summaryrefslogtreecommitdiff
path: root/Test/z3api/boog19.bpl
blob: 6e4f47ac5808494c396909cd3a6a52f88dd51e85 (plain)
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
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
type name;
type ref;
var alloc:[int]name;


function Field(int) returns (name);
function Base(int) returns (int);

// Constants
const unique UNALLOCATED:name;
const unique ALLOCATED: name;
const unique FREED:name;

const unique BYTE:name;

function Equal([int]bool, [int]bool) returns (bool);
function Subset([int]bool, [int]bool) returns (bool);
function Disjoint([int]bool, [int]bool) returns (bool);

function Empty() returns ([int]bool);
function Singleton(int) returns ([int]bool);
function Reachable([int,int]bool, int) returns ([int]bool);
function Union([int]bool, [int]bool) returns ([int]bool);
function Intersection([int]bool, [int]bool) returns ([int]bool);
function Difference([int]bool, [int]bool) returns ([int]bool);
function Dereference([int]bool, [int]int) returns ([int]bool);
function Inverse(f:[int]int, x:int) returns ([int]bool);

function AtLeast(int, int) returns ([int]bool);
function Rep(int, int) returns (int);
axiom(forall n:int, x:int, y:int :: {AtLeast(n,x)[y]} AtLeast(n,x)[y] ==> x <= y && Rep(n,x) == Rep(n,y));
axiom(forall n:int, x:int, y:int :: {AtLeast(n,x),Rep(n,x),Rep(n,y)} x <= y && Rep(n,x) == Rep(n,y) ==> AtLeast(n,x)[y]);
axiom(forall n:int, x:int :: {AtLeast(n,x)} AtLeast(n,x)[x]);
axiom(forall n:int, x:int, z:int :: {PLUS(x,n,z)} Rep(n,x) == Rep(n,PLUS(x,n,z)));
axiom(forall n:int, x:int :: {Rep(n,x)} (exists k:int :: Rep(n,x) - x  == n*k));


function Array(int, int, int) returns ([int]bool);
axiom(forall x:int, n:int, z:int :: {Array(x,n,z)} z <= 0 ==> Equal(Array(x,n,z), Empty()));
axiom(forall x:int, n:int, z:int :: {Array(x,n,z)} z > 0 ==> Equal(Array(x,n,z), Difference(AtLeast(n,x),AtLeast(n,PLUS(x,n,z)))));


axiom(forall x:int :: !Empty()[x]);

axiom(forall x:int, y:int :: {Singleton(y)[x]} Singleton(y)[x] <==> x == y);
axiom(forall y:int :: {Singleton(y)} Singleton(y)[y]);

/* this formulation of Union IS more complete than the earlier one */
/* (A U B)[e], A[d], A U B = Singleton(c), d != e */
axiom(forall x:int, S:[int]bool, T:[int]bool :: {Union(S,T)[x]} Union(S,T)[x] <==> S[x] || T[x]);
axiom(forall x:int, S:[int]bool, T:[int]bool :: {Union(S,T), S[x]} S[x] ==> Union(S,T)[x]);
axiom(forall x:int, S:[int]bool, T:[int]bool :: {Union(S,T), T[x]} T[x] ==> Union(S,T)[x]);

axiom(forall x:int, S:[int]bool, T:[int]bool :: {Intersection(S,T)[x]} Intersection(S,T)[x] <==>  S[x] && T[x]);
axiom(forall x:int, S:[int]bool, T:[int]bool :: {Intersection(S,T), S[x]} S[x] && T[x] ==> Intersection(S,T)[x]);
axiom(forall x:int, S:[int]bool, T:[int]bool :: {Intersection(S,T), T[x]} S[x] && T[x] ==> Intersection(S,T)[x]);

axiom(forall x:int, S:[int]bool, T:[int]bool :: {Difference(S,T)[x]} Difference(S,T)[x] <==> S[x] && !T[x]);
axiom(forall x:int, S:[int]bool, T:[int]bool :: {Difference(S,T), S[x]} S[x] ==> Difference(S,T)[x] || T[x]);

