test_ID
stringlengths 3
3
| test_file
stringlengths 14
119
| ground_truth
stringlengths 70
28.7k
| hints_removed
stringlengths 58
28.7k
|
|---|---|---|---|
000 | 630-dafny_tmp_tmpz2kokaiq_Solution.dfy |
function sorted(a: array<int>) : bool
reads a
{
forall i,j : int :: 0 <= i < j < a.Length ==> a[i] <= a[j]
}
method BinarySearch(a: array<int>, x: int) returns (index: int)
requires sorted(a)
ensures 0 <= index < a.Length ==> a[index] == x
ensures index == -1 ==> forall i : int :: 0 <= i < a.Length ==> a[i] != x
{
var low := 0;
var high := a.Length - 1;
var mid := 0;
while (low <= high)
invariant 0 <= low <= high + 1 <= a.Length
invariant x !in a[..low] && x !in a[high + 1..]
{
mid := (high + low) / 2;
if a[mid] < x {
low := mid + 1;
}
else if a[mid] > x {
high := mid - 1;
}
else {
return mid;
}
}
return -1;
}
|
function sorted(a: array<int>) : bool
reads a
{
forall i,j : int :: 0 <= i < j < a.Length ==> a[i] <= a[j]
}
method BinarySearch(a: array<int>, x: int) returns (index: int)
requires sorted(a)
ensures 0 <= index < a.Length ==> a[index] == x
ensures index == -1 ==> forall i : int :: 0 <= i < a.Length ==> a[i] != x
{
var low := 0;
var high := a.Length - 1;
var mid := 0;
while (low <= high)
{
mid := (high + low) / 2;
if a[mid] < x {
low := mid + 1;
}
else if a[mid] > x {
high := mid - 1;
}
else {
return mid;
}
}
return -1;
}
|
001 | 703FinalProject_tmp_tmpr_10rn4z_DP-GD.dfy | method DPGD_GradientPerturbation (size:int, learning_rate:real, noise_scale:real, gradient_norm_bound:real, iterations:int) returns (Para:real, PrivacyLost:real)
requires iterations>=0
requires size>=0
requires noise_scale >= 1.0
requires -1.0 <= gradient_norm_bound <= 1.0
{
var thetha:array<real> := new real[iterations+1];
thetha[0] := *;
var alpha:real := 0.0;
var tau:real := *;
assume(tau>=0.0);
var t :int := 0;
var constant:real := (size as real) * tau;
while (t < iterations)
invariant t <= iterations
invariant alpha == t as real * constant
{
var i :int := 0;
var beta:real := 0.0;
var summation_gradient:real := 0.0;
while (i< size)
invariant i <= size
invariant beta == i as real * tau
{
var gradient:real := *;
// Note: We do not need to clip the value of the gradient.
// Instead, we clip the sensitivity of the gradient by the gradient_norm_bound provided by the user
var eta:real := *;
beta := beta + tau;
var eta_hat:real := - gradient_norm_bound;
assert (gradient_norm_bound + eta_hat == 0.0);
summation_gradient := summation_gradient + gradient + eta;
i := i + 1;
}
alpha := alpha + beta;
thetha[t+1] := thetha[t] - learning_rate*summation_gradient;
t := t+1;
}
assert(t==iterations);
assert(alpha == iterations as real * constant);
Para := thetha[iterations];
PrivacyLost := alpha;
}
| method DPGD_GradientPerturbation (size:int, learning_rate:real, noise_scale:real, gradient_norm_bound:real, iterations:int) returns (Para:real, PrivacyLost:real)
requires iterations>=0
requires size>=0
requires noise_scale >= 1.0
requires -1.0 <= gradient_norm_bound <= 1.0
{
var thetha:array<real> := new real[iterations+1];
thetha[0] := *;
var alpha:real := 0.0;
var tau:real := *;
assume(tau>=0.0);
var t :int := 0;
var constant:real := (size as real) * tau;
while (t < iterations)
{
var i :int := 0;
var beta:real := 0.0;
var summation_gradient:real := 0.0;
while (i< size)
{
var gradient:real := *;
// Note: We do not need to clip the value of the gradient.
// Instead, we clip the sensitivity of the gradient by the gradient_norm_bound provided by the user
var eta:real := *;
beta := beta + tau;
var eta_hat:real := - gradient_norm_bound;
summation_gradient := summation_gradient + gradient + eta;
i := i + 1;
}
alpha := alpha + beta;
thetha[t+1] := thetha[t] - learning_rate*summation_gradient;
t := t+1;
}
Para := thetha[iterations];
PrivacyLost := alpha;
}
|
002 | 703FinalProject_tmp_tmpr_10rn4z_gaussian.dfy | // VERIFY USING DAFNY:
// /Applications/dafny/dafny /Users/apple/GaussianDP/Dafny/gaussian.dfy
method gaussian (size:int, q: array<real>, q_hat: array<real>) returns (out: array<real>)
requires q_hat.Length==size
requires q.Length==size
requires size > 0
requires arraySquaredSum(q_hat[..]) <= 1.0
{
var i : int := 0;
var alpha : real := arraySquaredSum(q_hat[..1]);
var eta: real := 0.0;
var eta_hat: real := 0.0;
out := new real[size];
while (i <size)
invariant 0 < i <= size ==> alpha <= arraySquaredSum(q_hat[..i])
invariant i<=size
{
eta := *;
eta_hat := - q_hat[i];
alpha := arraySquaredSum(q_hat[..i+1]);
assert (q_hat[i] + eta_hat ==0.0);
out[i] := q[i] + eta;
i := i+1;
}
assert i==size;
assert alpha <= arraySquaredSum(q_hat[..size]);
assert q_hat[..size] == q_hat[..];
assert alpha <= arraySquaredSum(q_hat[..]);
assert alpha <= 1.0;
}
function arraySquaredSum(a: seq<real>): real
requires |a| > 0
{
if |a| == 1 then
a[0]*a[0]
else
(a[0]*a[0]) + arraySquaredSum(a[1..])
}
| // VERIFY USING DAFNY:
// /Applications/dafny/dafny /Users/apple/GaussianDP/Dafny/gaussian.dfy
method gaussian (size:int, q: array<real>, q_hat: array<real>) returns (out: array<real>)
requires q_hat.Length==size
requires q.Length==size
requires size > 0
requires arraySquaredSum(q_hat[..]) <= 1.0
{
var i : int := 0;
var alpha : real := arraySquaredSum(q_hat[..1]);
var eta: real := 0.0;
var eta_hat: real := 0.0;
out := new real[size];
while (i <size)
{
eta := *;
eta_hat := - q_hat[i];
alpha := arraySquaredSum(q_hat[..i+1]);
out[i] := q[i] + eta;
i := i+1;
}
}
function arraySquaredSum(a: seq<real>): real
requires |a| > 0
{
if |a| == 1 then
a[0]*a[0]
else
(a[0]*a[0]) + arraySquaredSum(a[1..])
}
|
003 | AssertivePrograming_tmp_tmpwf43uz0e_DivMode_Unary.dfy | // Noa Leron 207131871
// Tsuri Farhana 315016907
// definitions borrowed from Rustan Leino's Program Proofs Chapter 7
// (https://program-proofs.com/code.html example code in Dafny; source file 7-Unary.dfy)
datatype Unary = Zero | Suc(pred: Unary)
ghost function UnaryToNat(x: Unary): nat {
match x
case Zero => 0
case Suc(x') => 1 + UnaryToNat(x')
}
ghost function NatToUnary(n: nat): Unary {
if n == 0 then Zero else Suc(NatToUnary(n-1))
}
lemma NatUnaryCorrespondence(n: nat, x: Unary)
ensures UnaryToNat(NatToUnary(n)) == n
ensures NatToUnary(UnaryToNat(x)) == x
{
}
predicate Less(x: Unary, y: Unary) {
y != Zero && (x.Suc? ==> Less(x.pred, y.pred))
}
predicate LessAlt(x: Unary, y: Unary) {
y != Zero && (x == Zero || Less(x.pred, y.pred))
}
lemma LessSame(x: Unary, y: Unary)
ensures Less(x, y) == LessAlt(x, y)
{
}
lemma LessCorrect(x: Unary, y: Unary)
ensures Less(x, y) <==> UnaryToNat(x) < UnaryToNat(y)
{
}
lemma LessTransitive(x: Unary, y: Unary, z: Unary)
requires Less(x, y) && Less(y, z)
ensures Less(x, z)
{
}
function Add(x: Unary, y: Unary): Unary {
match y
case Zero => x
case Suc(y') => Suc(Add(x, y'))
}
lemma {:induction false} SucAdd(x: Unary, y: Unary)
ensures Suc(Add(x, y)) == Add(Suc(x), y)
{
match y
case Zero =>
case Suc(y') =>
calc {
Suc(Add(x, Suc(y')));
== // def. Add
Suc(Suc(Add(x, y')));
== { SucAdd(x, y'); }
Suc(Add(Suc(x), y'));
== // def. Add
Add(Suc(x), Suc(y'));
}
}
lemma {:induction false} AddZero(x: Unary)
ensures Add(Zero, x) == x
{
match x
case Zero =>
case Suc(x') =>
calc {
Add(Zero, Suc(x'));
== // def. Add
Suc(Add(Zero, x'));
== { AddZero(x'); }
Suc(x');
}
}
function Sub(x: Unary, y: Unary): Unary
requires !Less(x, y)
{
match y
case Zero => x
case Suc(y') => Sub(x.pred, y')
}
function Mul(x: Unary, y: Unary): Unary {
match x
case Zero => Zero
case Suc(x') => Add(Mul(x', y), y)
}
lemma SubStructurallySmaller(x: Unary, y: Unary)
requires !Less(x, y) && y != Zero
ensures Sub(x, y) < x
{
}
lemma AddSub(x: Unary, y: Unary)
requires !Less(x, y)
ensures Add(Sub(x, y), y) == x
{
}
/*
Goal: implement correcly and clearly, using iterative code (no recursion), documenting the proof obligations
as we've learned, with assertions and a lemma for each proof goal
- DO NOT modify the specification or any of the definitions given in this file
- Not all definitions above are relevant, some are simply included as examples
- Feel free to use existing non-ghost functions/predicates in your code, and existing lemmas (for the proof) in your annotations
- New functions/predicates may be added ONLY as ghost
- If it helps you in any way, a recursive implementation + proof can be found in the book and the downloadable source file
[https://program-proofs.com/code.html example code in Dafny, source file 7-Unary.dfy]
*/
method{:verify false} IterativeDivMod'(x: Unary, y: Unary) returns (d: Unary, m: Unary)
requires y != Zero
ensures Add(Mul(d, y), m) == x && Less(m, y)
{
if (Less(x, y)) {
d := Zero;
m := x;
}
else{
var x0: Unary := x;
d := Zero;
while (!Less(x0, y))
invariant Add(Mul(d, y), x0) == x
decreases x0
{
d := Suc(d);
x0 := Sub(x0, y);
}
m := x0;
}
}
method IterativeDivMod(x: Unary, y: Unary) returns (d: Unary, m: Unary)
requires y != Zero
ensures Add(Mul(d, y), m) == x && Less(m, y)
{
if (Less(x, y)) {
assert Less(x, y);
AddZero(x);
assert Add(Zero, x) == x;
assert Mul(Zero, y) == Zero;
assert Add(Mul(Zero, y), x) == x;
d := Zero;
m := x;
assert Add(Mul(d, y), m) == m;
assert Less(m, y);
assert Add(Mul(d, y), m) == x && Less(m, y);
}
else{
assert !Less(x, y);
assert y != Zero;
var x0: Unary := x;
assert Mul(Zero, y) == Zero;
d := Zero;
assert Mul(d, y) == Zero;
AddZero(x);
assert Add(Zero, x) == x;
assert Add(Mul(d, y), x) == x;
assert Add(Mul(d, y), x0) == x;
while (!Less(x0, y))
invariant Add(Mul(d, y), x0) == x
decreases x0
{
assert Add(Mul(d, y), x0) == x;
assert !Less(x0, y);
assert y != Zero;
AddMulSucSubEqAddMul(d, y , x0);
assert Add(Mul(Suc(d), y), Sub(x0, y)) == Add(Mul(d, y), x0);
assert Add(Mul(Suc(d), y), Sub(x0, y)) == x;
d := Suc(d);
assert !Less(x0, y) && y != Zero;
SubStructurallySmaller(x0, y);
assert Sub(x0, y) < x0; // decreases
x0 := Sub(x0, y);
assert Add(Mul(d, y), x0) == x;
}
assert Add(Mul(d, y), x0) == x;
m := x0;
assert Add(Mul(d, y), m) == x;
}
assert Add(Mul(d, y), m) == x;
}
lemma AddMulEqMulSuc(a: Unary, b: Unary)
ensures Mul(Suc(a), b) == Add(Mul(a, b), b)
{
calc{
Mul(Suc(a), b);
== // def. Mul
Add(Mul(a, b), b);
}
}
lemma AddMulSucSubEqAddMul(d: Unary, y: Unary, x0: Unary)
requires !Less(x0, y)
requires y != Zero
ensures Add(Mul(Suc(d), y), Sub(x0, y)) == Add(Mul(d, y), x0)
{
calc{
Add(Mul(Suc(d), y), Sub(x0, y));
== {AddMulEqMulSuc(d, y);
assert Mul(Suc(d), y) == Add(Mul(d, y), y);}
Add(Add(Mul(d, y), y), Sub(x0, y));
== {AddTransitive(Mul(d, y), y, Sub(x0, y));
assert Add(Mul(d, y), Add(y, Sub(x0, y))) == Add(Add(Mul(d, y), y), Sub(x0, y));}
Add(Mul(d, y), Add(y, Sub(x0, y)));
== {AddCommutative(Sub(x0, y), y);
assert Add(Sub(x0, y), y) == Add(y, Sub(x0, y));}
Add(Mul(d, y), Add(Sub(x0, y), y));
== {assert !Less(x0, y);
AddSub(x0, y);
assert Add(Sub(x0, y), y) == x0;}
Add(Mul(d, y), x0);
}
}
lemma AddTransitive(a: Unary, b: Unary, c: Unary)
ensures Add(a, Add(b, c)) == Add(Add(a, b), c)
{//These assertions are only for documanting the proof obligations
match c
case Zero =>
calc{
Add(a, Add(b, c));
==
Add(a, Add(b, Zero));
== // def. Add
Add(a, b);
== // def. Add
Add(Add(a,b), Zero);
==
Add(Add(a,b), c);
}
case Suc(c') =>
match b
case Zero =>
calc{
Add(a, Add(b, c));
==
Add(a, Add(Zero, Suc(c')));
== {AddZero(Suc(c'));
assert Add(Zero, Suc(c')) == Suc(c');}
Add(a, Suc(c'));
== // def. Add
Add(Add(a, Zero), Suc(c'));
==
Add(Add(a, b), Suc(c'));
==
Add(Add(a,b), c);
}
case Suc(b') =>
match a
case Zero =>
calc{
Add(a, Add(b, c));
==
Add(Zero, Add(Suc(b'), Suc(c')));
== {AddZero(Add(Suc(b'), Suc(c')));
assert Add(Zero, Add(Suc(b'), Suc(c'))) == Add(Suc(b'), Suc(c'));}
Add(Suc(b'), Suc(c'));
== {AddZero(Suc(b'));
assert Add(Zero , Suc(b')) == Suc(b');}
Add(Add(Zero, Suc(b')), Suc(c'));
==
Add(Add(a, b), c);
}
case Suc(a') =>
calc{
Add(a, Add(b, c));
==
Add(Suc(a'), Add(Suc(b'), Suc(c')));
== // def. Add
Add(Suc(a'), Suc(Add(Suc(b'), c')));
== // def. Add
Suc(Add(Suc(a'), Add(Suc(b'), c')));
== {SucAdd(a', Add(Suc(b'), c'));
assert Suc(Add(a', Add(Suc(b'), c'))) == Add(Suc(a'), Add(Suc(b'), c'));}
Suc(Suc(Add(a', Add(Suc(b'), c'))));
== {SucAdd(b', c');
assert Suc(Add(b', c')) == Add(Suc(b'), c');}
Suc(Suc(Add(a', Suc(Add(b', c')))));
== // def. Add
Suc(Suc(Suc(Add(a', Add(b', c')))));
== {AddTransitive(a', b', c');
assert Add(a', Add(b',c')) == Add(Add(a',b'),c');}
Suc(Suc(Suc(Add(Add(a',b'), c'))));
== // def. Add
Suc(Suc(Add(Add(a', b'), Suc(c'))));
== {SucAdd(Add(a', b'), Suc(c'));
assert Suc(Add(Add(a', b'), Suc(c'))) == Add(Suc(Add(a', b')), Suc(c'));}
Suc(Add(Suc(Add(a', b')), Suc(c')));
== {SucAdd(a', b');
assert Suc(Add(a', b')) == Add(Suc(a'), b');}
Suc(Add(Add(Suc(a'), b'), Suc(c')));
== {SucAdd(Add(Suc(a'), b'), Suc(c'));
assert Suc(Add(Add(Suc(a'), b'), Suc(c'))) == Add(Suc(Add(Suc(a'), b')), Suc(c'));}
Add(Suc(Add(Suc(a'), b')), Suc(c'));
== // def. Add
Add(Add(Suc(a'), Suc(b')), Suc(c'));
==
Add(Add(a,b), c);
}
}
lemma AddCommutative(a: Unary, b: Unary)
ensures Add(a, b) == Add(b, a)
{
match b
case Zero =>
calc{
Add(a, b);
==
Add(a, Zero);
== // def. Add
a;
== {AddZero(a);
assert Add(Zero, a) == a;}
Add(Zero, a);
==
Add(b, a);
}
case Suc(b') =>
calc{
Add(a, b);
==
Add(a, Suc(b'));
== // def. Add
Suc(Add(a, b'));
== {AddCommutative(a, b');
assert Add(a, b') == Add(b', a);}
Suc(Add(b', a));
== {SucAdd(b', a);
assert Suc(Add(b',a)) == Add(Suc(b'),a);}
Add(Suc(b'), a);
==
Add(b, a);
}
}
method Main() {
var U3 := Suc(Suc(Suc(Zero)));
assert UnaryToNat(U3) == 3;
var U7 := Suc(Suc(Suc(Suc(U3))));
assert UnaryToNat(U7) == 7;
var d, m := IterativeDivMod(U7, U3);
assert Add(Mul(d, U3), m) == U7 && Less(m, U3);
print "Just as 7 divided by 3 is 2 with a remainder of 1, IterativeDivMod(", U7, ", ", U3, ") is ", d, " with a remainder of ", m;
}
| // Noa Leron 207131871
// Tsuri Farhana 315016907
// definitions borrowed from Rustan Leino's Program Proofs Chapter 7
// (https://program-proofs.com/code.html example code in Dafny; source file 7-Unary.dfy)
datatype Unary = Zero | Suc(pred: Unary)
ghost function UnaryToNat(x: Unary): nat {
match x
case Zero => 0
case Suc(x') => 1 + UnaryToNat(x')
}
ghost function NatToUnary(n: nat): Unary {
if n == 0 then Zero else Suc(NatToUnary(n-1))
}
lemma NatUnaryCorrespondence(n: nat, x: Unary)
ensures UnaryToNat(NatToUnary(n)) == n
ensures NatToUnary(UnaryToNat(x)) == x
{
}
predicate Less(x: Unary, y: Unary) {
y != Zero && (x.Suc? ==> Less(x.pred, y.pred))
}
predicate LessAlt(x: Unary, y: Unary) {
y != Zero && (x == Zero || Less(x.pred, y.pred))
}
lemma LessSame(x: Unary, y: Unary)
ensures Less(x, y) == LessAlt(x, y)
{
}
lemma LessCorrect(x: Unary, y: Unary)
ensures Less(x, y) <==> UnaryToNat(x) < UnaryToNat(y)
{
}
lemma LessTransitive(x: Unary, y: Unary, z: Unary)
requires Less(x, y) && Less(y, z)
ensures Less(x, z)
{
}
function Add(x: Unary, y: Unary): Unary {
match y
case Zero => x
case Suc(y') => Suc(Add(x, y'))
}
lemma {:induction false} SucAdd(x: Unary, y: Unary)
ensures Suc(Add(x, y)) == Add(Suc(x), y)
{
match y
case Zero =>
case Suc(y') =>
calc {
Suc(Add(x, Suc(y')));
== // def. Add
Suc(Suc(Add(x, y')));
== { SucAdd(x, y'); }
Suc(Add(Suc(x), y'));
== // def. Add
Add(Suc(x), Suc(y'));
}
}
