 3 - now validates
}
}
else // goes at front
{
r := InternalAdd2(xs, range);
assert SeqToSet(r) == SeqToSet(xs) + RangeToSet(range); // case 4 - validates
}
}
lemma SupersetOfPartsLemma(xs: seq<NeIntRange>, range: NeIntRange)
requires ValidSeq(xs)
requires range in xs
ensures SeqToSet(xs) >= RangeToSet(range)
{
}
method IndexAtOrBeforePlusOne(xs: seq<NeIntRange>, start: int) returns (i: nat)
requires ValidSeq(xs)
ensures i <= |xs|
ensures forall p | p in xs[..i] :: p.0 <= start
ensures forall j: nat | j < i :: xs[j].0 <= start
ensures forall j: nat | i <= j < |xs| :: start < xs[j].0
ensures i > 0 ==> xs[i-1].0 <= start
ensures forall j: nat :: j < |xs| && xs[j].0 == start ==> j + 1 == i
ensures forall j: nat | j < |xs| :: xs[j].0 == start ==> j + 1 == i
{
i := 0;
while i < |xs| && xs[i].0 <= start
{
i := i + 1;
}
}
ghost function RangeToSet(pair: IntRange): set<int>
{
set i {:autotriggers false} | pair.0 <= i <= pair.1 :: i
}
ghost function SeqToSet(sequence: seq<NeIntRange>): set<int>
decreases |sequence|
requires ValidSeq(sequence)
{
if |sequence| == 0 then {}
else if |sequence| == 1 then RangeToSet(sequence[0])
else RangeToSet(sequence[0]) + SeqToSet(sequence[1..])
}
method InternalAdd2(xs: seq<NeIntRange>, internalRange: NeIntRange) returns (r: seq<NeIntRange>)
requires ValidSeq(xs)
requires forall i : nat :: i < |xs| && xs[i].0 < internalRange.0 ==> xs[i].1+1 < internalRange.0
requires forall i: nat :: i < |xs| ==> xs[i].0 != internalRange.0
ensures ValidSeq(r)
ensures SeqToSet(r) == SeqToSet(xs) + RangeToSet(internalRange)
{
var wasThere: bool;
r, wasThere := InsertNe(xs, internalRange);
assert(!wasThere); // in Rust code
r := DeleteExtra(r, internalRange);
}
method DeleteExtra(xs: seq<NeIntRange>, internalRange: IntRange) returns (r: seq<NeIntRange>)
requires forall i:nat,j:nat :: i < j < |xs| ==> xs[i].0 < xs[j].0 // starts are sorted
requires exists i: nat :: i < |xs| && xs[i] == internalRange && ValidSeq(xs[..i+1]) && ValidSeq(xs[i+1..]) // each half is valid
requires forall i:nat,j:nat :: i < j < |xs| && xs[i] != internalRange && xs[j] != internalRange ==> !Touch(xs[i], xs[j]) // only special might touch
ensures ValidSeq(r) // result must be valid
ensures exists i: nat :: i < |xs| && xs[i] == internalRange && SeqToSet(xs[..i+1]) + SeqToSet(xs[i+1..]) == SeqToSet(r) // result must cover same set as the halves
{
var (start, end) := internalRange;
var indexAfter := IndexAtOrAfter(xs, start);
var (startAfter, endAfter) := xs[indexAfter];
var endNew := end;
var deleteList := [];
var indexDel := indexAfter+1;
while indexDel < |xs|
{
var (startDelete, endDelete) := xs[indexDel];
if startDelete <= end + 1 // e.g. touch?
{
endNew := Max(endNew, endDelete);
deleteList := deleteList + [startDelete];
indexDel := indexDel + 1;
}
else
{
break;
}
}
if endNew > end
{
endAfter := endNew;
r := xs[indexAfter := (start, endAfter)];
}
else
{
r := xs;
}
var r2 := DeleteFromList(r, deleteList, indexAfter+1, indexDel);
r := r2;
}
method InsertNe(s: seq<NeIntRange>, pair: NeIntRange) returns (r: seq<NeIntRange>, wasThere: bool)
requires ValidSeq(s)
requires forall i:nat | i < |s| :: s[i].0 < pair.0 ==> s[i].1+1 < pair.0
requires forall i:nat | i < |s| :: s[i].0 != pair.0
ensures SortedMapFlat.Valid(r)
ensures SortedMapFlat.KeyToSet(r) == SortedMapFlat.KeyToSet(s) + {pair.0}
ensures pair in r
ensures forall i:nat | i < |s| && s[i].0 != pair.0 :: s[i] in r
ensures wasThere == (pair.0 in SortedMapFlat.KeyToSet(s))
ensures exists i: nat :: i < |r| && r[i] == pair && ValidSeq(r[..i+1]) && ValidSeq(r[i+1..])
ensures exists i: nat :: i < |r| && r[i] == pair && SeqToSet(r[..i+1]) + SeqToSet(r[i+1..]) == SeqToSet(s) + RangeToSet(pair)
{
var i := IndexAtOrAfter(s, pair.0);
assert i <= |s|;
if i == |s| {
r := s[..i] + [pair] + s[i..];
}
else
{
r := s[..i] + [pair] + s[i..];
}
wasThere := false;
}
method IndexAtOrAfter(xs: seq<NeIntRange>, start: int) returns (i: nat)
requires forall i:nat,j:nat :: i < j < |xs| ==> xs[i].0 < xs[j].0
ensures i <= |xs|
ensures forall p | p in xs[..i] :: p.0 < start
ensures forall j: nat | j < i :: xs[j].0 < start
ensures forall j: nat | i <= j < |xs| :: start <= xs[j].0
{
i := 0;
while i < |xs| && xs[i].0 < start
{
i := i + 1;
}
}
function DeleteFromList(r: seq<NeIntRange>, deleteList: seq<int>,
indexAfterOne: nat, indexDel: nat
) : seq<NeIntRange>
requires 0 < indexAfterOne <= |r|
requires indexAfterOne <= indexDel <= |r|
{
r[..indexAfterOne] + (if indexDel < |r| then r[indexDel..] else [])
}
module SortedMapFlat {
// Check if a sequence of integer pairs is sorted and distinct (by key)
predicate Valid(sorted_seq: seq<(int,int)>) {
forall i:nat, j:nat | i < j < |sorted_seq| :: sorted_seq[i].0 < sorted_seq[j].0
}
function KeyToSet(m: seq<(int,int)>): set<int>
{
set i | i in m :: i.0
}
}