axiom(forall x:int, S:[int]bool, M:[int]int :: {Dereference(S,M)[x]} Dereference(S,M)[x] ==> (exists y:int :: x == M[y] && S[y]));
axiom(forall x:int, S:[int]bool, M:[int]int :: {M[x], S[x], Dereference(S,M)} S[x] ==> Dereference(S,M)[M[x]]);
axiom(forall x:int, y:int, S:[int]bool, M:[int]int :: {Dereference(S,M[x := y])} !S[x] ==> Equal(Dereference(S,M[x := y]), Dereference(S,M)));
axiom(forall x:int, y:int, S:[int]bool, M:[int]int :: {Dereference(S,M[x := y])} 
     S[x] &&  Equal(Intersection(Inverse(M,M[x]), S), Singleton(x)) ==> Equal(Dereference(S,M[x := y]), Union(Difference(Dereference(S,M), Singleton(M[x])), Singleton(y))));
axiom(forall x:int, y:int, S:[int]bool, M:[int]int :: {Dereference(S,M[x := y])} 
     S[x] && !Equal(Intersection(Inverse(M,M[x]), S), Singleton(x)) ==> Equal(Dereference(S,M[x := y]), Union(Dereference(S,M), Singleton(y))));

axiom(forall f:[int]int, x:int :: {Inverse(f,f[x])} Inverse(f,f[x])[x]);
axiom(forall f:[int]int, x:int, y:int :: {Inverse(f[x := y],y)} Equal(Inverse(f[x := y],y), Union(Inverse(f,y), Singleton(x))));
axiom(forall f:[int]int, x:int, y:int, z:int :: {Inverse(f[x := y],z)} y == z || Equal(Inverse(f[x := y],z), Difference(Inverse(f,z), Singleton(x))));

axiom(forall S:[int]bool, T:[int]bool :: {Equal(S,T)} Equal(S,T) <==> Subset(S,T) && Subset(T,S));
axiom(forall x:int, S:[int]bool, T:[int]bool :: {S[x], Subset(S,T)} S[x] && Subset(S,T) ==> T[x]);
axiom(forall S:[int]bool, T:[int]bool :: {Subset(S,T)} Subset(S,T) || (exists x:int :: S[x] && !T[x]));
axiom(forall x:int, S:[int]bool, T:[int]bool :: {S[x], Disjoint(S,T), T[x]} !(S[x] && Disjoint(S,T) && T[x]));
axiom(forall S:[int]bool, T:[int]bool :: {Disjoint(S,T)} Disjoint(S,T) || (exists x:int :: S[x] && T[x]));

function Unified([name][int]int) returns ([int]int);
axiom(forall M:[name][int]int, x:int :: {Unified(M)[x]} Unified(M)[x] == M[Field(x)][x]);
axiom(forall M:[name][int]int, x:int, y:int :: {Unified(M[Field(x) := M[Field(x)][x := y]])} Unified(M[Field(x) := M[Field(x)][x := y]]) == Unified(M)[x := y]);
// Memory model

var Mem: [name][int]int;

function Match(a:int, t:name) returns (bool);
function HasType(v:int, t:name) returns (bool);
function Values(t:name) returns ([int]bool);

axiom(forall v:int, t:name :: {Values(t)[v]} Values(t)[v] ==> HasType(v, t));
axiom(forall v:int, t:name :: {HasType(v, t), Values(t)} HasType(v, t) ==> Values(t)[v]);

// Field declarations


// Type declarations

const unique A100INT4_name:name;
const unique INT4_name:name;
const unique PA100INT4_name:name;
const unique PINT4_name:name;
const unique PPINT4_name:name;

// Field definitions

// Type definitions

axiom(forall a:int :: {Match(a, A100INT4_name)} Subset(Empty(), Array(a, 4, 100)));
axiom(forall a:int, e:int :: {Match(a, A100INT4_name), Array(a, 4, 100)[e]}
    Match(a, A100INT4_name) && Array(a, 4, 100)[e] ==> Match(e, INT4_name));

axiom(forall a:int :: {Match(a, INT4_name)}
    Match(a, INT4_name) <==> Field(a) == INT4_name);
axiom(forall v:int :: HasType(v, INT4_name));

axiom(forall a:int :: {Match(a, PA100INT4_name)}
    Match(a, PA100INT4_name) <==> Field(a) == PA100INT4_name);
axiom(forall v:int :: {HasType(v, PA100INT4_name)} {Match(v, A100INT4_name)}
    HasType(v, PA100INT4_name) <==> (v == 0 || (v > 0 && Match(v, A100INT4_name))));

axiom(forall a:int :: {Match(a, PINT4_name)}
    Match(a, PINT4_name) <==> Field(a) == PINT4_name);
axiom(forall v:int :: {HasType(v, PINT4_name)} {Match(v, INT4_name)}
    HasType(v, PINT4_name) <==> (v == 0 || (v > 0 && Match(v, INT4_name))));