lemma {:induction false} AddZero(x: Unary)
ensures Add(Zero, x) == x
{
match x
case Zero =>
case Suc(x') =>
calc {
Add(Zero, Suc(x'));
== // def. Add
Suc(Add(Zero, x'));
== { AddZero(x'); }
Suc(x');
}
}
function Sub(x: Unary, y: Unary): Unary
requires !Less(x, y)
{
match y
case Zero => x
case Suc(y') => Sub(x.pred, y')
}
function Mul(x: Unary, y: Unary): Unary {
match x
case Zero => Zero
case Suc(x') => Add(Mul(x', y), y)
}
lemma SubStructurallySmaller(x: Unary, y: Unary)
requires !Less(x, y) && y != Zero
ensures Sub(x, y) < x
{
}
lemma AddSub(x: Unary, y: Unary)
requires !Less(x, y)
ensures Add(Sub(x, y), y) == x
{
}
/*
Goal: implement correcly and clearly, using iterative code (no recursion), documenting the proof obligations
as we've learned, with assertions and a lemma for each proof goal
- DO NOT modify the specification or any of the definitions given in this file
- Not all definitions above are relevant, some are simply included as examples
- Feel free to use existing non-ghost functions/predicates in your code, and existing lemmas (for the proof) in your annotations
- New functions/predicates may be added ONLY as ghost
- If it helps you in any way, a recursive implementation + proof can be found in the book and the downloadable source file
[https://program-proofs.com/code.html example code in Dafny, source file 7-Unary.dfy]
*/
method{:verify false} IterativeDivMod'(x: Unary, y: Unary) returns (d: Unary, m: Unary)
requires y != Zero
ensures Add(Mul(d, y), m) == x && Less(m, y)
{
if (Less(x, y)) {
d := Zero;
m := x;
}
else{
var x0: Unary := x;
d := Zero;
while (!Less(x0, y))
{
d := Suc(d);
x0 := Sub(x0, y);
}
m := x0;
}
}
method IterativeDivMod(x: Unary, y: Unary) returns (d: Unary, m: Unary)
requires y != Zero
ensures Add(Mul(d, y), m) == x && Less(m, y)
{
if (Less(x, y)) {
AddZero(x);
d := Zero;
m := x;
}
else{
var x0: Unary := x;
d := Zero;
AddZero(x);
while (!Less(x0, y))
{
AddMulSucSubEqAddMul(d, y , x0);
d := Suc(d);
SubStructurallySmaller(x0, y);
x0 := Sub(x0, y);
}
m := x0;
}
}
lemma AddMulEqMulSuc(a: Unary, b: Unary)
ensures Mul(Suc(a), b) == Add(Mul(a, b), b)
{
calc{
Mul(Suc(a), b);
== // def. Mul
Add(Mul(a, b), b);
}
}
lemma AddMulSucSubEqAddMul(d: Unary, y: Unary, x0: Unary)
requires !Less(x0, y)
requires y != Zero
ensures Add(Mul(Suc(d), y), Sub(x0, y)) == Add(Mul(d, y), x0)
{
calc{
Add(Mul(Suc(d), y), Sub(x0, y));
== {AddMulEqMulSuc(d, y);
Add(Add(Mul(d, y), y), Sub(x0, y));
== {AddTransitive(Mul(d, y), y, Sub(x0, y));
Add(Mul(d, y), Add(y, Sub(x0, y)));
== {AddCommutative(Sub(x0, y), y);
Add(Mul(d, y), Add(Sub(x0, y), y));
== {assert !Less(x0, y);
AddSub(x0, y);
Add(Mul(d, y), x0);
}
}
lemma AddTransitive(a: Unary, b: Unary, c: Unary)
ensures Add(a, Add(b, c)) == Add(Add(a, b), c)
{//These assertions are only for documanting the proof obligations
match c
case Zero =>
calc{
Add(a, Add(b, c));
==
Add(a, Add(b, Zero));
== // def. Add
Add(a, b);
== // def. Add
Add(Add(a,b), Zero);
==
Add(Add(a,b), c);
}
case Suc(c') =>
match b
case Zero =>
calc{
Add(a, Add(b, c));
==
Add(a, Add(Zero, Suc(c')));
== {AddZero(Suc(c'));
Add(a, Suc(c'));
== // def. Add
Add(Add(a, Zero), Suc(c'));
==
Add(Add(a, b), Suc(c'));
==
Add(Add(a,b), c);
}
case Suc(b') =>
match a
case Zero =>
calc{
Add(a, Add(b, c));
==
Add(Zero, Add(Suc(b'), Suc(c')));
== {AddZero(Add(Suc(b'), Suc(c')));
Add(Suc(b'), Suc(c'));
== {AddZero(Suc(b'));
Add(Add(Zero, Suc(b')), Suc(c'));
==
Add(Add(a, b), c);
}
case Suc(a') =>
calc{
Add(a, Add(b, c));
==
Add(Suc(a'), Add(Suc(b'), Suc(c')));
== // def. Add
Add(Suc(a'), Suc(Add(Suc(b'), c')));
== // def. Add
Suc(Add(Suc(a'), Add(Suc(b'), c')));
== {SucAdd(a', Add(Suc(b'), c'));
Suc(Suc(Add(a', Add(Suc(b'), c'))));
== {SucAdd(b', c');
Suc(Suc(Add(a', Suc(Add(b', c')))));
== // def. Add
Suc(Suc(Suc(Add(a', Add(b', c')))));
== {AddTransitive(a', b', c');
Suc(Suc(Suc(Add(Add(a',b'), c'))));
== // def. Add
Suc(Suc(Add(Add(a', b'), Suc(c'))));
== {SucAdd(Add(a', b'), Suc(c'));
Suc(Add(Suc(Add(a', b')), Suc(c')));
== {SucAdd(a', b');
Suc(Add(Add(Suc(a'), b'), Suc(c')));
== {SucAdd(Add(Suc(a'), b'), Suc(c'));
Add(Suc(Add(Suc(a'), b')), Suc(c'));
== // def. Add
Add(Add(Suc(a'), Suc(b')), Suc(c'));
==
Add(Add(a,b), c);
}
}
lemma AddCommutative(a: Unary, b: Unary)
ensures Add(a, b) == Add(b, a)
{
match b
case Zero =>
calc{
Add(a, b);
==
Add(a, Zero);
== // def. Add
a;
== {AddZero(a);
Add(Zero, a);
==
Add(b, a);
}
case Suc(b') =>
calc{
Add(a, b);
==
Add(a, Suc(b'));
== // def. Add
Suc(Add(a, b'));
== {AddCommutative(a, b');
Suc(Add(b', a));
== {SucAdd(b', a);
Add(Suc(b'), a);
==
Add(b, a);
}
}
method Main() {
var U3 := Suc(Suc(Suc(Zero)));
var U7 := Suc(Suc(Suc(Suc(U3))));
var d, m := IterativeDivMod(U7, U3);
print "Just as 7 divided by 3 is 2 with a remainder of 1, IterativeDivMod(", U7, ", ", U3, ") is ", d, " with a remainder of ", m;
}
|
004 | AssertivePrograming_tmp_tmpwf43uz0e_Find_Substring.dfy | // Noa Leron 207131871
// Tsuri Farhana 315016907
ghost predicate ExistsSubstring(str1: string, str2: string) {
// string in Dafny is a sequence of characters (seq<char>) and <= on sequences is the prefix relation
exists offset :: 0 <= offset <= |str1| && str2 <= str1[offset..]
}
ghost predicate Post(str1: string, str2: string, found: bool, i: nat) {
(found <==> ExistsSubstring(str1, str2)) &&
(found ==> i + |str2| <= |str1| && str2 <= str1[i..])
}
/*
Goal: Verify correctness of the following code. Once done, remove the {:verify false} (or turn it into {:verify true}).
Feel free to add GHOST code, including calls to lemmas. But DO NOT modify the specification or the original (executable) code.
*/
method {:verify true} FindFirstOccurrence(str1: string, str2: string) returns (found: bool, i: nat)
ensures Post(str1, str2, found, i)
{
if |str2| == 0 {
found, i := true, 0;
assert Post(str1, str2, found, i); // this case is easy for dafny :)
}
else if |str1| < |str2| {
found, i := false, 0; // value of i irrelevant in this case
assert Post(str1, str2, found, i); // this case is easy for dafny :)
}
else {
found, i := false, |str2|-1;
assert |str2| > 0;
assert |str1| >= |str2|;
assert Outter_Inv_correctness(str1, str2, false, |str2|-1);
while !found && i < |str1|
invariant Outter_Inv_correctness(str1, str2, found, i);
decreases if !found then 1 else 0, |str1| - i;
{
assert Outter_Inv_correctness(str1, str2, found, i);
assert |str2| > 0;
assert !found && i < |str1|;
var j := |str2|-1;
ghost var old_i := i;
ghost var old_j := j;
while !found && str1[i] == str2[j]
invariant Inner_Inv_Termination(str1, str2, i, j, old_i, old_j);
invariant Inner_Inv_correctness (str1, str2, i, j, found);
decreases j, if !found then 1 else 0;
{
if j == 0 {
assert j==0 && str1[i] == str2[j];
found := true;
assert Inner_Inv_Termination(str1, str2, i, j, old_i, old_j);
assert Inner_Inv_correctness(str1, str2, i, j, found);
}
else {
assert j > 0;
assert Inner_Inv_Termination(str1, str2, i-1, j-1, old_i, old_j);
assert Inner_Inv_correctness(str1, str2, i-1, j-1, found);
i, j := i-1, j-1;
assert Inner_Inv_Termination(str1, str2, i, j, old_i, old_j);
assert Inner_Inv_correctness(str1, str2, i, j, found);
}
assert j >= 0;
assert Inner_Inv_Termination(str1, str2, i, j, old_i, old_j);
assert Inner_Inv_correctness(str1, str2, i, j, found);
}
assert Inner_Inv_Termination(str1, str2, i, j, old_i, old_j);
assert Inner_Inv_correctness(str1, str2, i, j, found);
assert found || str1[i] != str2[j]; // gaurd negation
assert found ==> i + |str2| <= |str1| && str2 <= str1[i..];
assert !found ==> str1[i] != str2[j];
if !found {
assert i < |str1|;
assert |str2| > 0;
assert old_j - j == old_i - i;
assert old_i < i+|str2|-j;
assert Outter_Inv_correctness(str1, str2, found, old_i);
assert i+|str2|-j == old_i + 1;
assert str1[i] != str2[j];
assert |str1[old_i+1 - |str2|..old_i+1]| == |str2|;
assert str1[old_i+1 - |str2|..old_i+1] != str2;
assert 0 < old_i <= |str1| ==> !(ExistsSubstring(str1[..old_i], str2));
Lemma1(str1, str2, i, j, old_i, old_j, found); // ==>
assert 0 < old_i+1 <= |str1| ==> !(ExistsSubstring(str1[..old_i+1], str2));
assert 0 < i+|str2|-j <= |str1| ==> !(ExistsSubstring(str1[..i+|str2|-j], str2));
assert Outter_Inv_correctness(str1, str2, found, i+|str2|-j);
i := i+|str2|-j;
assert old_i < i;
assert Outter_Inv_correctness(str1, str2, found, i);
assert i <= |str1|;
}
assert !found ==> i <= |str1|;
assert !found ==> old_i < i;
assert !found ==> Outter_Inv_correctness(str1, str2, found, i);
assert found ==> Outter_Inv_correctness(str1, str2, found, i);
assert Outter_Inv_correctness(str1, str2, found, i);
}
assert Outter_Inv_correctness(str1, str2, found, i);
assert (found ==> i + |str2| <= |str1| && str2 <= str1[i..]);
assert (!found && 0 < i <= |str1| ==> !(ExistsSubstring(str1[..i], str2)));
assert (!found ==> i <= |str1|);
assert found || i >= |str1|; // gaurd negation
assert (!found && i == |str1| ==> !(ExistsSubstring(str1[..i], str2)));
assert i == |str1| ==> str1[..i] == str1;
assert (!found && i == |str1| ==> !(ExistsSubstring(str1, str2)));
assert !found ==> i >= |str1|;
assert !found ==> i == |str1|;
assert (!found ==> !ExistsSubstring(str1, str2));
assert (found ==> ExistsSubstring(str1, str2));
assert (found <==> ExistsSubstring(str1, str2));
assert (found ==> i + |str2| <= |str1| && str2 <= str1[i..]);
assert Post(str1, str2, found, i);
}
assert Post(str1, str2, found, i);
}
method Main() {
var str1a, str1b := "string", " in Dafny is a sequence of characters (seq<char>)";
var str1, str2 := str1a + str1b, "ring";
var found, i := FindFirstOccurrence(str1, str2);
assert found by {
assert ExistsSubstring(str1, str2) by {
var offset := 2;
assert 0 <= offset <= |str1|;
assert str2 <= str1[offset..] by {
assert str2 == str1[offset..][..4];
}
}
}
print "\nfound, i := FindFirstOccurrence(\"", str1, "\", \"", str2, "\") returns found == ", found;
if found {
print " and i == ", i;
}
str1 := "<= on sequences is the prefix relation";
found, i := FindFirstOccurrence(str1, str2);
print "\nfound, i := FindFirstOccurrence(\"", str1, "\", \"", str2, "\") returns found == ", found;
if found {
print " and i == ", i;
}
}
//this is our lemmas, invatiants and presicats
ghost predicate Outter_Inv_correctness(str1: string, str2: string, found: bool, i : nat)
{
(found ==> (i + |str2| <= |str1| && str2 <= str1[i..])) // Second part of post condition
&&
(!found && 0 < i <= |str1| && i != |str2|-1 ==> !(ExistsSubstring(str1[..i], str2))) // First part of post condition
&&
(!found ==> i <= |str1|)
}
ghost predicate Inner_Inv_correctness(str1: string, str2: string, i : nat, j: int, found: bool){
0 <= j <= i && // index in range
j < |str2| && // index in range
i < |str1| &&// index in range
(str1[i] == str2[j] ==> str2[j..] <= str1[i..]) &&
(found ==> j==0 && str1[i] == str2[j])
}
ghost predicate Inner_Inv_Termination(str1: string, str2: string, i : nat, j: int, old_i: nat, old_j: nat){
old_j - j == old_i - i
}
lemma {:verify true} Lemma1 (str1: string, str2: string, i : nat, j: int, old_i: nat, old_j: nat, found: bool)
// requires old_j - j == old_i - i;
requires !found;
requires |str2| > 0;
requires Outter_Inv_correctness(str1, str2, found, old_i);
requires i+|str2|-j == old_i + 1;
requires old_i+1 >= |str2|;
requires old_i+1 <= |str1|;
requires 0 <= i < |str1| && 0 <= j < |str2|;
requires str1[i] != str2[j];
requires |str2| > 0;
requires 0 < old_i <= |str1| ==> !(ExistsSubstring(str1[..old_i], str2));
ensures 0 < old_i+1 <= |str1| ==> !(ExistsSubstring(str1[..old_i+1], str2));
{
assert |str1[old_i+1 - |str2|..old_i+1]| == |str2|;
assert str1[old_i+1 - |str2|..old_i+1] != str2;
assert !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
assert 0 <= old_i < old_i+1 <= |str1|;
assert old_i+1 >= |str2|;
calc{
0 < old_i+1 <= |str1|
&& (ExistsSubstring(str1[..old_i+1], str2))
&& !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
==>
(!(ExistsSubstring(str1[..old_i], str2)))
&& (ExistsSubstring(str1[..old_i+1], str2))
&& !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
==> {Lemma2(str1, str2, old_i, found);}
((0 < old_i < old_i+1 <= |str1| && old_i != |str2|-1) ==>
(|str1[old_i+1 - |str2|..old_i+1]| == |str2|) && (str2 <= str1[old_i+1 - |str2| .. old_i+1]))
&& !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
==>
((0 < old_i < old_i+1 <= |str1| && old_i != |str2|-1) ==> false);
}
}
lemma {:verify true} Lemma2 (str1: string, str2: string, i : nat, found: bool)
requires 0 <= i < |str1|;
requires 0 < |str2| <= i+1;
requires !ExistsSubstring(str1[..i], str2);
requires ExistsSubstring(str1[..i+1], str2);
ensures str2 <= str1[i+1 - |str2| .. i+1];
{
assert exists offset :: 0 <= offset <= i+1 && str2 <= str1[offset..i+1]
&& ((offset <= i) || (offset == i+1));
calc{
(0 < |str2|)
&& (!exists offset :: 0 <= offset <= i && str2 <= str1[offset..i])
&& (exists offset :: 0 <= offset <= i+1 && str2 <= str1[offset..i+1]);
==>
(0 < |str2|)
&& (forall offset :: 0 <= offset <= i ==> !(str2 <= str1[offset..i]))
&& (exists offset :: 0 <= offset <= i+1 && str2 <= str1[offset..i+1]);
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && str2 <= str1[offset..i+1])
&& (forall offset :: 0 <= offset <= i+1 ==>
(offset <= i ==> !(str2 <= str1[offset..i])));
==> {Lemma3(str1, str2, i);}
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1]) && (offset <= i ==> !(str2 <= str1[offset..i])));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& (offset <= i ==> !(str2 <= str1[offset..i])) && (offset == i+1 ==> |str2| == 0));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& (offset <= i ==> !(str2 <= str1[offset..i])) && (offset == i+1 ==> |str2| == 0) && (offset != i+1));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& (offset <= i ==> !(str2 <= str1[offset..i])) && (offset <= i));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& !(str2 <= str1[offset..i]));
==>
str2 <= str1[i+1 - |str2| .. i+1];
}
}
lemma Lemma3(str1: string, str2: string, i : nat)
requires 0 <= i < |str1|;
requires 0 < |str2| <= i+1;
requires exists offset :: (0 <= offset <= i+1) && str2 <= str1[offset..i+1];
requires forall offset :: 0 <= offset <= i+1 ==> (offset <= i ==> !(str2 <= str1[offset..i]));
ensures exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1]) && (offset <= i ==> !(str2 <= str1[offset..i]));
{
var offset :| (0 <= offset <= i+1) && str2 <= str1[offset..i+1];
assert 0 <= offset <= i+1 ==> (offset <= i ==> !(str2 <= str1[offset..i]));
}
| // Noa Leron 207131871
// Tsuri Farhana 315016907
ghost predicate ExistsSubstring(str1: string, str2: string) {
// string in Dafny is a sequence of characters (seq<char>) and <= on sequences is the prefix relation
exists offset :: 0 <= offset <= |str1| && str2 <= str1[offset..]
}
ghost predicate Post(str1: string, str2: string, found: bool, i: nat) {
(found <==> ExistsSubstring(str1, str2)) &&
(found ==> i + |str2| <= |str1| && str2 <= str1[i..])
}
/*
Goal: Verify correctness of the following code. Once done, remove the {:verify false} (or turn it into {:verify true}).
Feel free to add GHOST code, including calls to lemmas. But DO NOT modify the specification or the original (executable) code.
*/
method {:verify true} FindFirstOccurrence(str1: string, str2: string) returns (found: bool, i: nat)
ensures Post(str1, str2, found, i)
{
if |str2| == 0 {
found, i := true, 0;
}
else if |str1| < |str2| {
found, i := false, 0; // value of i irrelevant in this case
}
else {
found, i := false, |str2|-1;
while !found && i < |str1|
{
var j := |str2|-1;
ghost var old_i := i;
ghost var old_j := j;
while !found && str1[i] == str2[j]
{
if j == 0 {
found := true;
}
else {
i, j := i-1, j-1;
}
}
if !found {
Lemma1(str1, str2, i, j, old_i, old_j, found); // ==>
i := i+|str2|-j;
}
}
}
}
method Main() {
var str1a, str1b := "string", " in Dafny is a sequence of characters (seq<char>)";
var str1, str2 := str1a + str1b, "ring";
var found, i := FindFirstOccurrence(str1, str2);
var offset := 2;
}
}
}
print "\nfound, i := FindFirstOccurrence(\"", str1, "\", \"", str2, "\") returns found == ", found;
if found {
print " and i == ", i;
}
str1 := "<= on sequences is the prefix relation";
found, i := FindFirstOccurrence(str1, str2);
print "\nfound, i := FindFirstOccurrence(\"", str1, "\", \"", str2, "\") returns found == ", found;
if found {
print " and i == ", i;
}
}
//this is our lemmas, invatiants and presicats
ghost predicate Outter_Inv_correctness(str1: string, str2: string, found: bool, i : nat)
{
(found ==> (i + |str2| <= |str1| && str2 <= str1[i..])) // Second part of post condition
&&
(!found && 0 < i <= |str1| && i != |str2|-1 ==> !(ExistsSubstring(str1[..i], str2))) // First part of post condition
&&
(!found ==> i <= |str1|)
}
ghost predicate Inner_Inv_correctness(str1: string, str2: string, i : nat, j: int, found: bool){
0 <= j <= i && // index in range
j < |str2| && // index in range
i < |str1| &&// index in range
(str1[i] == str2[j] ==> str2[j..] <= str1[i..]) &&
(found ==> j==0 && str1[i] == str2[j])
}
ghost predicate Inner_Inv_Termination(str1: string, str2: string, i : nat, j: int, old_i: nat, old_j: nat){
old_j - j == old_i - i
}
lemma {:verify true} Lemma1 (str1: string, str2: string, i : nat, j: int, old_i: nat, old_j: nat, found: bool)
// requires old_j - j == old_i - i;
requires !found;
requires |str2| > 0;
requires Outter_Inv_correctness(str1, str2, found, old_i);
requires i+|str2|-j == old_i + 1;
requires old_i+1 >= |str2|;
requires old_i+1 <= |str1|;
requires 0 <= i < |str1| && 0 <= j < |str2|;
requires str1[i] != str2[j];
requires |str2| > 0;
requires 0 < old_i <= |str1| ==> !(ExistsSubstring(str1[..old_i], str2));
ensures 0 < old_i+1 <= |str1| ==> !(ExistsSubstring(str1[..old_i+1], str2));
{
calc{
0 < old_i+1 <= |str1|
&& (ExistsSubstring(str1[..old_i+1], str2))
&& !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
==>
(!(ExistsSubstring(str1[..old_i], str2)))
&& (ExistsSubstring(str1[..old_i+1], str2))
&& !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
==> {Lemma2(str1, str2, old_i, found);}
((0 < old_i < old_i+1 <= |str1| && old_i != |str2|-1) ==>
(|str1[old_i+1 - |str2|..old_i+1]| == |str2|) && (str2 <= str1[old_i+1 - |str2| .. old_i+1]))
&& !(str2 <= str1[old_i+1 - |str2|..old_i+1]);
==>
((0 < old_i < old_i+1 <= |str1| && old_i != |str2|-1) ==> false);
}
}
lemma {:verify true} Lemma2 (str1: string, str2: string, i : nat, found: bool)
requires 0 <= i < |str1|;
requires 0 < |str2| <= i+1;
requires !ExistsSubstring(str1[..i], str2);
requires ExistsSubstring(str1[..i+1], str2);
ensures str2 <= str1[i+1 - |str2| .. i+1];
{
&& ((offset <= i) || (offset == i+1));
calc{
(0 < |str2|)
&& (!exists offset :: 0 <= offset <= i && str2 <= str1[offset..i])
&& (exists offset :: 0 <= offset <= i+1 && str2 <= str1[offset..i+1]);
==>
(0 < |str2|)
&& (forall offset :: 0 <= offset <= i ==> !(str2 <= str1[offset..i]))
&& (exists offset :: 0 <= offset <= i+1 && str2 <= str1[offset..i+1]);
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && str2 <= str1[offset..i+1])
&& (forall offset :: 0 <= offset <= i+1 ==>
(offset <= i ==> !(str2 <= str1[offset..i])));
==> {Lemma3(str1, str2, i);}
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1]) && (offset <= i ==> !(str2 <= str1[offset..i])));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& (offset <= i ==> !(str2 <= str1[offset..i])) && (offset == i+1 ==> |str2| == 0));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& (offset <= i ==> !(str2 <= str1[offset..i])) && (offset == i+1 ==> |str2| == 0) && (offset != i+1));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& (offset <= i ==> !(str2 <= str1[offset..i])) && (offset <= i));
==>
(0 < |str2|)
&& (exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1])
&& !(str2 <= str1[offset..i]));
==>
str2 <= str1[i+1 - |str2| .. i+1];
}
}
lemma Lemma3(str1: string, str2: string, i : nat)
requires 0 <= i < |str1|;
requires 0 < |str2| <= i+1;
requires exists offset :: (0 <= offset <= i+1) && str2 <= str1[offset..i+1];
requires forall offset :: 0 <= offset <= i+1 ==> (offset <= i ==> !(str2 <= str1[offset..i]));
ensures exists offset :: (0 <= offset <= i+1) && (str2 <= str1[offset..i+1]) && (offset <= i ==> !(str2 <= str1[offset..i]));
{
var offset :| (0 <= offset <= i+1) && str2 <= str1[offset..i+1];
}
|
005 | AssertivePrograming_tmp_tmpwf43uz0e_MergeSort.dfy | // Noa Leron 207131871
// Tsuri Farhana 315016907
predicate Sorted(q: seq<int>) {
forall i,j :: 0 <= i <= j < |q| ==> q[i] <= q[j]
}
/*
Goal: Implement the well known merge sort algorithm in O(a.Length X log_2(a.Length)) time, recursively.
- Divide the contents of the original array into two local arrays
- After sorting the local arrays (recursively), merge the contents of the two returned arrays using the Merge method (see below)
- DO NOT modify the specification or any other part of the method's signature
- DO NOT introduce any further methods
*/
method MergeSort(a: array<int>) returns (b: array<int>)
ensures b.Length == a.Length && Sorted(b[..]) && multiset(a[..]) == multiset(b[..])
decreases a.Length
{
if (a.Length <= 1) {b := a;}
else{
var mid: nat := a.Length / 2;
var a1: array<int> := new int[mid];
var a2: array<int> := new int[a.Length - mid];
assert a1.Length <= a2.Length;
assert a.Length == a1.Length + a2.Length;
var i: nat := 0;
while (i < a1.Length )
invariant Inv(a[..], a1[..], a2[..], i, mid)
decreases a1.Length - i
{
a1[i] := a[i];
a2[i] := a[i+mid];
i:=i+1;
}
assert !(i < a1.Length);
assert (i >= a1.Length);
assert i == a1.Length;
assert Inv(a[..], a1[..], a2[..], i, mid);
assert (i <= |a1[..]|) && (i <= |a2[..]|) && (i+mid <= |a[..]|);
assert (a1[..i] == a[..i]) && (a2[..i] == a[mid..(i+mid)]);
if(a1.Length < a2.Length) {
a2[i] := a[i+mid];
assert i+1 == a2.Length;
assert (a2[..i+1] == a[mid..(i+1+mid)]);
assert (a1[..i] == a[..i]) && (a2[..i+1] == a[mid..(i+1+mid)]);
assert a[..i] + a[i..i+1+mid] == a1[..i] + a2[..i+1];
assert a[..i] + a[i..i+1+mid] == a1[..] + a2[..];
assert a[..] == a1[..] + a2[..];
} // If a.Length is odd.
else{
assert i == a2.Length;
assert (a1[..i] == a[..i]) && (a2[..i] == a[mid..(i+mid)]);
assert a[..i] + a[i..i+mid] == a1[..i] + a2[..i];
assert a[..i] + a[i..i+mid] == a1[..] + a2[..];
assert a[..] == a1[..] + a2[..];
}
assert a1.Length < a.Length;
a1:= MergeSort(a1);
assert a2.Length < a.Length;
a2:= MergeSort(a2);
b := new int [a.Length];
Merge(b, a1, a2);
assert multiset(b[..]) == multiset(a1[..]) + multiset(a2[..]);
assert Sorted(b[..]);
}
assert b.Length == a.Length && Sorted(b[..]) && multiset(a[..]) == multiset(b[..]);
}
ghost predicate Inv(a: seq<int>, a1: seq<int>, a2: seq<int>, i: nat, mid: nat){
(i <= |a1|) && (i <= |a2|) && (i+mid <= |a|) &&
(a1[..i] == a[..i]) && (a2[..i] == a[mid..(i+mid)])
}
/*
Goal: Implement iteratively, correctly, efficiently, clearly
DO NOT modify the specification or any other part of the method's signature
*/
method Merge(b: array<int>, c: array<int>, d: array<int>)
requires b != c && b != d && b.Length == c.Length + d.Length
requires Sorted(c[..]) && Sorted(d[..])
ensures Sorted(b[..]) && multiset(b[..]) == multiset(c[..])+multiset(d[..])
modifies b
{
var i: nat, j: nat := 0, 0;
while i + j < b.Length
invariant i <= c.Length && j <= d.Length && i + j <= b.Length
invariant InvSubSet(b[..],c[..],d[..],i,j)
invariant InvSorted(b[..],c[..],d[..],i,j)
decreases c.Length-i, d.Length-j
{
i,j := MergeLoop (b,c,d,i,j);
assert InvSubSet(b[..],c[..],d[..],i,j);
assert InvSorted(b[..],c[..],d[..],i,j);
}
assert InvSubSet(b[..],c[..],d[..],i,j);
LemmaMultysetsEquals(b[..],c[..],d[..],i,j);
assert multiset(b[..]) == multiset(c[..])+multiset(d[..]);
}
//This is a method that replace the loop body
method {:verify true} MergeLoop(b: array<int>, c: array<int>, d: array<int>,i0: nat , j0: nat) returns (i: nat, j: nat)
requires b != c && b != d && b.Length == c.Length + d.Length
requires Sorted(c[..]) && Sorted(d[..])
requires i0 <= c.Length && j0 <= d.Length && i0 + j0 <= b.Length
requires InvSubSet(b[..],c[..],d[..],i0,j0)
requires InvSorted(b[..],c[..],d[..],i0,j0)
requires i0 + j0 < b.Length
modifies b
ensures i <= c.Length && j <= d.Length && i + j <= b.Length
ensures InvSubSet(b[..],c[..],d[..],i,j)
ensures InvSorted(b[..],c[..],d[..],i,j)
//decreases ensures
ensures 0 <= c.Length - i < c.Length - i0 || (c.Length - i == c.Length - i0 && 0 <= d.Length - j < d.Length - j0)
{
i,j := i0,j0;
if(i == c.Length || (j< d.Length && d[j] < c[i])){
// in this case we take the next value from d
assert InvSorted(b[..][i+j:=d[j]],c[..],d[..],i,j+1);
b[i+j] := d[j];
lemmaInvSubsetTakeValueFromD(b[..],c[..],d[..],i,j);
assert InvSubSet(b[..],c[..],d[..],i,j+1);
assert InvSorted(b[..],c[..],d[..],i,j+1);
j := j + 1;
}
else{
assert j == d.Length || (i < c.Length && c[i] <= d[j]);
// in this case we take the next value from c
assert InvSorted(b[..][i+j:=c[i]],c[..],d[..],i+1,j);
b[i+j] := c[i];
lemmaInvSubsetTakeValueFromC(b[..],c[..],d[..],i,j);
assert InvSubSet(b[..],c[..],d[..],i+1,j);
assert InvSorted(b[..],c[..],d[..],i+1,j);
i := i + 1;
}
}
//Loop invariant - b is sprted so far and the next two potential values that will go into b are bigger then the biggest value in b.
ghost predicate InvSorted(b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat){
i <= |c| && j <= |d| && i + j <= |b| &&
((i+j > 0 && i < |c|) ==> (b[j + i - 1] <= c[i])) &&
((i+j > 0 && j < |d|) ==> (b[j + i - 1] <= d[j])) &&
Sorted(b[..i+j])
}
//Loop invariant - the multiset of the prefix of b so far is the same multiset as the prefixes of c and d so far.
ghost predicate InvSubSet(b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat){
i <= |c| && j <= |d| && i + j <= |b| &&
multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
}
//This lemma helps dafny see that if the prefixs of arrays are the same multiset until the end of the arrays,
//all the arrays are the same multiset.
lemma LemmaMultysetsEquals (b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat)
requires i == |c|;
requires j == |d|;
requires i + j == |b|;
requires multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
ensures multiset(b[..]) == multiset(c[..])+multiset(d[..]);
{
assert b[..] == b[..i+j];
assert c[..] == c[..i];
assert d[..] == d[..j];
}
//This lemma helps dafny see that after adding the next value from c to b the prefixes are still the same subsets.
lemma lemmaInvSubsetTakeValueFromC (b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat)
requires i < |c|;
requires j <= |d|;
requires i + j < |b|;
requires |c| + |d| == |b|;
requires multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
requires b[i+j] == c[i]
ensures multiset(b[..i+j+1]) == multiset(c[..i+1])+multiset(d[..j])
{
assert c[..i]+[c[i]] == c[..i+1];
assert b[..i+j+1] == b[..i+j] + [b[i+j]];
assert multiset(b[..i+j+1]) == multiset(c[..i+1])+multiset(d[..j]);
}
//This lemma helps dafny see that after adding the next value from d to b the prefixes are still the same subsets.
lemma{:verify true} lemmaInvSubsetTakeValueFromD (b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat)
requires i <= |c|;
requires j < |d|;
requires i + j < |b|;
requires |c| + |d| == |b|;
requires multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
requires b[i+j] == d[j]
ensures multiset(b[..i+j+1]) == multiset(c[..i])+multiset(d[..j+1])
{
assert d[..j]+[d[j]] == d[..j+1];
assert b[..i+j+1] == b[..i+j] + [b[i+j]];
assert multiset(b[..i+j+1]) == multiset(c[..i])+multiset(d[..j+1]);
}
method Main() {
var a := new int[3] [4, 8, 6];
var q0 := a[..];
assert q0 == [4,8,6];
a := MergeSort(a);
assert a.Length == |q0| && multiset(a[..]) == multiset(q0);
print "\nThe sorted version of ", q0, " is ", a[..];
assert Sorted(a[..]);
assert a[..] == [4, 6, 8];
a := new int[5] [3, 8, 5, -1, 10];
q0 := a[..];
assert q0 == [3, 8, 5, -1, 10];
a := MergeSort(a);
assert a.Length == |q0| && multiset(a[..]) == multiset(q0);
print "\nThe sorted version of ", q0, " is ", a[..];
assert Sorted(a[..]);
//assert a[..] == [-1, 3, 5, 8, 10];
}
| // Noa Leron 207131871
// Tsuri Farhana 315016907
predicate Sorted(q: seq<int>) {
forall i,j :: 0 <= i <= j < |q| ==> q[i] <= q[j]
}
/*
Goal: Implement the well known merge sort algorithm in O(a.Length X log_2(a.Length)) time, recursively.
- Divide the contents of the original array into two local arrays
- After sorting the local arrays (recursively), merge the contents of the two returned arrays using the Merge method (see below)
- DO NOT modify the specification or any other part of the method's signature
- DO NOT introduce any further methods
*/
method MergeSort(a: array<int>) returns (b: array<int>)
ensures b.Length == a.Length && Sorted(b[..]) && multiset(a[..]) == multiset(b[..])
{
if (a.Length <= 1) {b := a;}
else{
var mid: nat := a.Length / 2;
var a1: array<int> := new int[mid];
var a2: array<int> := new int[a.Length - mid];
var i: nat := 0;
while (i < a1.Length )
{
a1[i] := a[i];
a2[i] := a[i+mid];
i:=i+1;
}
if(a1.Length < a2.Length) {
a2[i] := a[i+mid];
} // If a.Length is odd.
else{
}
a1:= MergeSort(a1);
a2:= MergeSort(a2);
b := new int [a.Length];
Merge(b, a1, a2);
}
}
ghost predicate Inv(a: seq<int>, a1: seq<int>, a2: seq<int>, i: nat, mid: nat){
(i <= |a1|) && (i <= |a2|) && (i+mid <= |a|) &&
(a1[..i] == a[..i]) && (a2[..i] == a[mid..(i+mid)])
}
/*
Goal: Implement iteratively, correctly, efficiently, clearly
DO NOT modify the specification or any other part of the method's signature
*/
method Merge(b: array<int>, c: array<int>, d: array<int>)
requires b != c && b != d && b.Length == c.Length + d.Length
requires Sorted(c[..]) && Sorted(d[..])
ensures Sorted(b[..]) && multiset(b[..]) == multiset(c[..])+multiset(d[..])
modifies b
{
var i: nat, j: nat := 0, 0;
while i + j < b.Length
{
i,j := MergeLoop (b,c,d,i,j);
}
LemmaMultysetsEquals(b[..],c[..],d[..],i,j);
}
//This is a method that replace the loop body
method {:verify true} MergeLoop(b: array<int>, c: array<int>, d: array<int>,i0: nat , j0: nat) returns (i: nat, j: nat)
requires b != c && b != d && b.Length == c.Length + d.Length
requires Sorted(c[..]) && Sorted(d[..])
requires i0 <= c.Length && j0 <= d.Length && i0 + j0 <= b.Length
requires InvSubSet(b[..],c[..],d[..],i0,j0)
requires InvSorted(b[..],c[..],d[..],i0,j0)
requires i0 + j0 < b.Length
modifies b
ensures i <= c.Length && j <= d.Length && i + j <= b.Length
ensures InvSubSet(b[..],c[..],d[..],i,j)
ensures InvSorted(b[..],c[..],d[..],i,j)
//decreases ensures
ensures 0 <= c.Length - i < c.Length - i0 || (c.Length - i == c.Length - i0 && 0 <= d.Length - j < d.Length - j0)
{
i,j := i0,j0;
if(i == c.Length || (j< d.Length && d[j] < c[i])){
// in this case we take the next value from d
b[i+j] := d[j];
lemmaInvSubsetTakeValueFromD(b[..],c[..],d[..],i,j);
j := j + 1;
}
else{
// in this case we take the next value from c
b[i+j] := c[i];
lemmaInvSubsetTakeValueFromC(b[..],c[..],d[..],i,j);
i := i + 1;
}
}
//Loop invariant - b is sprted so far and the next two potential values that will go into b are bigger then the biggest value in b.
ghost predicate InvSorted(b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat){
i <= |c| && j <= |d| && i + j <= |b| &&
((i+j > 0 && i < |c|) ==> (b[j + i - 1] <= c[i])) &&
((i+j > 0 && j < |d|) ==> (b[j + i - 1] <= d[j])) &&
Sorted(b[..i+j])
}
//Loop invariant - the multiset of the prefix of b so far is the same multiset as the prefixes of c and d so far.
ghost predicate InvSubSet(b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat){
i <= |c| && j <= |d| && i + j <= |b| &&
multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
}
//This lemma helps dafny see that if the prefixs of arrays are the same multiset until the end of the arrays,
//all the arrays are the same multiset.
lemma LemmaMultysetsEquals (b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat)
requires i == |c|;
requires j == |d|;
requires i + j == |b|;
requires multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
ensures multiset(b[..]) == multiset(c[..])+multiset(d[..]);
{
}
//This lemma helps dafny see that after adding the next value from c to b the prefixes are still the same subsets.
lemma lemmaInvSubsetTakeValueFromC (b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat)
requires i < |c|;
requires j <= |d|;
requires i + j < |b|;
requires |c| + |d| == |b|;
requires multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
requires b[i+j] == c[i]
ensures multiset(b[..i+j+1]) == multiset(c[..i+1])+multiset(d[..j])
{
}
//This lemma helps dafny see that after adding the next value from d to b the prefixes are still the same subsets.