axiom(forall a:int :: {Match(a, PPINT4_name)}
    Match(a, PPINT4_name) <==> Field(a) == PPINT4_name);
axiom(forall v:int :: {HasType(v, PPINT4_name)} {Match(v, PINT4_name)}
    HasType(v, PPINT4_name) <==> (v == 0 || (v > 0 && Match(v, PINT4_name))));

function MINUS_BOTH_PTR_OR_BOTH_INT(a:int, b:int, size:int) returns (int); 
axiom(forall a:int, b:int, size:int :: {MINUS_BOTH_PTR_OR_BOTH_INT(a,b,size)} 
size * MINUS_BOTH_PTR_OR_BOTH_INT(a,b,size) <= a - b && a - b < size * (MINUS_BOTH_PTR_OR_BOTH_INT(a,b,size) + 1));

function MINUS_LEFT_PTR(a:int, a_size:int, b:int) returns (int);
axiom(forall a:int, a_size:int, b:int :: {MINUS_LEFT_PTR(a,a_size,b)} MINUS_LEFT_PTR(a,a_size,b) == a - a_size * b);

function PLUS(a:int, a_size:int, b:int) returns (int);
axiom(forall a:int, a_size:int, b:int :: {PLUS(a,a_size,b)} PLUS(a,a_size,b) == a + a_size * b);

function MULT(a:int, b:int) returns (int); // a*b
axiom(forall a:int, b:int :: {MULT(a,b)} MULT(a,b) == a * b);

function DIV(a:int, b:int) returns (int); // a/b	
	      
axiom(forall a:int, b:int :: {DIV(a,b)}
a >= 0 && b > 0 ==> b * DIV(a,b) <= a && a < b * (DIV(a,b) + 1)
); 

axiom(forall a:int, b:int :: {DIV(a,b)}
a >= 0 && b < 0 ==> b * DIV(a,b) <= a && a < b * (DIV(a,b) - 1)
); 

axiom(forall a:int, b:int :: {DIV(a,b)}
a < 0 && b > 0 ==> b * DIV(a,b) >= a && a > b * (DIV(a,b) - 1)
); 

axiom(forall a:int, b:int :: {DIV(a,b)}
a < 0 && b < 0 ==> b * DIV(a,b) >= a && a > b * (DIV(a,b) + 1)
); 

function BINARY_BOTH_INT(a:int, b:int) returns (int);
/*
function POW2(a:int) returns (bool);
axiom POW2(1);
axiom POW2(2);
axiom POW2(4);
axiom POW2(8);
axiom POW2(16);
axiom POW2(32);
axiom POW2(64);
axiom POW2(128);
axiom POW2(256);
axiom POW2(512);
axiom POW2(1024);
axiom POW2(2048);
axiom POW2(4096);
axiom POW2(8192);
axiom POW2(16384);
axiom POW2(32768);
axiom POW2(65536);
axiom POW2(131072);
axiom POW2(262144);
axiom POW2(524288);
axiom POW2(1048576);
axiom POW2(2097152);
axiom POW2(4194304);
axiom POW2(8388608);
axiom POW2(16777216);
axiom POW2(33554432);
*/
function LIFT(a:bool) returns (int);
axiom(forall a:bool :: {LIFT(a)} a <==> LIFT(a) != 0);

function NOT(a:int) returns (int);
axiom(forall a:int :: {NOT(a)} a == 0 ==> NOT(a) != 0);
axiom(forall a:int :: {NOT(a)} a != 0 ==> NOT(a) == 0);

function NULL_CHECK(a:int) returns (int);
axiom(forall a:int :: {NULL_CHECK(a)} a == 0 ==> NULL_CHECK(a) != 0);
axiom(forall a:int :: {NULL_CHECK(a)} a != 0 ==> NULL_CHECK(a) == 0);

const unique g : int;
axiom(g != 0);


procedure  main ( ) returns ($result.main$11.5$1$:int)

//TAG: requires __objectOf(g) != 0
requires(Base(g) != 0);

//TAG: requires __allocated(g)
requires(alloc[Base(g)] == ALLOCATED);

//TAG: requires __allocated(g + 55)
requires(alloc[Base(PLUS(g, 4, 55))] == ALLOCATED);

//TAG: Type Safety Precondition
requires(forall a:int :: {Mem[Field(a)][a]} HasType(Mem[Field(a)][a], Field(a)));
requires(HasType(g, PA100INT4_name));

{
var p : int;

assume(HasType(p, PINT4_name));
p := PLUS(g, 4, 55) ;
assert(HasType(p, PINT4_name));

}