lemma{:verify true} lemmaInvSubsetTakeValueFromD (b: seq<int>, c: seq<int>, d: seq<int>, i: nat, j: nat)
requires i <= |c|;
requires j < |d|;
requires i + j < |b|;
requires |c| + |d| == |b|;
requires multiset(b[..i+j]) == multiset(c[..i]) + multiset(d[..j])
requires b[i+j] == d[j]
ensures multiset(b[..i+j+1]) == multiset(c[..i])+multiset(d[..j+1])
{
}
method Main() {
var a := new int[3] [4, 8, 6];
var q0 := a[..];
a := MergeSort(a);
print "\nThe sorted version of ", q0, " is ", a[..];
a := new int[5] [3, 8, 5, -1, 10];
q0 := a[..];
a := MergeSort(a);
print "\nThe sorted version of ", q0, " is ", a[..];
//assert a[..] == [-1, 3, 5, 8, 10];
}
|
006 | BPTree-verif_tmp_tmpq1z6xm1d_Utils.dfy |
// method CountLessThan(numbers: set<int>, threshold: int) returns (count: int)
// // ensures count == |set i | i in numbers && i < threshold|
// ensures count == |SetLessThan(numbers, threshold)|
// {
// count := 0;
// var ss := numbers;
// while ss != {}
// decreases |ss|
// {
// var i: int :| i in ss;
// ss := ss - {i};
// if i < threshold {
// count := count + 1;
// }
// }
// assert count == |SetLessThan(numbers, threshold)|;
// // assert count == |set i | i in numbers && i < threshold|;
// }
function SetLessThan(numbers: set<int>, threshold: int): set<int>
{
set i | i in numbers && i < threshold
}
method Main()
{
// var s: set<int> := {1, 2, 3, 4, 5};
// var c: int := CountLessThan(s, 4);
// print c;
// assert c == 3;
// if you manualy create set and sequence with same elements, |s|==|t| works
var t: seq<int> := [1, 2, 3];
var s: set<int> := {1, 2, 3};
assert |s| == 3;
assert |s| == |t|;
// but if you create set from the sequence with distinct elements it does not understand that |s|==|t|
// Dafny has problems when reasoning about set sizes ==>
s := set x | x in t;
assert forall x :: x in t ==> x in s;
assert forall x :: x in s ==> x in t;
assert forall x :: x in s <==> x in t;
assert forall i, j :: 0 <= i < |t| && 0 <= j < |t| && i !=j ==> t[i] != t[j];
assert |t| == 3;
// assert |s| == |t|; // not verifying
// assert |s| == 3; // not verifying
// other expriments
set_memebrship_implies_cardinality(s, set x | x in t); // s and the other argument is the same thing
var s2 : set<int> := set x | x in t;
assert |s| == |s2|;
s2 := {1, 2, 3};
// assert |s| == |s2|; // may not hold
set_memebrship_implies_cardinality(s, s2);
assert |s| == |s2|; // after lemma it holds
}
lemma set_memebrship_implies_cardinality_helper<A>(s: set<A>, t: set<A>, s_size: int)
requires s_size >= 0 && s_size == |s|
requires forall x :: x in s <==> x in t
ensures |s| == |t|
decreases s_size {
if s_size == 0 {
} else {
var s_hd;
// assign s_hd to a value *such that* s_hd is in s (see such_that expressions)
s_hd :| s_hd in s;
set_memebrship_implies_cardinality_helper(s - {s_hd}, t - {s_hd}, s_size - 1);
}
}
lemma set_memebrship_implies_cardinality<A>(s: set<A>, t: set<A>)
requires forall x :: x in s <==> x in t
ensures |s| == |t| {
set_memebrship_implies_cardinality_helper(s, t, |s|);
}
/*
lemma Bijection(arr: seq<int>, s: set<int>) // returns (bool)
requires sorted(arr)
// requires forall x, y :: x in s && y in s && x != y ==> x < y
ensures |s| == |arr|
{
var mapping: map<int, int> := map[];
// Establish the bijection
for i := 0 to |arr| {
mapping := mapping[arr[i]:= arr[i]];
}
// Prove injectiveness
assert forall i, j :: (0 <= i < |arr|-1 && 0 <= j < |arr|-1 && i != j )==> mapping[arr[i]] != mapping[arr[j]];
// Prove surjectiveness
// assert forall x :: x in s ==> exists i :: 0 <= i < |arr|-1 && arr[i] == x;
// Conclude equinumerosity
// assert |s| == |arr|;
// return true;
}
*/
function seqSet(nums: seq<int>, index: nat): set<int> {
set x | 0 <= x < index < |nums| :: nums[x]
}
lemma containsDuplicateI(nums: seq<int>) returns (containsDuplicate: bool)
ensures containsDuplicate ==> exists i,j :: 0 <= i < j < |nums| && nums[i] == nums[j]
{
var windowGhost: set<int> := {};
var windowSet: set<int> := {};
for i:= 0 to |nums|
invariant 0 <= i <= |nums|
invariant forall j :: 0 <= j < i < |nums| ==> nums[j] in windowSet
// invariant forall x :: x in windowSet ==> x in nums
invariant forall x :: x in windowSet ==> x in nums[0..i]
invariant seqSet(nums, i) <= windowSet
{
windowGhost := windowSet;
if nums[i] in windowSet { // does not verify
// if nums[i] in seqSet(nums, i) { //verifies
return true;
}
windowSet := windowSet + {nums[i]};
}
return false;
}
// lemma numElemsOfSet(a: seq<int>)
// requires sorted(a)
// {
// assert distinct(a);
// var s := set x | x in a;
// assert forall x :: x in s ==> x in a[..];
// assert forall x :: x in a ==> x in s;
// assert |s| == |a|;
// }
// lemma CardinalitySetEqualsArray(a: seq<int>, s: set<int>)
// requires s == set x | x in a
// requires distinct(a)
// ensures |s| == |a|
// {
// assert forall x :: x in s ==> exists i :: 0 <= i < |a| && a[i] == x;
// assert forall i, j :: 0 <= i < |a| && 0 <= j < |a| && i != j ==> a[i] != a[j];
// // Assert that each element in the array is in the set
// assert forall i :: 0 <= i < |a| ==> a[i] in s;
// // Assert that the set contains exactly the elements in the array
// assert s == set x | x in a;
// // Assert that the set is a subset of the array
// assert forall x :: x in s <==> x in a;
// // Conclude the equivalence
// assert |s| == |a|;
// }
/*
lemma memebrship_implies_cardinality_helper<A>(s: set<A>, t: seq<A>, s_size: int)
requires s_size >= 0 && s_size == |s|
requires forall x :: x in s <==> x in t
requires forall i, j :: (0 <= i < |t| && 0 <= j < |t| && i != j ) ==> t[i] != t[j]
requires |set x | x in t| == |t|
ensures |s| == |t|
decreases s_size {
if s_size == 0 {
} else {
var t_hd;
t_hd := t[0];
assert t_hd in s;
ghost var t_h := set x | x in t[1..];
assert |t_h| == |t[1..]|;
memebrship_implies_cardinality_helper(s - {t_hd}, t[1..], s_size - 1);
}
}
lemma memebrship_implies_cardinality<A>(s: set<A>, t: seq<A>)
requires forall x :: x in s <==> x in t
ensures |s| == |t| {
memebrship_implies_cardinality_helper(s, t, |s|);
}
*/
lemma set_memebrship_implies_equality_helper<A>(s: set<A>, t: set<A>, s_size: int)
requires s_size >= 0 && s_size == |s|
requires forall x :: x in s <==> x in t
ensures s == t
decreases s_size {
if s_size == 0 {
} else {
var s_hd;
// assign s_hd to a value *such that* s_hd is in s (see such_that expressions)
s_hd :| s_hd in s;
set_memebrship_implies_equality_helper(s - {s_hd}, t - {s_hd}, s_size - 1);
}
}
lemma set_memebrship_implies_equality<A>(s: set<A>, t: set<A>)
requires forall x :: x in s <==> x in t
ensures s == t {
set_memebrship_implies_equality_helper(s, t, |s|);
}
// TODO play with this for keys==Contents
lemma set_seq_equality(s: set<int>, t: seq<int>)
requires distinct(t)
requires forall x :: x in t <==> x in s
{
var s2 : set<int> := set x | x in t;
set_memebrship_implies_equality_helper(s, s2, |s|);
assert |s2| == |s|;
// assert |s2| == |t|;
// assert |s| == |t|;
}
ghost predicate SortedSeq(a: seq<int>)
//sequence is sorted from left to right
{
(forall i,j :: 0<= i< j < |a| ==> ( a[i] < a[j] ))
}
method GetInsertIndex(a: array<int>, limit: int, x:int) returns (idx:int)
// get index so that array stays sorted
requires x !in a[..]
requires 0 <= limit <= a.Length
requires SortedSeq(a[..limit])
ensures 0<= idx <= limit
ensures SortedSeq(a[..limit])
ensures idx > 0 ==> a[idx-1]< x
ensures idx < limit ==> x < a[idx]
{
idx := limit;
for i := 0 to limit
invariant i>0 ==> x > a[i-1]
{
if x < a[i] {
idx := i;
break;
}
}
}
predicate sorted(a: seq<int>)
{
forall i,j :: 0 <= i < j < |a| ==> a[i] < a[j]
}
predicate distinct(a: seq<int>)
{
forall i,j :: (0 <= i < |a| && 0 <= j < |a| && i != j) ==> a[i] != a[j]
}
predicate sorted_eq(a: seq<int>)
{
forall i,j :: 0 <= i < j < |a| ==> a[i] <= a[j]
}
predicate lessThan(a:seq<int>, key:int) {
forall i :: 0 <= i < |a| ==> a[i] < key
}
predicate greaterThan(a:seq<int>, key:int) {
forall i :: 0 <= i < |a| ==> a[i] > key
}
predicate greaterEqualThan(a:seq<int>, key:int) {
forall i :: 0 <= i < |a| ==> a[i] >= key
}
/*
method InsertSorted(a: array<int>, key: int ) returns (b: array<int>)
requires sorted_eq(a[..])
ensures sorted_eq(b[..])
{
b:= new int[a.Length + 1];
ghost var k := 0;
b[0] := key;
ghost var a' := a[..];
var i:= 0;
while (i < a.Length)
modifies b
invariant 0 <= k <= i <= a.Length
invariant b.Length == a.Length + 1
invariant a[..] == a'
invariant lessThan(a[..i], key) ==> i == k
invariant lessThan(a[..k], key)
invariant b[..k] == a[..k]
invariant b[k] == key
invariant k < i ==> b[k+1..i+1] == a[k..i]
invariant k < i ==> greaterEqualThan(b[k+1..i+1], key)
invariant 0 <= k < b.Length && b[k] == key
{
if(a[i]<key)
{
b[i]:= a[i];
b[i+1] := key;
k := i+1;
}
else if (a[i] >= key)
{
b[i+1] := a[i];
}
i := i+1;
}
assert b[..] == a[..k] + [key] + a[k..];
}
*/
lemma DistributiveLemma(a: seq<bool>, b: seq<bool>)
ensures count(a + b) == count(a) + count(b)
{
if a == [] {
assert a + b == b;
} else {
DistributiveLemma(a[1..], b);
assert a + b == [a[0]] + (a[1..] + b);
}
}
function count(a: seq<bool>): nat
{
if |a| == 0 then 0 else
(if a[0] then 1 else 0) + count(a[1..])
}
lemma DistributiveIn(a: seq<int>, b:seq<int>, k:int, key:int)
requires |a| + 1 == |b|
requires 0 <= k <= |a|
requires b == a[..k] + [key] + a[k..]
ensures forall i :: 0 <= i < |a| ==> a[i] in b
{
assert forall j :: 0 <= j < k ==> a[j] in b;
assert forall j :: k <= j < |a| ==> a[j] in b;
assert ((forall j :: 0 <= j < k ==> a[j] in b) && (forall j :: k <= j < |a| ==> a[j] in b)) ==> (forall j :: 0 <= j < |a| ==> a[j] in b);
assert forall j :: 0 <= j < |a| ==> a[j] in b;
}
lemma DistributiveGreater(a: seq<int>, b:seq<int>, k:int, key:int)
requires |a| + 1 == |b|
requires 0 <= k <= |a|
requires b == a[..k] + [key] + a[k..]
requires forall j :: 0 <= j < |a| ==> a[j] > 0
requires key > 0
ensures forall i :: 0 <= i < |b| ==> b[i] > 0
{
// assert ((forall j :: 0 <= j < k ==> b[j] > 0) && (forall j :: k <= j < |a| ==> b[j] > 0)) ==> (forall j :: 0 <= j < |b| ==> b[j] > 0);
assert forall j :: 0 <= j < |b| ==> b[j] > 0;
}
// verifies in more than 45 seconds, but less than 100 seconds
method InsertIntoSorted(a: array<int>, limit:int, key:int) returns (b: array<int>)
requires key > 0
requires key !in a[..]
requires 0 <= limit < a.Length
requires forall i :: 0 <= i < limit ==> a[i] > 0
requires forall i :: limit <= i < a.Length ==> a[i] == 0
requires sorted(a[..limit])
ensures b.Length == a.Length
ensures sorted(b[..(limit+ 1)])
ensures forall i :: limit + 1 <= i < b.Length ==> b[i] == 0
ensures forall i :: 0 <= i < limit ==> a[i] in b[..]
ensures forall i :: 0 <= i < limit + 1 ==> b[i] > 0
{
b:= new int[a.Length];
ghost var k := 0;
b[0] := key;
ghost var a' := a[..];
var i:= 0;
while (i < limit)
modifies b
invariant 0 <= k <= i <= limit
invariant b.Length == a.Length
invariant a[..] == a'
invariant lessThan(a[..i], key) ==> i == k
invariant lessThan(a[..k], key)
invariant b[..k] == a[..k]
invariant b[k] == key
invariant k < i ==> b[k+1..i+1] == a[k..i]
invariant k < i ==> greaterThan(b[k+1..i+1], key)
invariant 0 <= k < b.Length && b[k] == key
{
if(a[i]<key)
{
b[i]:= a[i];
b[i+1] := key;
k := i+1;
}
else if (a[i] >= key)
{
b[i+1] := a[i];
}
i := i+1;
}
assert b[..limit+1] == a[..k] + [key] + a[k..limit];
assert sorted(b[..limit+1]);
// assert b[..limit+1] == a[..k] + [key] + a[k..limit];
DistributiveIn(a[..limit], b[..limit+1], k, key);
assert forall i :: 0 <= i < limit ==> a[i] in b[..limit+1];
DistributiveGreater(a[..limit], b[..limit+1], k, key);
// assert forall i :: 0 <= i < limit + 1 ==> b[i] > 0;
ghost var b' := b[..];
i := limit + 1;
while i < b.Length
invariant limit + 1 <= i <= b.Length
invariant forall j :: limit + 1 <= j < i ==> b[j] == 0
invariant b[..limit+1] == b'[..limit+1]
invariant sorted(b[..limit+1])
{
b[i] := 0;
i := i + 1;
}
assert forall i :: limit + 1 <= i < b.Length ==> b[i] == 0;
}
|
// method CountLessThan(numbers: set<int>, threshold: int) returns (count: int)
// // ensures count == |set i | i in numbers && i < threshold|
// ensures count == |SetLessThan(numbers, threshold)|
// {
// count := 0;
// var ss := numbers;
// while ss != {}
// decreases |ss|
// {
// var i: int :| i in ss;
// ss := ss - {i};
// if i < threshold {
// count := count + 1;
// }
// }
// assert count == |SetLessThan(numbers, threshold)|;
// // assert count == |set i | i in numbers && i < threshold|;
// }
function SetLessThan(numbers: set<int>, threshold: int): set<int>
{
set i | i in numbers && i < threshold
}
method Main()
{
// var s: set<int> := {1, 2, 3, 4, 5};
// var c: int := CountLessThan(s, 4);
// print c;
// assert c == 3;
// if you manualy create set and sequence with same elements, |s|==|t| works
var t: seq<int> := [1, 2, 3];
var s: set<int> := {1, 2, 3};
// but if you create set from the sequence with distinct elements it does not understand that |s|==|t|
// Dafny has problems when reasoning about set sizes ==>
s := set x | x in t;
// assert |s| == |t|; // not verifying
// assert |s| == 3; // not verifying
// other expriments
set_memebrship_implies_cardinality(s, set x | x in t); // s and the other argument is the same thing
var s2 : set<int> := set x | x in t;
s2 := {1, 2, 3};
// assert |s| == |s2|; // may not hold
set_memebrship_implies_cardinality(s, s2);
}
lemma set_memebrship_implies_cardinality_helper<A>(s: set<A>, t: set<A>, s_size: int)
requires s_size >= 0 && s_size == |s|
requires forall x :: x in s <==> x in t
ensures |s| == |t|
if s_size == 0 {
} else {
var s_hd;
// assign s_hd to a value *such that* s_hd is in s (see such_that expressions)
s_hd :| s_hd in s;
set_memebrship_implies_cardinality_helper(s - {s_hd}, t - {s_hd}, s_size - 1);
}
}
lemma set_memebrship_implies_cardinality<A>(s: set<A>, t: set<A>)
requires forall x :: x in s <==> x in t
ensures |s| == |t| {
set_memebrship_implies_cardinality_helper(s, t, |s|);
}
/*
lemma Bijection(arr: seq<int>, s: set<int>) // returns (bool)
requires sorted(arr)
// requires forall x, y :: x in s && y in s && x != y ==> x < y
ensures |s| == |arr|
{
var mapping: map<int, int> := map[];
// Establish the bijection
for i := 0 to |arr| {
mapping := mapping[arr[i]:= arr[i]];
}
// Prove injectiveness
// Prove surjectiveness
// assert forall x :: x in s ==> exists i :: 0 <= i < |arr|-1 && arr[i] == x;
// Conclude equinumerosity
// assert |s| == |arr|;
// return true;
}
*/
function seqSet(nums: seq<int>, index: nat): set<int> {
set x | 0 <= x < index < |nums| :: nums[x]
}
lemma containsDuplicateI(nums: seq<int>) returns (containsDuplicate: bool)
ensures containsDuplicate ==> exists i,j :: 0 <= i < j < |nums| && nums[i] == nums[j]
{
var windowGhost: set<int> := {};
var windowSet: set<int> := {};
for i:= 0 to |nums|
// invariant forall x :: x in windowSet ==> x in nums
{
windowGhost := windowSet;
if nums[i] in windowSet { // does not verify
// if nums[i] in seqSet(nums, i) { //verifies
return true;
}
windowSet := windowSet + {nums[i]};
}
return false;
}
// lemma numElemsOfSet(a: seq<int>)
// requires sorted(a)
// {
// assert distinct(a);
// var s := set x | x in a;
// assert forall x :: x in s ==> x in a[..];
// assert forall x :: x in a ==> x in s;
// assert |s| == |a|;
// }
// lemma CardinalitySetEqualsArray(a: seq<int>, s: set<int>)
// requires s == set x | x in a
// requires distinct(a)
// ensures |s| == |a|
// {
// assert forall x :: x in s ==> exists i :: 0 <= i < |a| && a[i] == x;
// assert forall i, j :: 0 <= i < |a| && 0 <= j < |a| && i != j ==> a[i] != a[j];
// // Assert that each element in the array is in the set
// assert forall i :: 0 <= i < |a| ==> a[i] in s;
// // Assert that the set contains exactly the elements in the array
// assert s == set x | x in a;
// // Assert that the set is a subset of the array
// assert forall x :: x in s <==> x in a;
// // Conclude the equivalence
// assert |s| == |a|;
// }
/*
lemma memebrship_implies_cardinality_helper<A>(s: set<A>, t: seq<A>, s_size: int)
requires s_size >= 0 && s_size == |s|
requires forall x :: x in s <==> x in t
requires forall i, j :: (0 <= i < |t| && 0 <= j < |t| && i != j ) ==> t[i] != t[j]
requires |set x | x in t| == |t|
ensures |s| == |t|
if s_size == 0 {
} else {
var t_hd;
t_hd := t[0];
ghost var t_h := set x | x in t[1..];
memebrship_implies_cardinality_helper(s - {t_hd}, t[1..], s_size - 1);
}
}
lemma memebrship_implies_cardinality<A>(s: set<A>, t: seq<A>)
requires forall x :: x in s <==> x in t
ensures |s| == |t| {
memebrship_implies_cardinality_helper(s, t, |s|);
}
*/
lemma set_memebrship_implies_equality_helper<A>(s: set<A>, t: set<A>, s_size: int)
requires s_size >= 0 && s_size == |s|
requires forall x :: x in s <==> x in t
ensures s == t
if s_size == 0 {
} else {
var s_hd;
// assign s_hd to a value *such that* s_hd is in s (see such_that expressions)
s_hd :| s_hd in s;
set_memebrship_implies_equality_helper(s - {s_hd}, t - {s_hd}, s_size - 1);
}
}
lemma set_memebrship_implies_equality<A>(s: set<A>, t: set<A>)
requires forall x :: x in s <==> x in t
ensures s == t {
set_memebrship_implies_equality_helper(s, t, |s|);
}
// TODO play with this for keys==Contents
lemma set_seq_equality(s: set<int>, t: seq<int>)
requires distinct(t)
requires forall x :: x in t <==> x in s
{
var s2 : set<int> := set x | x in t;
set_memebrship_implies_equality_helper(s, s2, |s|);
// assert |s2| == |t|;
// assert |s| == |t|;
}
ghost predicate SortedSeq(a: seq<int>)
//sequence is sorted from left to right
{
(forall i,j :: 0<= i< j < |a| ==> ( a[i] < a[j] ))
}
method GetInsertIndex(a: array<int>, limit: int, x:int) returns (idx:int)
// get index so that array stays sorted
requires x !in a[..]
requires 0 <= limit <= a.Length
requires SortedSeq(a[..limit])
ensures 0<= idx <= limit
ensures SortedSeq(a[..limit])
ensures idx > 0 ==> a[idx-1]< x
ensures idx < limit ==> x < a[idx]
{
idx := limit;
for i := 0 to limit
{
if x < a[i] {
idx := i;
break;
}
}
}
predicate sorted(a: seq<int>)
{
forall i,j :: 0 <= i < j < |a| ==> a[i] < a[j]
}
predicate distinct(a: seq<int>)
{
forall i,j :: (0 <= i < |a| && 0 <= j < |a| && i != j) ==> a[i] != a[j]
}
predicate sorted_eq(a: seq<int>)
{
forall i,j :: 0 <= i < j < |a| ==> a[i] <= a[j]
}
predicate lessThan(a:seq<int>, key:int) {
forall i :: 0 <= i < |a| ==> a[i] < key
}
predicate greaterThan(a:seq<int>, key:int) {
forall i :: 0 <= i < |a| ==> a[i] > key
}
predicate greaterEqualThan(a:seq<int>, key:int) {
forall i :: 0 <= i < |a| ==> a[i] >= key
}
/*
method InsertSorted(a: array<int>, key: int ) returns (b: array<int>)
requires sorted_eq(a[..])
ensures sorted_eq(b[..])
{
b:= new int[a.Length + 1];
ghost var k := 0;
b[0] := key;
ghost var a' := a[..];
var i:= 0;
while (i < a.Length)
modifies b
{
if(a[i]<key)
{
b[i]:= a[i];
b[i+1] := key;
k := i+1;
}
else if (a[i] >= key)
{
b[i+1] := a[i];
}
i := i+1;
}
}
*/
lemma DistributiveLemma(a: seq<bool>, b: seq<bool>)
ensures count(a + b) == count(a) + count(b)
{
if a == [] {
} else {
DistributiveLemma(a[1..], b);
}
}
function count(a: seq<bool>): nat
{
if |a| == 0 then 0 else
(if a[0] then 1 else 0) + count(a[1..])
}
lemma DistributiveIn(a: seq<int>, b:seq<int>, k:int, key:int)
requires |a| + 1 == |b|
requires 0 <= k <= |a|
requires b == a[..k] + [key] + a[k..]
ensures forall i :: 0 <= i < |a| ==> a[i] in b
{
}
lemma DistributiveGreater(a: seq<int>, b:seq<int>, k:int, key:int)
requires |a| + 1 == |b|
requires 0 <= k <= |a|
requires b == a[..k] + [key] + a[k..]
requires forall j :: 0 <= j < |a| ==> a[j] > 0
requires key > 0
ensures forall i :: 0 <= i < |b| ==> b[i] > 0
{
// assert ((forall j :: 0 <= j < k ==> b[j] > 0) && (forall j :: k <= j < |a| ==> b[j] > 0)) ==> (forall j :: 0 <= j < |b| ==> b[j] > 0);
}
// verifies in more than 45 seconds, but less than 100 seconds
method InsertIntoSorted(a: array<int>, limit:int, key:int) returns (b: array<int>)
requires key > 0
requires key !in a[..]
requires 0 <= limit < a.Length
requires forall i :: 0 <= i < limit ==> a[i] > 0
requires forall i :: limit <= i < a.Length ==> a[i] == 0
requires sorted(a[..limit])
ensures b.Length == a.Length
ensures sorted(b[..(limit+ 1)])
ensures forall i :: limit + 1 <= i < b.Length ==> b[i] == 0
ensures forall i :: 0 <= i < limit ==> a[i] in b[..]
ensures forall i :: 0 <= i < limit + 1 ==> b[i] > 0
{
b:= new int[a.Length];
ghost var k := 0;
b[0] := key;
ghost var a' := a[..];
var i:= 0;
while (i < limit)
modifies b
{
if(a[i]<key)
{
b[i]:= a[i];
b[i+1] := key;
k := i+1;
}
else if (a[i] >= key)
{
b[i+1] := a[i];
}
i := i+1;
}
// assert b[..limit+1] == a[..k] + [key] + a[k..limit];
DistributiveIn(a[..limit], b[..limit+1], k, key);
DistributiveGreater(a[..limit], b[..limit+1], k, key);
// assert forall i :: 0 <= i < limit + 1 ==> b[i] > 0;
ghost var b' := b[..];
i := limit + 1;
while i < b.Length
{
b[i] := 0;
i := i + 1;
}
}
|
007 | BinarySearchTree_tmp_tmp_bn2twp5_bst4copy.dfy | datatype Tree = Empty | Node(left: Tree, value: int, right: Tree)
predicate BinarySearchTree(tree: Tree)
decreases tree
{
match tree
case Empty => true
case Node(_,_,_) =>
(tree.left == Empty || tree.left.value < tree.value)
&& (tree.right == Empty || tree.right.value > tree.value)
&& BinarySearchTree(tree.left) && BinarySearchTree(tree.right)
&& minValue(tree.right, tree.value) && maxValue(tree.left, tree.value)
}
predicate maxValue(tree: Tree, max: int)
decreases tree
{
match tree
case Empty => true
case Node(left,v,right) => (max > v) && maxValue(left, max) && maxValue(right, max)
}
predicate minValue(tree: Tree, min: int)
decreases tree
{
match tree
case Empty => true
case Node(left,v,right) => (min < v) && minValue(left, min) && minValue(right, min)
}
method GetMin(tree: Tree) returns (res: int)
{
match tree {
case Empty => res := 0;
case Node (Empty, value, Empty) => res := tree.value;
case Node (Empty, value, right) => res := tree.value;
case Node (left, value, right) =>
var minval := tree.value;
minval := GetMin(tree.left);
var tmp := Node(tree.left, minval, tree.right);
res := tmp.value;
}
}
method GetMax(tree: Tree) returns (res: int){
match tree {
case Empty => res := 0;
case Node (Empty, value, Empty) => res := tree.value;
case Node (left, value, Empty) => res := tree.value;
case Node (left, value, right) =>
var minval := tree.value;
minval := GetMax(tree.right);
var tmp := Node(tree.left, minval, tree.right);
res := tmp.value;
}
}
method insert(tree: Tree, value : int) returns (res: Tree)
requires BinarySearchTree(tree)
decreases tree;
ensures BinarySearchTree(res)
{
res := insertRecursion(tree, value);
}
method insertRecursion(tree: Tree, value: int) returns (res: Tree)
requires BinarySearchTree(tree)
decreases tree;
ensures res != Empty ==> BinarySearchTree(res)
ensures forall x :: minValue(tree, x) && x < value ==> minValue(res, x)
ensures forall x :: maxValue(tree, x) && x > value ==> maxValue(res, x)
{
match tree {
case Empty => res := Node(Empty, value, Empty);
case Node(_,_,_) =>
var temp: Tree;
if(value == tree.value) {
return tree;
}
if(value < tree.value){
temp := insertRecursion(tree.left, value);
res := Node(temp, tree.value, tree.right);
}else if (value > tree.value){
temp := insertRecursion(tree.right, value);
res := Node(tree.left, tree.value, temp);
}
}
}
method delete(tree: Tree, value: int) returns (res: Tree)
requires BinarySearchTree(tree)
decreases tree;
//ensures BinarySearchTree(res)
//ensures res != Empty ==> BinarySearchTree(res)
{
match tree {
case Empty => return tree;
case Node(_,_ ,_) =>
var temp: Tree;
if (value < tree.value){
temp := delete(tree.left, value);
res := Node(temp, tree.value, tree.right);
} else if (value > tree.value){
temp := delete(tree.right, value);
res := Node(tree.left, tree.value, temp);
} else {
if (tree.left == Empty){
return tree.right;
} else if (tree.right == Empty) {
return tree.left;
}
var minVal := GetMin(tree.right);
temp := delete(tree.right, minVal);
res := Node(tree.left, minVal, temp);
//assert BinarySearchTree(res);
}
}
}
method Inorder(tree: Tree)
{
match tree {
case Empty =>
case Node(left, value, right) =>
Inorder(tree.left);
print tree.value, ", ";
Inorder(tree.right);
}
}
method Postorder(tree: Tree)
{
match tree {
case Empty =>
case Node(left, value, right) =>
Postorder(tree.left);
Postorder(tree.right);
print tree.value, ", ";
}
}
method Main() {
var tree := insert(Empty, 3);
var u := insert(tree, 2);
u := insert(u, 7);
u := insert(u, 6);
u := insert(u, 9);
print "This is Inorder: ";
Inorder(u);
print "\n";
//u := delete (u, 1);
print "This is Postorder: ";
Postorder(u);
print "\n";
print "tree before delete: ", u, "\n";
u := delete(u, 7);
print "tree after delete: ", u, "\n";
print "This is Inorder: ";
Inorder(u);
print "\n";
print "This is Postorder: ";
Postorder(u);
// var res := GetMin(u);
// var max := GetMax(u);
// print "this is max: ", max;
//print "this is res: ", res;
//print u;
}
| datatype Tree = Empty | Node(left: Tree, value: int, right: Tree)
predicate BinarySearchTree(tree: Tree)
{
match tree
case Empty => true
case Node(_,_,_) =>
(tree.left == Empty || tree.left.value < tree.value)
&& (tree.right == Empty || tree.right.value > tree.value)
&& BinarySearchTree(tree.left) && BinarySearchTree(tree.right)
&& minValue(tree.right, tree.value) && maxValue(tree.left, tree.value)
}
predicate maxValue(tree: Tree, max: int)
{
match tree
case Empty => true
case Node(left,v,right) => (max > v) && maxValue(left, max) && maxValue(right, max)
}
predicate minValue(tree: Tree, min: int)
{
match tree
case Empty => true
case Node(left,v,right) => (min < v) && minValue(left, min) && minValue(right, min)
}
method GetMin(tree: Tree) returns (res: int)
{
match tree {
case Empty => res := 0;
case Node (Empty, value, Empty) => res := tree.value;
case Node (Empty, value, right) => res := tree.value;
case Node (left, value, right) =>
var minval := tree.value;
minval := GetMin(tree.left);
var tmp := Node(tree.left, minval, tree.right);
res := tmp.value;
}
}
method GetMax(tree: Tree) returns (res: int){
match tree {
case Empty => res := 0;
case Node (Empty, value, Empty) => res := tree.value;
case Node (left, value, Empty) => res := tree.value;
case Node (left, value, right) =>
var minval := tree.value;
minval := GetMax(tree.right);
var tmp := Node(tree.left, minval, tree.right);
res := tmp.value;
}
}
method insert(tree: Tree, value : int) returns (res: Tree)
requires BinarySearchTree(tree)
ensures BinarySearchTree(res)
{
res := insertRecursion(tree, value);
}
method insertRecursion(tree: Tree, value: int) returns (res: Tree)
requires BinarySearchTree(tree)
ensures res != Empty ==> BinarySearchTree(res)
ensures forall x :: minValue(tree, x) && x < value ==> minValue(res, x)
ensures forall x :: maxValue(tree, x) && x > value ==> maxValue(res, x)
{
match tree {
case Empty => res := Node(Empty, value, Empty);
case Node(_,_,_) =>
var temp: Tree;
if(value == tree.value) {
return tree;
}
if(value < tree.value){
temp := insertRecursion(tree.left, value);
res := Node(temp, tree.value, tree.right);
}else if (value > tree.value){
temp := insertRecursion(tree.right, value);
res := Node(tree.left, tree.value, temp);
}
}
}
method delete(tree: Tree, value: int) returns (res: Tree)
requires BinarySearchTree(tree)
//ensures BinarySearchTree(res)
//ensures res != Empty ==> BinarySearchTree(res)
{
match tree {
case Empty => return tree;
case Node(_,_ ,_) =>
var temp: Tree;
if (value < tree.value){
temp := delete(tree.left, value);
res := Node(temp, tree.value, tree.right);
} else if (value > tree.value){
temp := delete(tree.right, value);
res := Node(tree.left, tree.value, temp);
} else {
if (tree.left == Empty){
return tree.right;
} else if (tree.right == Empty) {
return tree.left;
}
var minVal := GetMin(tree.right);
temp := delete(tree.right, minVal);
res := Node(tree.left, minVal, temp);
//assert BinarySearchTree(res);
}
}
}
method Inorder(tree: Tree)
{
match tree {
case Empty =>
case Node(left, value, right) =>
Inorder(tree.left);
print tree.value, ", ";
Inorder(tree.right);
}
}
method Postorder(tree: Tree)
{
match tree {
case Empty =>
case Node(left, value, right) =>
Postorder(tree.left);
Postorder(tree.right);
print tree.value, ", ";
}
}
method Main() {
var tree := insert(Empty, 3);
var u := insert(tree, 2);
u := insert(u, 7);
u := insert(u, 6);
u := insert(u, 9);
print "This is Inorder: ";
Inorder(u);
print "\n";
//u := delete (u, 1);
print "This is Postorder: ";
Postorder(u);
print "\n";
print "tree before delete: ", u, "\n";
u := delete(u, 7);
print "tree after delete: ", u, "\n";
print "This is Inorder: ";
Inorder(u);
print "\n";
print "This is Postorder: ";
Postorder(u);
// var res := GetMin(u);
// var max := GetMax(u);
// print "this is max: ", max;
//print "this is res: ", res;
//print u;
}
|
008 | CO3408-Advanced-Software-Modelling-Assignment-2022-23-Part-2-A-Specification-Spectacular_tmp_tmp4pj4p2zx_car_park.dfy | class {:autocontracts} CarPark {
const totalSpaces: nat := 10;
const normalSpaces: nat:= 7;
const reservedSpaces: nat := 3;
const badParkingBuffer: int := 5;
var weekend: bool;
var subscriptions: set<string>;
var carPark: set<string>;
var reservedCarPark: set<string>;
constructor()
requires true
ensures this.subscriptions == {} && this.carPark == {} && this.reservedCarPark == {} && this.weekend == false;
{
this.subscriptions := {};
this.carPark := {};
this.reservedCarPark := {};
this.weekend := false;
}
// This predicate checks if the car park is in a valid state at all times.
// It checks if the sets of cars in the car park and the reserved car park are disjoint and share no values,
// the total number of cars in the car park is less than or equal to the total number of spaces in
// the car park plus the bad parking buffer, the number of normal spaces plus reserved spaces is
// equal to the total number of spaces, and the number of cars in the reserved car park is less than or equal
// to the number of reserved spaces
ghost predicate Valid()
reads this
{
carPark * reservedCarPark == {} && |carPark| <= totalSpaces + badParkingBuffer && (normalSpaces + reservedSpaces) == totalSpaces && |reservedCarPark| <= reservedSpaces
}
// The method maintains the invariant that if success is true, then the car parameter is removed from either
// the carPark or the reservedCarPark set. Otherwise, neither set is modified
method leaveCarPark(car: string) returns (success: bool)
requires true
modifies this
ensures success ==> (((car in old(carPark)) && carPark == old(carPark) - {car} && reservedCarPark == old(reservedCarPark)) || ((car in old(reservedCarPark)) && reservedCarPark == old(reservedCarPark) - {car} && carPark == old(carPark)));
ensures success ==> (car !in carPark) && (car !in reservedCarPark);
ensures !success ==> carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && (car !in old(carPark)) && (car !in old(reservedCarPark));
ensures subscriptions == old(subscriptions) && weekend == old(weekend);
{
success := false;
if car in carPark {
carPark := carPark - {car};
success := true;
} else if car in reservedCarPark {
reservedCarPark := reservedCarPark - {car};
success := true;
}
}
// The method maintains the invariant that the number of available spaces availableSpaces is updated correctly
// based on the current state of the car park and whether it is a weekend or not
method checkAvailability() returns (availableSpaces: int)
requires true
modifies this
ensures weekend ==> availableSpaces == (normalSpaces - old(|carPark|)) + (reservedSpaces - old(|reservedCarPark|)) - badParkingBuffer;
ensures !weekend ==> availableSpaces == (normalSpaces - old(|carPark|)) - badParkingBuffer;
ensures carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && weekend == old(weekend) && subscriptions == old(subscriptions);
{
if (weekend){
availableSpaces := (normalSpaces - |carPark|) + (reservedSpaces - |reservedCarPark|) - badParkingBuffer;
} else{
availableSpaces := (normalSpaces - |carPark|) - badParkingBuffer;
}
}
// The method maintains the invariant that if success is true, then the car parameter is added to the
// subscriptions set. Otherwise, the subscriptions set is not modified
method makeSubscription(car: string) returns (success: bool)
requires true
modifies this
ensures success ==> old(|subscriptions|) < reservedSpaces && car !in old(subscriptions) && subscriptions == old(subscriptions) + {car};
ensures !success ==> subscriptions == old(subscriptions) && (car in old(subscriptions) || old(|subscriptions|) >= reservedSpaces);
ensures carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && weekend == old(weekend);
{
if |subscriptions| >= reservedSpaces || car in subscriptions {
success := false;
} else {
subscriptions := subscriptions + {car};
success := true;
}
}
// The method maintains the invariant that the weekend variable is set to true
method openReservedArea()
requires true
modifies this
ensures carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && weekend == true && subscriptions == old(subscriptions);
{
weekend := true;
}
// The method maintains the invariant that the carPark, reservedCarPark, and subscriptions sets are all cleared
method closeCarPark()
requires true
modifies this
ensures carPark == {} && reservedCarPark == {} && subscriptions == {}
ensures weekend == old(weekend);
{
carPark := {};
reservedCarPark := {};
subscriptions := {};
}
// The method maintains the invariant that if success is true, then the car parameter is added to the carPark
// set and the number of cars in the carPark set is less than the number of normal spaces minus the bad parking
// buffer. Otherwise, the carPark and reservedCarPark sets are not modified
method enterCarPark(car: string) returns (success: bool)
requires true
modifies this;
ensures success ==> (car !in old(carPark)) && (car !in old(reservedCarPark)) && (old(|carPark|) < normalSpaces - badParkingBuffer);
ensures success ==> carPark == old(carPark) + {car};
ensures !success ==> carPark == old(carPark) && reservedCarPark == old(reservedCarPark);
ensures !success ==> (car in old(carPark)) || (car in old(reservedCarPark) || (old(|carPark|) >= normalSpaces - badParkingBuffer));
ensures subscriptions == old(subscriptions) && reservedCarPark == old(reservedCarPark) && weekend == old(weekend);
{
if (|carPark| >= normalSpaces - badParkingBuffer || car in carPark || car in reservedCarPark) {
return false;
}
else
{
carPark := carPark + {car};
return true;
}
}
// The method maintains the invariant that if success is true, then the car parameter is added to the
// reservedCarPark set and the number of cars in the reservedCarPark set is less than the number of
// reserved spaces and either the weekend variable is true or the car parameter is in the subscriptions set.
// Otherwise, the carPark and reservedCarPark sets are not modified
method enterReservedCarPark(car: string) returns (success: bool)
requires true
modifies this;
ensures success ==> (car !in old(carPark)) && (car !in old(reservedCarPark)) && (old(|reservedCarPark|) < reservedSpaces) && (car in subscriptions || weekend == true);
ensures success ==> reservedCarPark == old(reservedCarPark) + {car};
ensures !success ==> carPark == old(carPark) && reservedCarPark == old(reservedCarPark);
ensures !success ==> (car in old(carPark)) || (car in old(reservedCarPark) || (old(|reservedCarPark|) >= reservedSpaces) || (car !in subscriptions && weekend == false));
ensures subscriptions == old(subscriptions) && carPark == old(carPark) && weekend == old(weekend);
ensures weekend == old(weekend) && subscriptions == old(subscriptions);
{
if (|reservedCarPark| >= reservedSpaces || car in carPark || car in reservedCarPark || (car !in subscriptions && weekend == false)) {
return false;
}
else
{
reservedCarPark := reservedCarPark + {car};
return true;
}
}
}
method Main() {
// Initialises car park with 10 spaces, 3 of which are reserved and therefore 7 are normal
var carPark := new CarPark();
// As we are allowing 5 spaces for idiots who can't park within the lines 7 - 5 == 2
var availableSpaces := carPark.checkAvailability();
assert availableSpaces == 2;
// Test entering the car park with one car, One space should now be left
var success := carPark.enterCarPark("car1");
availableSpaces := carPark.checkAvailability();
assert success && carPark.carPark == {"car1"} && availableSpaces == 1;
// Test entering the car with another car, No spaces should be left
success := carPark.enterCarPark("car2");
availableSpaces := carPark.checkAvailability();
assert success && "car2" in carPark.carPark && carPark.carPark == {"car1", "car2"} && availableSpaces == 0;
// Test entering with another car, should return invalid as carpark is full
// normalSpaces = 7, minus 5 spaces because of the bad parking buffer, therefore 2 spaces max
success := carPark.enterCarPark("car3");
assert !success && carPark.carPark == {"car1", "car2"} && carPark.reservedCarPark == {};
// Test creating car subscription
success := carPark.makeSubscription("car4");
assert success && carPark.subscriptions == {"car4"};
// Test entering the reserved carPark with a valid and an invalid option
success := carPark.enterReservedCarPark("car4");
assert success && carPark.reservedCarPark == {"car4"};
// This car doesn't have a subscription so it should not be successful
success := carPark.enterReservedCarPark("car5");
assert !success && carPark.reservedCarPark == {"car4"};
// Test filling the car subscription list
success := carPark.makeSubscription("car6");
assert success && carPark.subscriptions == {"car4", "car6"};
success := carPark.makeSubscription("car7");
assert success && carPark.subscriptions == {"car4", "car6", "car7"};
// This won't add as reserved spaces are 3 and we can't have more subscriptions than reserved spaces
success := carPark.makeSubscription("car8");
assert !success && carPark.subscriptions == {"car4", "car6", "car7"};
// Test filling reserved car park
success := carPark.enterReservedCarPark("car6");
assert success && carPark.reservedCarPark == {"car4", "car6"};
success := carPark.enterReservedCarPark("car7");
assert success && carPark.reservedCarPark == {"car4", "car6", "car7"};
// Test leaving car park
assert carPark.carPark == {"car1", "car2"};
success := carPark.leaveCarPark("car1");
assert success && carPark.carPark == {"car2"} && carPark.reservedCarPark == {"car4", "car6", "car7"};
// Test leaving with car that doesn't exist
assert "car9" !in carPark.carPark && "car9" !in carPark.reservedCarPark;
success := carPark.leaveCarPark("car9");
assert !success && carPark.carPark == {"car2"} && carPark.reservedCarPark == {"car4", "car6", "car7"};
// Test leaving reserved car park
success := carPark.leaveCarPark("car6");
assert success && carPark.carPark == {"car2"} && carPark.reservedCarPark == {"car4", "car7"};
// Testing closing car park, all cars should be destroyed
carPark.closeCarPark();
assert carPark.carPark == {} && carPark.reservedCarPark == {} && carPark.subscriptions == {};
}
// Added due to timeout in Main
method MainB () {
var carPark := new CarPark();
// Test opening the reserved carPark
assert carPark.weekend == false;
carPark.openReservedArea();
assert carPark.weekend == true;
// Test joining carPark on weekend with car without subscription
var success := carPark.enterReservedCarPark("car3");
assert "car3" !in carPark.subscriptions && success && carPark.carPark == {} && carPark.reservedCarPark == {"car3"};
// Testing closing car park, all cars should be destroyed
carPark.closeCarPark();
assert carPark.carPark == {} && carPark.reservedCarPark == {} && carPark.subscriptions == {};
}
| class {:autocontracts} CarPark {
const totalSpaces: nat := 10;
const normalSpaces: nat:= 7;
const reservedSpaces: nat := 3;
const badParkingBuffer: int := 5;
var weekend: bool;
var subscriptions: set<string>;
var carPark: set<string>;
var reservedCarPark: set<string>;
constructor()
requires true
ensures this.subscriptions == {} && this.carPark == {} && this.reservedCarPark == {} && this.weekend == false;
{
this.subscriptions := {};
this.carPark := {};
this.reservedCarPark := {};
this.weekend := false;
}
// This predicate checks if the car park is in a valid state at all times.
// It checks if the sets of cars in the car park and the reserved car park are disjoint and share no values,
// the total number of cars in the car park is less than or equal to the total number of spaces in
// the car park plus the bad parking buffer, the number of normal spaces plus reserved spaces is
// equal to the total number of spaces, and the number of cars in the reserved car park is less than or equal
// to the number of reserved spaces
ghost predicate Valid()
reads this
{
carPark * reservedCarPark == {} && |carPark| <= totalSpaces + badParkingBuffer && (normalSpaces + reservedSpaces) == totalSpaces && |reservedCarPark| <= reservedSpaces
}
// The method maintains the invariant that if success is true, then the car parameter is removed from either
// the carPark or the reservedCarPark set. Otherwise, neither set is modified
method leaveCarPark(car: string) returns (success: bool)
requires true
modifies this
ensures success ==> (((car in old(carPark)) && carPark == old(carPark) - {car} && reservedCarPark == old(reservedCarPark)) || ((car in old(reservedCarPark)) && reservedCarPark == old(reservedCarPark) - {car} && carPark == old(carPark)));
ensures success ==> (car !in carPark) && (car !in reservedCarPark);
ensures !success ==> carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && (car !in old(carPark)) && (car !in old(reservedCarPark));
ensures subscriptions == old(subscriptions) && weekend == old(weekend);
{
success := false;
if car in carPark {
carPark := carPark - {car};
success := true;
} else if car in reservedCarPark {
reservedCarPark := reservedCarPark - {car};
success := true;
}
}
// The method maintains the invariant that the number of available spaces availableSpaces is updated correctly
// based on the current state of the car park and whether it is a weekend or not
method checkAvailability() returns (availableSpaces: int)
requires true
modifies this
ensures weekend ==> availableSpaces == (normalSpaces - old(|carPark|)) + (reservedSpaces - old(|reservedCarPark|)) - badParkingBuffer;
ensures !weekend ==> availableSpaces == (normalSpaces - old(|carPark|)) - badParkingBuffer;
ensures carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && weekend == old(weekend) && subscriptions == old(subscriptions);
{
if (weekend){
availableSpaces := (normalSpaces - |carPark|) + (reservedSpaces - |reservedCarPark|) - badParkingBuffer;
} else{
availableSpaces := (normalSpaces - |carPark|) - badParkingBuffer;
}
}
// The method maintains the invariant that if success is true, then the car parameter is added to the
// subscriptions set. Otherwise, the subscriptions set is not modified
method makeSubscription(car: string) returns (success: bool)
requires true
modifies this
ensures success ==> old(|subscriptions|) < reservedSpaces && car !in old(subscriptions) && subscriptions == old(subscriptions) + {car};
ensures !success ==> subscriptions == old(subscriptions) && (car in old(subscriptions) || old(|subscriptions|) >= reservedSpaces);
ensures carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && weekend == old(weekend);
{
if |subscriptions| >= reservedSpaces || car in subscriptions {
success := false;
} else {
subscriptions := subscriptions + {car};
success := true;
}
}
// The method maintains the invariant that the weekend variable is set to true
method openReservedArea()
requires true
modifies this
ensures carPark == old(carPark) && reservedCarPark == old(reservedCarPark) && weekend == true && subscriptions == old(subscriptions);
{
weekend := true;
}
// The method maintains the invariant that the carPark, reservedCarPark, and subscriptions sets are all cleared
method closeCarPark()
requires true
modifies this
ensures carPark == {} && reservedCarPark == {} && subscriptions == {}
ensures weekend == old(weekend);
{
carPark := {};
reservedCarPark := {};
subscriptions := {};
}
// The method maintains the invariant that if success is true, then the car parameter is added to the carPark
// set and the number of cars in the carPark set is less than the number of normal spaces minus the bad parking
// buffer. Otherwise, the carPark and reservedCarPark sets are not modified
method enterCarPark(car: string) returns (success: bool)
requires true
modifies this;
ensures success ==> (car !in old(carPark)) && (car !in old(reservedCarPark)) && (old(|carPark|) < normalSpaces - badParkingBuffer);
ensures success ==> carPark == old(carPark) + {car};
ensures !success ==> carPark == old(carPark) && reservedCarPark == old(reservedCarPark);
ensures !success ==> (car in old(carPark)) || (car in old(reservedCarPark) || (old(|carPark|) >= normalSpaces - badParkingBuffer));
ensures subscriptions == old(subscriptions) && reservedCarPark == old(reservedCarPark) && weekend == old(weekend);
{
if (|carPark| >= normalSpaces - badParkingBuffer || car in carPark || car in reservedCarPark) {
return false;
}
else
{
carPark := carPark + {car};
return true;
}
}
// The method maintains the invariant that if success is true, then the car parameter is added to the
// reservedCarPark set and the number of cars in the reservedCarPark set is less than the number of
// reserved spaces and either the weekend variable is true or the car parameter is in the subscriptions set.
// Otherwise, the carPark and reservedCarPark sets are not modified
method enterReservedCarPark(car: string) returns (success: bool)
requires true
modifies this;
ensures success ==> (car !in old(carPark)) && (car !in old(reservedCarPark)) && (old(|reservedCarPark|) < reservedSpaces) && (car in subscriptions || weekend == true);
ensures success ==> reservedCarPark == old(reservedCarPark) + {car};
ensures !success ==> carPark == old(carPark) && reservedCarPark == old(reservedCarPark);
ensures !success ==> (car in old(carPark)) || (car in old(reservedCarPark) || (old(|reservedCarPark|) >= reservedSpaces) || (car !in subscriptions && weekend == false));
ensures subscriptions == old(subscriptions) && carPark == old(carPark) && weekend == old(weekend);
ensures weekend == old(weekend) && subscriptions == old(subscriptions);
{
if (|reservedCarPark| >= reservedSpaces || car in carPark || car in reservedCarPark || (car !in subscriptions && weekend == false)) {
return false;
}
else
{
reservedCarPark := reservedCarPark + {car};
return true;
}
}
}
method Main() {
// Initialises car park with 10 spaces, 3 of which are reserved and therefore 7 are normal
var carPark := new CarPark();
// As we are allowing 5 spaces for idiots who can't park within the lines 7 - 5 == 2
var availableSpaces := carPark.checkAvailability();
// Test entering the car park with one car, One space should now be left
var success := carPark.enterCarPark("car1");
availableSpaces := carPark.checkAvailability();
// Test entering the car with another car, No spaces should be left
success := carPark.enterCarPark("car2");
availableSpaces := carPark.checkAvailability();
// Test entering with another car, should return invalid as carpark is full
// normalSpaces = 7, minus 5 spaces because of the bad parking buffer, therefore 2 spaces max
success := carPark.enterCarPark("car3");
// Test creating car subscription
success := carPark.makeSubscription("car4");
// Test entering the reserved carPark with a valid and an invalid option
success := carPark.enterReservedCarPark("car4");
// This car doesn't have a subscription so it should not be successful
success := carPark.enterReservedCarPark("car5");
// Test filling the car subscription list
success := carPark.makeSubscription("car6");
success := carPark.makeSubscription("car7");
// This won't add as reserved spaces are 3 and we can't have more subscriptions than reserved spaces
success := carPark.makeSubscription("car8");
// Test filling reserved car park
success := carPark.enterReservedCarPark("car6");
success := carPark.enterReservedCarPark("car7");
// Test leaving car park
success := carPark.leaveCarPark("car1");
// Test leaving with car that doesn't exist
success := carPark.leaveCarPark("car9");
// Test leaving reserved car park
success := carPark.leaveCarPark("car6");
// Testing closing car park, all cars should be destroyed
carPark.closeCarPark();
}
// Added due to timeout in Main
method MainB () {
var carPark := new CarPark();
// Test opening the reserved carPark
carPark.openReservedArea();
// Test joining carPark on weekend with car without subscription
var success := carPark.enterReservedCarPark("car3");
// Testing closing car park, all cars should be destroyed
carPark.closeCarPark();
}
|
009 | CS494-final-project_tmp_tmp7nof55uq_bubblesort.dfy | //Bubblesort CS 494 submission
//References: https://stackoverflow.com/questions/69364687/how-to-prove-time-complexity-of-bubble-sort-using-dafny/69365785#69365785
// predicate checks if elements of a are in ascending order, two additional conditions are added to allow us to sort in specific range within array
predicate sorted(a:array<int>, from:int, to:int)
requires a != null; // requires array to have n amount of elements
reads a;
requires 0 <= from <= to <= a.Length; // pre condition checks that from is the start of the range and to is the end of the range, requires values to be within 0 - a.Length
{
forall x, y :: from <= x < y < to ==> a[x] <= a[y]
}
//helps ensure swapping is valid, it is used inside the nested while loop to make sure linear order is being kept
predicate pivot(a:array<int>, to:int, pvt:int)
requires a != null; // requires array to have n amount of elements
reads a;
requires 0 <= pvt < to <= a.Length;
{
forall x, y :: 0 <= x < pvt < y < to ==> a[x] <= a[y] // all values within the array should be in ascending order
}
// Here having the algorithm for the bubblesort
method BubbleSort (a: array<int>)
requires a != null && a.Length > 0; // makes sure a is not empty and length is greater than 0
modifies a; // as method runs, we are changing a
ensures sorted(a, 0, a.Length); // makes sure elements of array a are sorted from 0 - a.Length
ensures multiset(a[..]) == multiset(old(a[..])); // Since a is being modified, we deference the heap
//and compare the previous elements to current elements.
{
var i := 1;
while (i < a.Length)
invariant i <= a.Length; // more-or-less validates while loop condition during coputations
invariant sorted(a, 0, i); // Checks that for each increment of i, the array stays sorted, causing the
invariant multiset(a[..]) == multiset(old(a[..])); //makes sure elements that existed in previous heap for a are presnt in current run
{
var j := i;
//this while loop inherits any previous pre/post conditions. It checks that
while (j > 0)
invariant multiset(a[..]) == multiset(old(a[..]));
invariant sorted(a, 0, j); // O(n^2) runtime. Makes sure that a[0] - a[j] is sorted
invariant sorted(a, j, i+1); // then makes sure from a[j] - a[i+1] is sorted
invariant pivot(a, i+1, j); // important for ensuring that each computation is correct after swapping
{
// Here it also simplifies the remaining invariants to handle the empty array.
if (a[j-1] > a[j]) { // reverse iterate through range within the array
a[j - 1], a[j] := a[j], a[j - 1]; // swaps objects if the IF condition is met
}
j := j - 1; // decrement j
}
i := i+1; // increment i
}
}
| //Bubblesort CS 494 submission
//References: https://stackoverflow.com/questions/69364687/how-to-prove-time-complexity-of-bubble-sort-using-dafny/69365785#69365785
// predicate checks if elements of a are in ascending order, two additional conditions are added to allow us to sort in specific range within array
predicate sorted(a:array<int>, from:int, to:int)
requires a != null; // requires array to have n amount of elements
reads a;
requires 0 <= from <= to <= a.Length; // pre condition checks that from is the start of the range and to is the end of the range, requires values to be within 0 - a.Length
{
forall x, y :: from <= x < y < to ==> a[x] <= a[y]
}
//helps ensure swapping is valid, it is used inside the nested while loop to make sure linear order is being kept
predicate pivot(a:array<int>, to:int, pvt:int)
requires a != null; // requires array to have n amount of elements
reads a;
requires 0 <= pvt < to <= a.Length;
{
forall x, y :: 0 <= x < pvt < y < to ==> a[x] <= a[y] // all values within the array should be in ascending order
}
// Here having the algorithm for the bubblesort
method BubbleSort (a: array<int>)
requires a != null && a.Length > 0; // makes sure a is not empty and length is greater than 0
modifies a; // as method runs, we are changing a
ensures sorted(a, 0, a.Length); // makes sure elements of array a are sorted from 0 - a.Length
ensures multiset(a[..]) == multiset(old(a[..])); // Since a is being modified, we deference the heap
//and compare the previous elements to current elements.
{
var i := 1;
while (i < a.Length)
{
var j := i;
//this while loop inherits any previous pre/post conditions. It checks that
while (j > 0)
{
// Here it also simplifies the remaining invariants to handle the empty array.
if (a[j-1] > a[j]) { // reverse iterate through range within the array
a[j - 1], a[j] := a[j], a[j - 1]; // swaps objects if the IF condition is met
}
j := j - 1; // decrement j
}
i := i+1; // increment i
}
}
|
010 | CS5232_Project_tmp_tmpai_cfrng_LFUSimple.dfy | class LFUCache {
var capacity : int;
var cacheMap : map<int, (int, int)>; //key -> {value, freq}
constructor(capacity: int)
requires capacity > 0;
ensures Valid();
{
this.capacity := capacity;
this.cacheMap := map[];
}
predicate Valid()
reads this;
// reads this.freqMap.Values;
{
// general value check
this.capacity > 0 &&
0 <= |cacheMap| <= capacity &&
(|cacheMap| > 0 ==> (forall e :: e in cacheMap ==> cacheMap[e].1 >= 1)) && // frequency should always larger than 0
(|cacheMap| > 0 ==> (forall e :: e in cacheMap ==> cacheMap[e].0 >= 0)) // only allow positive values
}
method getLFUKey() returns (lfuKey : int)
requires Valid();
requires |cacheMap| > 0;
ensures Valid();
ensures lfuKey in cacheMap;
ensures forall k :: k in cacheMap.Items ==> cacheMap[lfuKey].1 <= cacheMap[k.0].1;
{
var items := cacheMap.Items;
var seenItems := {};
var anyItem :| anyItem in items;
var minFreq := anyItem.1.1;
lfuKey := anyItem.0;
while items != {}
decreases |items|;
invariant cacheMap.Items >= items;
invariant cacheMap.Items >= seenItems;
invariant cacheMap.Items == seenItems + items;
invariant lfuKey in cacheMap;
invariant cacheMap[lfuKey].1 == minFreq;
invariant forall e :: e in seenItems ==> minFreq <= e.1.1;
invariant forall e :: e in seenItems ==> minFreq <= cacheMap[e.0].1;
invariant forall e :: e in seenItems ==> cacheMap[lfuKey].1 <= cacheMap[e.0].1;
invariant exists e :: e in seenItems + items ==> minFreq == e.1.1;
{
var item :| item in items;
if (item.1.1 < minFreq) {
lfuKey := item.0;
minFreq := item.1.1;
}
items := items - { item };
seenItems := seenItems + { item };
}
assert seenItems == cacheMap.Items;
assert cacheMap[lfuKey].1 == minFreq;
assert forall e :: e in seenItems ==> minFreq <= e.1.1;
assert forall e :: e in cacheMap.Items ==> minFreq <= e.1.1;
assert forall k :: k in seenItems ==> cacheMap[lfuKey].1 <= cacheMap[k.0].1;
assert forall k :: k in cacheMap.Items ==> cacheMap[lfuKey].1 <= cacheMap[k.0].1;
// assert forall k :: k in cacheMap ==> cacheMap[lfuKey].1 <= cacheMap[k].1; // ????
return lfuKey;
}
method get(key: int) returns (value: int)
requires Valid();
modifies this;
ensures Valid();
ensures key !in cacheMap ==> value == -1;
ensures forall e :: e in old(cacheMap) <==> e in cacheMap;
ensures forall e :: e in old(cacheMap) ==> (old(cacheMap[e].0) == cacheMap[e].0);
ensures key in cacheMap ==> value == cacheMap[key].0 && old(cacheMap[key].1) == cacheMap[key].1-1;
{
assert key in cacheMap ==> cacheMap[key].0 >= 0;
if(key !in cacheMap) {
value := -1;
}
else{
assert key in cacheMap;
assert cacheMap[key].0 >= 0;
value := cacheMap[key].0;
var oldFreq := cacheMap[key].1;
var newV := (value, oldFreq + 1);
cacheMap := cacheMap[key := newV];
}
print "after get: ";
print cacheMap;
print "\n";
return value;
}
method put(key: int, value: int)
requires Valid();
requires value > 0;
modifies this
ensures Valid();
{
if (key in cacheMap) {
var currFreq := cacheMap[key].1;
cacheMap := cacheMap[key := (value, currFreq)];
} else {
if (|cacheMap| < capacity) {
cacheMap := cacheMap[key := (value, 1)];
} else {
var LFUKey := getLFUKey();
assert LFUKey in cacheMap;
assert |cacheMap| == capacity;
ghost var oldMap := cacheMap;
var newMap := cacheMap - {LFUKey};
cacheMap := newMap;
assert newMap == cacheMap - {LFUKey};
assert LFUKey !in cacheMap;
assert LFUKey in oldMap;
ghost var oldCard := |oldMap|;
ghost var newCard := |newMap|;
assert |cacheMap.Keys| < |oldMap|; // ????
cacheMap := cacheMap[key := (value, 1)];
}
}
print "after put: ";
print cacheMap;
print "\n";
}
}
method Main()
{
var LFUCache := new LFUCache(5);
print "Cache Capacity = 5 \n";
print "PUT (1, 1) - ";
LFUCache.put(1,1);
print "PUT (2, 2) - ";
LFUCache.put(2,2);
print "PUT (3, 3) - ";
LFUCache.put(3,3);
print "GET (1) - ";
var val := LFUCache.get(1);
print "get(1) = ";
print val;
print "\n";
print "PUT (3, 5) - ";
LFUCache.put(3,5);
print "GET (3) - ";
val := LFUCache.get(3);
print "get(3) = ";
print val;
print "\n";
print "PUT (4, 6) - ";
LFUCache.put(4,6);
print "PUT (5, 7) - ";
LFUCache.put(5,7);
print "PUT (10, 100) - ";
LFUCache.put(10,100);
print "GET (2) - ";
val := LFUCache.get(2);
print "get(2) = ";
print val;
print "\n";
}
| class LFUCache {
var capacity : int;
var cacheMap : map<int, (int, int)>; //key -> {value, freq}
constructor(capacity: int)
requires capacity > 0;
ensures Valid();
{
this.capacity := capacity;
this.cacheMap := map[];
}
predicate Valid()
reads this;
// reads this.freqMap.Values;
{
// general value check
this.capacity > 0 &&
0 <= |cacheMap| <= capacity &&
(|cacheMap| > 0 ==> (forall e :: e in cacheMap ==> cacheMap[e].1 >= 1)) && // frequency should always larger than 0
(|cacheMap| > 0 ==> (forall e :: e in cacheMap ==> cacheMap[e].0 >= 0)) // only allow positive values
}
method getLFUKey() returns (lfuKey : int)
requires Valid();
requires |cacheMap| > 0;
ensures Valid();
ensures lfuKey in cacheMap;
ensures forall k :: k in cacheMap.Items ==> cacheMap[lfuKey].1 <= cacheMap[k.0].1;
{
var items := cacheMap.Items;
var seenItems := {};
var anyItem :| anyItem in items;
var minFreq := anyItem.1.1;
lfuKey := anyItem.0;
while items != {}
{
var item :| item in items;
if (item.1.1 < minFreq) {
lfuKey := item.0;
minFreq := item.1.1;
}
items := items - { item };
seenItems := seenItems + { item };
}
// assert forall k :: k in cacheMap ==> cacheMap[lfuKey].1 <= cacheMap[k].1; // ????
return lfuKey;
}
method get(key: int) returns (value: int)
requires Valid();
modifies this;
ensures Valid();
ensures key !in cacheMap ==> value == -1;
ensures forall e :: e in old(cacheMap) <==> e in cacheMap;
ensures forall e :: e in old(cacheMap) ==> (old(cacheMap[e].0) == cacheMap[e].0);
ensures key in cacheMap ==> value == cacheMap[key].0 && old(cacheMap[key].1) == cacheMap[key].1-1;
{
if(key !in cacheMap) {
value := -1;
}
else{
value := cacheMap[key].0;
var oldFreq := cacheMap[key].1;
var newV := (value, oldFreq + 1);
cacheMap := cacheMap[key := newV];
}
print "after get: ";
print cacheMap;
print "\n";
return value;
}
method put(key: int, value: int)
requires Valid();
requires value > 0;
modifies this
ensures Valid();
{
if (key in cacheMap) {
var currFreq := cacheMap[key].1;
cacheMap := cacheMap[key := (value, currFreq)];
} else {
if (|cacheMap| < capacity) {
cacheMap := cacheMap[key := (value, 1)];
} else {
var LFUKey := getLFUKey();
ghost var oldMap := cacheMap;
var newMap := cacheMap - {LFUKey};
cacheMap := newMap;
ghost var oldCard := |oldMap|;
ghost var newCard := |newMap|;
cacheMap := cacheMap[key := (value, 1)];
}
}
print "after put: ";
print cacheMap;
print "\n";
}
}
method Main()
{
var LFUCache := new LFUCache(5);
print "Cache Capacity = 5 \n";
print "PUT (1, 1) - ";
LFUCache.put(1,1);
print "PUT (2, 2) - ";
LFUCache.put(2,2);
print "PUT (3, 3) - ";
LFUCache.put(3,3);
print "GET (1) - ";
var val := LFUCache.get(1);
print "get(1) = ";
print val;
print "\n";
print "PUT (3, 5) - ";
LFUCache.put(3,5);
print "GET (3) - ";
val := LFUCache.get(3);
print "get(3) = ";
print val;
print "\n";
print "PUT (4, 6) - ";
LFUCache.put(4,6);
print "PUT (5, 7) - ";
LFUCache.put(5,7);
print "PUT (10, 100) - ";
LFUCache.put(10,100);
print "GET (2) - ";
val := LFUCache.get(2);
print "get(2) = ";
print val;
print "\n";
}
|
011 | CS5232_Project_tmp_tmpai_cfrng_test.dfy | iterator Gen(start: int) yields (x: int)
yield ensures |xs| <= 10 && x == start + |xs| - 1
{
var i := 0;
while i < 10 invariant |xs| == i {
x := start + i;
yield;
i := i + 1;
}
}
method Main() {
var i := new Gen(30);
while true
invariant i.Valid() && fresh(i._new)
decreases 10 - |i.xs|
{
var m := i.MoveNext();
if (!m) {break; }
print i.x;
}
}
| iterator Gen(start: int) yields (x: int)
yield ensures |xs| <= 10 && x == start + |xs| - 1
{
var i := 0;
while i < 10 invariant |xs| == i {
x := start + i;
yield;
i := i + 1;
}
}
method Main() {
var i := new Gen(30);
while true
{
var m := i.MoveNext();
if (!m) {break; }
print i.x;
}
}
|
012 | CSC8204-Dafny_tmp_tmp11yhjb53_stack.dfy | /*
Dafny Tutorial 2: Sequences and Stacks, Predicates and Assertions
In this tutorial we introduce a simple stack model using the functional
style of programming.
*/
type intStack = seq<int>
function isEmpty(s: intStack): bool
{
|s| == 0
}
function push(s: intStack, x: int): intStack
{
s + [x]
}
function pop(s: intStack): intStack
requires !isEmpty(s)
{
s[..|s|-1]
}
method testStack() returns (r: intStack)
{
var s: intStack := [20, 30, 15, 40, 60, 100, 80];
assert pop(push(s,100)) == s;
assert forall e: int :: 0 <= e < |s| ==> s[e] > 5;
r:= s;
}
method Main()
{
var t:=testStack();
print "Stack tested\nStack is ", t, "\n";
}
| /*
Dafny Tutorial 2: Sequences and Stacks, Predicates and Assertions
In this tutorial we introduce a simple stack model using the functional
style of programming.
*/
type intStack = seq<int>
function isEmpty(s: intStack): bool
{
|s| == 0
}
function push(s: intStack, x: int): intStack
{
s + [x]
}
function pop(s: intStack): intStack
requires !isEmpty(s)
{
s[..|s|-1]
}
method testStack() returns (r: intStack)
{
var s: intStack := [20, 30, 15, 40, 60, 100, 80];
r:= s;
}
method Main()
{
var t:=testStack();
print "Stack tested\nStack is ", t, "\n";
}
|
013 | CSU55004---Formal-Verification_tmp_tmp4ki9iaqy_Project_Project_Part_1_project_pt_1.dfy | //This method should return true iff pre is a prefix of str. That is, str starts with pre
method isPrefix(pre:string, str:string) returns(res:bool)
requires 0 < |pre| <= |str| //This line states that this method requires that pre is less than or equal in length to str. Without this line, an out of bounds error is shown on line 14: "str[i] != pre[i]"
{
//Initialising the index variable
var i := 0;
//Iterating through the first |pre| elements in str
while (i < |pre|)
invariant 0 <= i <= |pre| //Specifying the range of the while loop
decreases |pre| - i //Specifying that the while loop will terminate
{
//If an element does not match, return false
if (str[i] != pre[i]) {
//Debug print
print str[i], " != ", pre[i], "\n";
//Return once mismatch detected, no point in iterating any further
return false;
}
//Else loop until mismatch found or we have reached the end of pre
else{
//Debug pront
print str[i], " == ", pre[i], "\n";
i := i + 1;
}
}
return true;
}
//This method should return true iff sub is a substring of str. That is, str contains sub
method isSubstring(sub:string, str:string) returns(res:bool)
requires 0 < |sub| <= |str| //This method requires that sub is less than or equal in length to str
{
//Initialising the index variable
var i := 0;
//This variable stores the difference in length between the two strings
var n := (|str| - |sub|);
//Here, we want to re-use the "isPrefix" method above, so with each iteration of the search, we are passing an offset of str - effectively trimming a character off the front of str and passing it to isPrefix
//example 1 (sub found in str):
//str = door & sub = or
//iteration 1: isPrefix(or, door), returns false, trim & iterate again
//iteration 2: isprefix(or, oor), returns false, trim & iterate again
//iteration 3: isPrefix(or, or), returns true, stop iterating
//example 2 (sub not found in str):
//str = doom & sub = or
//iteration 1: isPrefix(or, doom), returns false, trim & iterate again
//iteration 2: isprefix(or, oom), returns false, trim & iterate again
//iteration 3: isPrefix(or, om), returns false, str is has not been "trimmed" to the same length as sub, so we stop iterating
while(i < n+1)
invariant 0 <= i <= n+1 //Specifying the range of the while loop
decreases n - i //Specifying that the while loop will terminate
{
//Debug print to show what is being passed to isPrefix with each iteration
print "\n", sub, ", ", str[i..|str|], "\n";
var result:= isPrefix(sub, str[i..|str|]);
//Return once the substring is found, no point in iterating any further
if(result == true){
return true;
}
//Else loop until sub is found, or we have reached the end of str
else{
i := i+1;
}
}
return false;
}
//This method should return true iff str1 and str1 have a common substring of length k
method haveCommonKSubstring(k:nat, str1:string, str2:string) returns(found:bool)
requires 0 < k <= |str1| && 0 < k <= |str2| //This method requires that k > 0 and k is less than or equal to in length to str1 and str2
{
//Initialising the index variable
var i := 0;
//This variable is used to define the end condition of the while loop
var n := |str1|-k;
//Here, we want to re-use the "isSubstring" method above, so with each iteration of the search, we are passing a substring of str1 with length k and searching for this substring in str2. If the k-length substring is not found, we "slide" the length-k substring "window" along and search again
//example:
//str1 = operation, str2 = rational, k = 5
//Iteration 1: isSubstring(opera, rational), returns false, slide the substring & iterate again
//Iteration 2: isSubstring(perat, rational), returns false, slide the substring & iterate again
//Iteration 3: isSubstring(erati, rational), returns false, slide the substring & iterate again
//Iteration 4: isSubstring(ratio, rational), returns true, stop iterating
while(i < n)
decreases n - i //Specifying that the loop will terminate
{
//Debug print to show what is being passed to isSubstring with each iteration
print "\n", str1[i..i+k], ", ", str2, "\n";
var result := isSubstring(str1[i..i+k], str2);
//Return once the length-k substring is found, no point in iterating any further
if(result == true){
return true;
}
//Else loop until the length-k substring is found, or we have reached the end condition
else{
i:=i+1;
}
}
return false;
}
//This method should return the natural number len which is equal to the length of the longest common substring of str1 and str2. Note that every two strings have a common substring of length zero.
method maxCommonSubstringLength(str1:string, str2:string) returns(len:nat)
requires 0 < |str1| && 0 < |str1|
{
//This variable is used to store the result of calling haveCommonKSubstring
var result:bool;
//We want the longest common substring between str1 and str2, so the starting point is going to be the shorter of the two strings.
var i:= |str1|;
if(|str2| < |str1|){
i := |str2|;
}
//Here, we want to re-use the "haveKCommonSubstring" method above, so with each iteration of the search, we pass a decreasing value of k until a common substring of this length is found. If no common substring is found, we return 0.
while (i > 0)
decreases i - 0
{
print str1, ", ", str2, " k = ", i, "\n";
result := haveCommonKSubstring(i, str1, str2);
if(result == true){
return i;
}
else{
i := i - 1;
}
}
return 0;
}
//Main to test each method
method Main(){
// isPrefix test
var prefix:string := "pre";
var str_1:string := "prehistoric";
var result:bool;
/*
result := isPrefix(prefix, str_1);
if(result == true){
print "TRUE: ", prefix, " is a prefix of the string ", str_1, "\n";
}
else{
print "FALSE: ", prefix, " is not a prefix of the string ", str_1, "\n";
}
*/
// isSubstring test
var substring := "and";
var str_2 := "operand";
/*
result := isSubstring(substring, str_2);
if(result == true){
print "TRUE: ", substring, " is a substring of the string ", str_2, "\n";
}
else{
print "FALSE: ", substring, " is not a substring of the string ", str_2, "\n";
}
*/
// haveCommonKSubstring test
//these 2 strings share the common substring "ratio" of length 5
var string1 := "operation";
var string2 := "irrational";
var k:nat := 5;
/*
result := haveCommonKSubstring(k, string1, string2);
if(result == true){
print "TRUE: ", string1, " and ", string2, " have a common substring of length ", k, "\n";
}
else{
print "FALSE: ", string1, " and ", string2, " do not have a common substring of length ", k, "\n";
}
*/
var x := maxCommonSubstringLength(string1, string2);
print "Result: ", x, "\n";
}
| //This method should return true iff pre is a prefix of str. That is, str starts with pre
method isPrefix(pre:string, str:string) returns(res:bool)
requires 0 < |pre| <= |str| //This line states that this method requires that pre is less than or equal in length to str. Without this line, an out of bounds error is shown on line 14: "str[i] != pre[i]"
{
//Initialising the index variable
var i := 0;
//Iterating through the first |pre| elements in str
while (i < |pre|)
{
//If an element does not match, return false
if (str[i] != pre[i]) {
//Debug print
print str[i], " != ", pre[i], "\n";
//Return once mismatch detected, no point in iterating any further
return false;
}
//Else loop until mismatch found or we have reached the end of pre
else{
//Debug pront
print str[i], " == ", pre[i], "\n";
i := i + 1;
}
}
return true;
}
//This method should return true iff sub is a substring of str. That is, str contains sub
method isSubstring(sub:string, str:string) returns(res:bool)
requires 0 < |sub| <= |str| //This method requires that sub is less than or equal in length to str
{
//Initialising the index variable
var i := 0;
//This variable stores the difference in length between the two strings
var n := (|str| - |sub|);
//Here, we want to re-use the "isPrefix" method above, so with each iteration of the search, we are passing an offset of str - effectively trimming a character off the front of str and passing it to isPrefix
//example 1 (sub found in str):
//str = door & sub = or
//iteration 1: isPrefix(or, door), returns false, trim & iterate again
//iteration 2: isprefix(or, oor), returns false, trim & iterate again
//iteration 3: isPrefix(or, or), returns true, stop iterating
//example 2 (sub not found in str):
//str = doom & sub = or
//iteration 1: isPrefix(or, doom), returns false, trim & iterate again
//iteration 2: isprefix(or, oom), returns false, trim & iterate again
//iteration 3: isPrefix(or, om), returns false, str is has not been "trimmed" to the same length as sub, so we stop iterating
while(i < n+1)
{
//Debug print to show what is being passed to isPrefix with each iteration
print "\n", sub, ", ", str[i..|str|], "\n";
var result:= isPrefix(sub, str[i..|str|]);
//Return once the substring is found, no point in iterating any further
if(result == true){
return true;
}
//Else loop until sub is found, or we have reached the end of str
else{
i := i+1;
}
}
return false;
}
//This method should return true iff str1 and str1 have a common substring of length k
method haveCommonKSubstring(k:nat, str1:string, str2:string) returns(found:bool)
requires 0 < k <= |str1| && 0 < k <= |str2| //This method requires that k > 0 and k is less than or equal to in length to str1 and str2
{
//Initialising the index variable
var i := 0;
//This variable is used to define the end condition of the while loop
var n := |str1|-k;
//Here, we want to re-use the "isSubstring" method above, so with each iteration of the search, we are passing a substring of str1 with length k and searching for this substring in str2. If the k-length substring is not found, we "slide" the length-k substring "window" along and search again
//example:
//str1 = operation, str2 = rational, k = 5
//Iteration 1: isSubstring(opera, rational), returns false, slide the substring & iterate again
//Iteration 2: isSubstring(perat, rational), returns false, slide the substring & iterate again
//Iteration 3: isSubstring(erati, rational), returns false, slide the substring & iterate again
//Iteration 4: isSubstring(ratio, rational), returns true, stop iterating
while(i < n)
{
//Debug print to show what is being passed to isSubstring with each iteration
print "\n", str1[i..i+k], ", ", str2, "\n";
var result := isSubstring(str1[i..i+k], str2);
//Return once the length-k substring is found, no point in iterating any further
if(result == true){
return true;
}
//Else loop until the length-k substring is found, or we have reached the end condition
else{
i:=i+1;
}
}
return false;
}
//This method should return the natural number len which is equal to the length of the longest common substring of str1 and str2. Note that every two strings have a common substring of length zero.
method maxCommonSubstringLength(str1:string, str2:string) returns(len:nat)
requires 0 < |str1| && 0 < |str1|
{
//This variable is used to store the result of calling haveCommonKSubstring
var result:bool;
//We want the longest common substring between str1 and str2, so the starting point is going to be the shorter of the two strings.
var i:= |str1|;
if(|str2| < |str1|){
i := |str2|;
}
//Here, we want to re-use the "haveKCommonSubstring" method above, so with each iteration of the search, we pass a decreasing value of k until a common substring of this length is found. If no common substring is found, we return 0.
while (i > 0)
{
print str1, ", ", str2, " k = ", i, "\n";
result := haveCommonKSubstring(i, str1, str2);
if(result == true){
return i;
}
else{
i := i - 1;
}
}
return 0;
}
//Main to test each method
method Main(){
// isPrefix test
var prefix:string := "pre";
var str_1:string := "prehistoric";
var result:bool;
/*
result := isPrefix(prefix, str_1);
if(result == true){
print "TRUE: ", prefix, " is a prefix of the string ", str_1, "\n";
}
else{
print "FALSE: ", prefix, " is not a prefix of the string ", str_1, "\n";
}
*/
// isSubstring test
var substring := "and";
var str_2 := "operand";
/*
result := isSubstring(substring, str_2);
if(result == true){
print "TRUE: ", substring, " is a substring of the string ", str_2, "\n";
}
else{
print "FALSE: ", substring, " is not a substring of the string ", str_2, "\n";
}
*/
// haveCommonKSubstring test
//these 2 strings share the common substring "ratio" of length 5
var string1 := "operation";
var string2 := "irrational";
var k:nat := 5;
/*
result := haveCommonKSubstring(k, string1, string2);
if(result == true){
print "TRUE: ", string1, " and ", string2, " have a common substring of length ", k, "\n";
}
else{
print "FALSE: ", string1, " and ", string2, " do not have a common substring of length ", k, "\n";
}
*/
var x := maxCommonSubstringLength(string1, string2);
print "Result: ", x, "\n";
}
|
014 | CVS-Projto1_tmp_tmpb1o0bu8z_Hoare.dfy | method Max (x: nat, y:nat) returns (r:nat)
ensures (r >= x && r >=y)
ensures (r == x || r == y)
{
if (x >= y) { r := x;}
else { r := y;}
}
method Test ()
{
var result := Max(42, 73);
assert result == 73;
}
method m1 (x: int, y: int) returns (z: int)
requires 0 < x < y
ensures z >= 0 && z <= y && z != x
{
//assume 0 < x < y
z := 0;
}
function fib (n: nat) : nat
{
if n == 0 then 1 else
if n == 1 then 1 else
fib(n -1) + fib (n-2)
}
method Fib (n: nat) returns (r:nat)
ensures r == fib(n)
{
if (n == 0) {
return 1;
}
r := 1;
var next:=2;
var i := 1;
while i < n
invariant 1 <= i <= n
invariant r == fib(i)
invariant next == fib(i+1)
{
var tmp:=next;
next:= next + r;
r:= tmp;
i:= i + 1;
}
assert r == fib(n);
return r;
}
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
function add(l:List<int>) : int
{
match l
case Nil => 0
case Cons(x, xs) => x + add(xs)
}
method addImp (l: List<int>) returns (s: int)
ensures s == add(l)
{
var ll := l;
s := 0;
while ll != Nil
decreases ll
invariant add(l) == s + add(ll)
{
s := s + ll.head;
ll:= ll.tail;
}
assert s == add(l);
}
method MaxA (a: array<int>) returns (m: int)
requires a.Length > 0
ensures forall i :: 0 <= i < a.Length ==> a[i] <= m
ensures exists i :: 0 <= i < a.Length && a[i] == m
{
m := a[0];
var i := 1;
while i< a.Length
invariant 1 <= i <= a.Length
invariant forall j :: 0 <= j < i ==> a[j] <=m
invariant exists j :: 0 <= j < i && a[j] ==m
{
if a[i] > m {
m:= a[i];
}
i := i +1;
}
}
| method Max (x: nat, y:nat) returns (r:nat)
ensures (r >= x && r >=y)
ensures (r == x || r == y)
{
if (x >= y) { r := x;}
else { r := y;}
}
method Test ()
{
var result := Max(42, 73);
}
method m1 (x: int, y: int) returns (z: int)
requires 0 < x < y
ensures z >= 0 && z <= y && z != x
{
//assume 0 < x < y
z := 0;
}
function fib (n: nat) : nat
{
if n == 0 then 1 else
if n == 1 then 1 else
fib(n -1) + fib (n-2)
}
method Fib (n: nat) returns (r:nat)
ensures r == fib(n)
{
if (n == 0) {
return 1;
}
r := 1;
var next:=2;
var i := 1;
while i < n
{
var tmp:=next;
next:= next + r;
r:= tmp;
i:= i + 1;
}
return r;
}
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
function add(l:List<int>) : int
{
match l
case Nil => 0
case Cons(x, xs) => x + add(xs)
}
method addImp (l: List<int>) returns (s: int)
ensures s == add(l)
{
var ll := l;
s := 0;
while ll != Nil
{
s := s + ll.head;
ll:= ll.tail;
}
}
method MaxA (a: array<int>) returns (m: int)
requires a.Length > 0
ensures forall i :: 0 <= i < a.Length ==> a[i] <= m
ensures exists i :: 0 <= i < a.Length && a[i] == m
{
m := a[0];
var i := 1;
while i< a.Length
{
if a[i] > m {
m:= a[i];
}
i := i +1;
}
}
|
015 | CVS-Projto1_tmp_tmpb1o0bu8z_fact.dfy | function fact (n:nat): nat
decreases n
{if n == 0 then 1 else n * fact(n-1)}
function factAcc (n:nat, a:int): int
decreases n
{if (n==0) then a else factAcc(n-1,n*a)}
function factAlt(n:nat):int
{factAcc(n,1)}
lemma factAcc_correct (n:nat, a:int)
ensures factAcc(n, a) == a*fact(n)
{
}
lemma factAlt_correct (n:nat)
ensures factAlt(n) == fact(n)
{
factAcc_correct(n,1);
assert factAcc(n,1) == 1 * fact(n);
assert 1 * fact(n) == fact(n);
assert factAlt(n) == factAcc(n, 1);
}
datatype List<T> = Nil | Cons(T, List<T>)
function length<T> (l: List<T>) : nat
decreases l
{
match l
case Nil => 0
case Cons(_, r) => 1 + length(r)
}
lemma {:induction false} length_non_neg<T> (l:List<T>)
ensures length(l) >= 0
{
match l
case Nil =>
case Cons(_, r) =>
length_non_neg(r);
assert length(r) >= 0;
// assert forall k : int :: k >= 0 ==> 1 + k >= 0;
assert 1 + length(r) >= 0;
assert 1 + length(r) == length(l);
}
function lengthTL<T> (l: List<T>, acc: nat) : nat
{
match l
case Nil => acc
case Cons(_, r) => lengthTL(r, 1 + acc)
}
lemma {:induction false}lengthTL_aux<T> (l: List<T>, acc: nat)
ensures lengthTL(l, acc) == acc + length(l)
{
match l
case Nil => assert acc + length<T>(Nil) == acc;
case Cons(_, r) =>
lengthTL_aux(r, acc + 1);
}
lemma lengthEq<T> (l: List<T>)
ensures length(l) == lengthTL(l,0)
{
lengthTL_aux(l, 0);
}
| function fact (n:nat): nat
{if n == 0 then 1 else n * fact(n-1)}
function factAcc (n:nat, a:int): int
{if (n==0) then a else factAcc(n-1,n*a)}
function factAlt(n:nat):int
{factAcc(n,1)}
lemma factAcc_correct (n:nat, a:int)
ensures factAcc(n, a) == a*fact(n)
{
}
lemma factAlt_correct (n:nat)
ensures factAlt(n) == fact(n)
{
factAcc_correct(n,1);
}
datatype List<T> = Nil | Cons(T, List<T>)
function length<T> (l: List<T>) : nat
{
match l
case Nil => 0
case Cons(_, r) => 1 + length(r)
}
lemma {:induction false} length_non_neg<T> (l:List<T>)
ensures length(l) >= 0
{
match l
case Nil =>
case Cons(_, r) =>
length_non_neg(r);
// assert forall k : int :: k >= 0 ==> 1 + k >= 0;
}
function lengthTL<T> (l: List<T>, acc: nat) : nat
{
match l
case Nil => acc
case Cons(_, r) => lengthTL(r, 1 + acc)
}
lemma {:induction false}lengthTL_aux<T> (l: List<T>, acc: nat)
ensures lengthTL(l, acc) == acc + length(l)
{
match l
case Nil => assert acc + length<T>(Nil) == acc;
case Cons(_, r) =>
lengthTL_aux(r, acc + 1);
}
lemma lengthEq<T> (l: List<T>)
ensures length(l) == lengthTL(l,0)
{
lengthTL_aux(l, 0);
}
|
016 | CVS-Projto1_tmp_tmpb1o0bu8z_proj1_proj1.dfy | //Exercicio 1.a)
function sum (a:array<int>, i:int, j:int) :int
decreases j
reads a
requires 0 <= i <= j <= a.Length
{
if i == j then
0
else
a[j-1] + sum(a, i, j-1)
}
//Exercicio 1.b)
method query (a:array<int>, i:int, j:int) returns (s:int)
requires 0 <= i <= j <= a.Length
ensures s == sum(a, i, j)
{
s := 0;
var aux := i;
while (aux < j)
invariant i <= aux <= j
invariant s == sum(a, i, aux)
decreases j - aux
{
s := s + a[aux];
aux := aux + 1;
}
return s;
}
//Exercicio 1.c)
lemma queryLemma(a:array<int>, i:int, j:int, k:int)
requires 0 <= i <= k <= j <= a.Length
ensures sum(a,i,k) + sum(a,k,j) == sum(a,i,j)
{
}
method queryFast (a:array<int>, c:array<int>, i:int, j:int) returns (r:int)
requires is_prefix_sum_for(a,c) && 0 <= i <= j <= a.Length < c.Length
ensures r == sum(a, i,j)
{
r := c[j] - c[i];
queryLemma(a,0,j,i);
return r;
}
predicate is_prefix_sum_for (a:array<int>, c:array<int>)
reads c, a
{
a.Length + 1 == c.Length
&& c[0] == 0
&& forall j :: 1 <= j <= a.Length ==> c[j] == sum(a,0,j)
}
///Exercicio 2.
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
method from_array<T>(a: array<T>) returns (l: List<T>)
requires a.Length > 0
ensures forall j::0 <= j < a.Length ==> mem(a[j],l)
{
var i:= a.Length-1;
l:= Nil;
while (i >= 0)
invariant -1 <= i < a. Length
invariant forall j:: i+1 <= j < a.Length ==> mem(a[j],l)
{
l := Cons(a[i], l);
i := i - 1;
}
return l;
}
function mem<T(==)> (x: T, l:List<T>) : bool
decreases l
{
match l
case Nil => false
case Cons(y,r)=> if (x==y) then true else mem(x,r)
}
| //Exercicio 1.a)
function sum (a:array<int>, i:int, j:int) :int
reads a
requires 0 <= i <= j <= a.Length
{
if i == j then
0
else
a[j-1] + sum(a, i, j-1)
}
//Exercicio 1.b)
method query (a:array<int>, i:int, j:int) returns (s:int)
requires 0 <= i <= j <= a.Length
ensures s == sum(a, i, j)
{
s := 0;
var aux := i;
while (aux < j)
{
s := s + a[aux];
aux := aux + 1;
}
return s;
}
//Exercicio 1.c)
lemma queryLemma(a:array<int>, i:int, j:int, k:int)
requires 0 <= i <= k <= j <= a.Length
ensures sum(a,i,k) + sum(a,k,j) == sum(a,i,j)
{
}
method queryFast (a:array<int>, c:array<int>, i:int, j:int) returns (r:int)
requires is_prefix_sum_for(a,c) && 0 <= i <= j <= a.Length < c.Length
ensures r == sum(a, i,j)
{
r := c[j] - c[i];
queryLemma(a,0,j,i);
return r;
}
predicate is_prefix_sum_for (a:array<int>, c:array<int>)
reads c, a
{
a.Length + 1 == c.Length
&& c[0] == 0
&& forall j :: 1 <= j <= a.Length ==> c[j] == sum(a,0,j)
}
///Exercicio 2.
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
method from_array<T>(a: array<T>) returns (l: List<T>)
requires a.Length > 0
ensures forall j::0 <= j < a.Length ==> mem(a[j],l)
{
var i:= a.Length-1;
l:= Nil;
while (i >= 0)
{
l := Cons(a[i], l);
i := i - 1;
}
return l;
}
function mem<T(==)> (x: T, l:List<T>) : bool
{
match l
case Nil => false
case Cons(y,r)=> if (x==y) then true else mem(x,r)
}
|
017 | CVS-Projto1_tmp_tmpb1o0bu8z_searchSort.dfy | method fillK(a: array<int>, n: int, k: int, c: int) returns (b: bool)
requires 0 <= c <= n
requires n == a.Length
{
if c == 0 {
return true;
}
var p := 0;
while p < c
invariant 0 <= p <= c
{
if a[p] != k
{
return false;
}
p := p + 1;
}
return true;
}
method containsSubString(a: array<char>, b: array<char>) returns (pos: int)
requires 0 <= b.Length <= a.Length
{
pos := -1;
if b.Length == 0 {
return pos;
}
var p := 0;
while p < a.Length
invariant 0 <= p <= a.Length
{
if a.Length - p < b.Length
{
return pos;
}
if a[p] == b[0] {
var i := 0;
while i < b.Length
{
if a[i + p] != b[i] {
return -1;
}
i:= i + 1;
}
pos := p;
return pos;
}
p:= p +1;
}
}
| method fillK(a: array<int>, n: int, k: int, c: int) returns (b: bool)
requires 0 <= c <= n
requires n == a.Length
{
if c == 0 {
return true;
}
var p := 0;
while p < c
{
if a[p] != k
{
return false;
}
p := p + 1;
}
return true;
}
method containsSubString(a: array<char>, b: array<char>) returns (pos: int)
requires 0 <= b.Length <= a.Length
{
pos := -1;
if b.Length == 0 {
return pos;
}
var p := 0;
while p < a.Length
{
if a.Length - p < b.Length
{
return pos;
}
if a[p] == b[0] {
var i := 0;
while i < b.Length
{
if a[i + p] != b[i] {
return -1;
}
i:= i + 1;
}
pos := p;
return pos;
}
p:= p +1;
}
}
|
018 | CVS-handout1_tmp_tmptm52no3k_1.dfy | /* Cumulative Sums over Arrays */
/*
Daniel Cavalheiro 57869
Pedro Nunes 57854
*/
//(a)
function sum(a: array<int>, i: int, j: int): int
reads a
requires 0 <= i <= j <= a.Length
decreases j - i
{
if (i == j) then 0
else a[i] + sum(a, i+1, j)
}
//(b)
method query(a: array<int>, i: int, j: int) returns (res:int)
requires 0 <= i <= j <= a.Length
ensures res == sum(a, i, j)
{
res := 0;
var k := i;
while(k < j)
invariant i <= k <= j <= a.Length
invariant res + sum(a, k, j) == sum(a, i, j)
{
res := res + a[k];
k := k + 1;
}
}
//(c)
predicate is_prefix_sum_for (a: array<int>, c: array<int>)
requires a.Length + 1 == c.Length
requires c[0] == 0
reads c, a
{
forall i: int :: 0 <= i < a.Length ==> c[i+1] == c[i] + a[i]
}
lemma aux(a: array<int>, c: array<int>, i: int, j: int)
requires 0 <= i <= j <= a.Length
requires a.Length + 1 == c.Length
requires c[0] == 0
requires is_prefix_sum_for(a, c)
decreases j - i
ensures forall k: int :: i <= k <= j ==> sum(a, i, k) + sum(a, k, j) == c[k] - c[i] + c[j] - c[k] //sum(a, i, j) == c[j] - c[i]
{}
method queryFast(a: array<int>, c: array<int>, i: int, j: int) returns (r: int)
requires a.Length + 1 == c.Length && c[0] == 0
requires 0 <= i <= j <= a.Length
requires is_prefix_sum_for(a,c)
ensures r == sum(a, i, j)
{
aux(a, c, i, j);
r := c[j] - c[i];
}
method Main()
{
var x := new int[10];
x[0], x[1], x[2], x[3] := 2, 2, 1, 5;
var y := sum(x, 0, x.Length);
//assert y == 10;
var c := new int[11];
c[0], c[1], c[2], c[3], c[4] := 0, 2, 4, 5, 10;
// var r := queryFast(x, c, 0, x.Length);
}
| /* Cumulative Sums over Arrays */
/*
Daniel Cavalheiro 57869
Pedro Nunes 57854
*/
//(a)
function sum(a: array<int>, i: int, j: int): int
reads a
requires 0 <= i <= j <= a.Length
{
if (i == j) then 0
else a[i] + sum(a, i+1, j)
}
//(b)
method query(a: array<int>, i: int, j: int) returns (res:int)
requires 0 <= i <= j <= a.Length
ensures res == sum(a, i, j)
{
res := 0;
var k := i;
while(k < j)
{
res := res + a[k];
k := k + 1;
}
}
//(c)
predicate is_prefix_sum_for (a: array<int>, c: array<int>)
requires a.Length + 1 == c.Length
requires c[0] == 0
reads c, a
{
forall i: int :: 0 <= i < a.Length ==> c[i+1] == c[i] + a[i]
}
lemma aux(a: array<int>, c: array<int>, i: int, j: int)
requires 0 <= i <= j <= a.Length
requires a.Length + 1 == c.Length
requires c[0] == 0
requires is_prefix_sum_for(a, c)
ensures forall k: int :: i <= k <= j ==> sum(a, i, k) + sum(a, k, j) == c[k] - c[i] + c[j] - c[k] //sum(a, i, j) == c[j] - c[i]
{}
method queryFast(a: array<int>, c: array<int>, i: int, j: int) returns (r: int)
requires a.Length + 1 == c.Length && c[0] == 0
requires 0 <= i <= j <= a.Length
requires is_prefix_sum_for(a,c)
ensures r == sum(a, i, j)
{
aux(a, c, i, j);
r := c[j] - c[i];
}
method Main()
{
var x := new int[10];
x[0], x[1], x[2], x[3] := 2, 2, 1, 5;
var y := sum(x, 0, x.Length);
//assert y == 10;
var c := new int[11];
c[0], c[1], c[2], c[3], c[4] := 0, 2, 4, 5, 10;
// var r := queryFast(x, c, 0, x.Length);
}
|
019 | CVS-handout1_tmp_tmptm52no3k_2.dfy | /* Functional Lists and Imperative Arrays */
/*
Daniel Cavalheiro 57869
Pedro Nunes 57854
*/
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
function length<T>(l: List<T>): nat
{
match l
case Nil => 0
case Cons(_, t) => 1 + length(t)
}
predicate mem<T(==)> (l: List<T>, x: T)
{
match l
case Nil => false
case Cons(h, t) => if(h == x) then true else mem(t, x)
}
function at<T>(l: List<T>, i: nat): T
requires i < length(l)
{
if i == 0 then l.head else at(l.tail, i - 1)
}
method from_array<T>(a: array<T>) returns (l: List<T>)
requires a.Length >= 0
ensures length(l) == a.Length
ensures forall i: int :: 0 <= i < length(l) ==> at(l, i) == a[i]
ensures forall x :: mem(l, x) ==> exists i: int :: 0 <= i < length(l) && a[i] == x
{
l := Nil;
var i: int := a.Length - 1;
while(i >= 0)
invariant -1 <= i <= a.Length - 1
invariant length(l) == a.Length - 1 - i
invariant forall j: int :: i < j < a.Length ==> at(l,j-i-1) == a[j]
invariant forall x :: mem(l, x) ==> exists k: int :: i < k < a.Length && a[k] == x
{
l := Cons(a[i], l);
i := i-1;
}
}
method Main() {
var l: List<int> := List.Cons(1, List.Cons(2, List.Cons(3, Nil)));
var arr: array<int> := new int [3](i => i + 1);
var t: List<int> := from_array(arr);
print l;
print "\n";
print t;
print "\n";
print t == l;
}
| /* Functional Lists and Imperative Arrays */
/*
Daniel Cavalheiro 57869
Pedro Nunes 57854
*/
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
function length<T>(l: List<T>): nat
{
match l
case Nil => 0
case Cons(_, t) => 1 + length(t)
}
predicate mem<T(==)> (l: List<T>, x: T)
{
match l
case Nil => false
case Cons(h, t) => if(h == x) then true else mem(t, x)
}
function at<T>(l: List<T>, i: nat): T
requires i < length(l)
{
if i == 0 then l.head else at(l.tail, i - 1)
}
method from_array<T>(a: array<T>) returns (l: List<T>)
requires a.Length >= 0
ensures length(l) == a.Length
ensures forall i: int :: 0 <= i < length(l) ==> at(l, i) == a[i]
ensures forall x :: mem(l, x) ==> exists i: int :: 0 <= i < length(l) && a[i] == x
{
l := Nil;
var i: int := a.Length - 1;
while(i >= 0)
{
l := Cons(a[i], l);
i := i-1;
}
}
method Main() {
var l: List<int> := List.Cons(1, List.Cons(2, List.Cons(3, Nil)));
var arr: array<int> := new int [3](i => i + 1);
var t: List<int> := from_array(arr);
print l;
print "\n";
print t;
print "\n";
print t == l;
}
|
020 | Clover_abs.dfy | method Abs(x: int) returns (y: int)
ensures x>=0 ==> x==y
ensures x<0 ==> x+y==0
{
if x < 0 {
return -x;
} else {
return x;
}
}
| method Abs(x: int) returns (y: int)
ensures x>=0 ==> x==y
ensures x<0 ==> x+y==0
{
if x < 0 {
return -x;
} else {
return x;
}
}
|
021 | Clover_all_digits.dfy | method allDigits(s: string) returns (result: bool)
ensures result <==> (forall i :: 0 <= i < |s| ==> s[i] in "0123456789")
{
result:=true ;
for i := 0 to |s|
invariant result <==> (forall ii :: 0 <= ii < i ==> s[ii] in "0123456789")
{
if ! (s[i] in "0123456789"){
return false;
}
}
}
| method allDigits(s: string) returns (result: bool)
ensures result <==> (forall i :: 0 <= i < |s| ==> s[i] in "0123456789")
{
result:=true ;
for i := 0 to |s|
{
if ! (s[i] in "0123456789"){
return false;
}
}
}
|
022 | Clover_array_append.dfy | method append(a:array<int>, b:int) returns (c:array<int>)
ensures a[..] + [b] == c[..]
{
c := new int[a.Length+1];
var i:= 0;
while (i < a.Length)
invariant 0 <= i <= a.Length
invariant forall ii::0<=ii<i ==> c[ii]==a[ii]
{
c[i] := a[i];
i:=i+1;
}
c[a.Length]:=b;
}
| method append(a:array<int>, b:int) returns (c:array<int>)
ensures a[..] + [b] == c[..]
{
c := new int[a.Length+1];
var i:= 0;
while (i < a.Length)
{
c[i] := a[i];
i:=i+1;
}
c[a.Length]:=b;
}
|
023 | Clover_array_concat.dfy | method concat(a:array<int>, b:array<int>) returns (c:array<int>)
ensures c.Length==b.Length+a.Length
ensures forall k :: 0 <= k < a.Length ==> c[k] == a[k]
ensures forall k :: 0 <= k < b.Length ==> c[k+a.Length] == b[k]
{
c := new int[a.Length+b.Length];
var i:= 0;
while (i < c.Length)
invariant 0 <= i <= c.Length
invariant if i<a.Length then c[..i]==a[..i] else c[..i]==a[..]+b[..(i-a.Length)]
{
c[i] := if i<a.Length then a[i] else b[i-a.Length];
i:=i+1;
}
}
| method concat(a:array<int>, b:array<int>) returns (c:array<int>)
ensures c.Length==b.Length+a.Length
ensures forall k :: 0 <= k < a.Length ==> c[k] == a[k]
ensures forall k :: 0 <= k < b.Length ==> c[k+a.Length] == b[k]
{
c := new int[a.Length+b.Length];
var i:= 0;
while (i < c.Length)
{
c[i] := if i<a.Length then a[i] else b[i-a.Length];
i:=i+1;
}
}
|
024 | Clover_array_copy.dfy | method iter_copy<T(0)>(s: array<T>) returns (t: array<T>)
ensures s.Length==t.Length
ensures forall i::0<=i<s.Length ==> s[i]==t[i]
{
t := new T[s.Length];
var i:= 0;
while (i < s.Length)
invariant 0 <= i <= s.Length
invariant forall x :: 0 <= x < i ==> s[x] == t[x]
{
t[i] := s[i];
i:=i+1;
}
}
| method iter_copy<T(0)>(s: array<T>) returns (t: array<T>)
ensures s.Length==t.Length
ensures forall i::0<=i<s.Length ==> s[i]==t[i]
{
t := new T[s.Length];
var i:= 0;
while (i < s.Length)
{
t[i] := s[i];
i:=i+1;
}
}
|
025 | Clover_array_product.dfy | method arrayProduct(a: array<int>, b: array<int>) returns (c: array<int> )
requires a.Length==b.Length
ensures c.Length==a.Length
ensures forall i:: 0 <= i< a.Length==> a[i] * b[i]==c[i]
{
c:= new int[a.Length];
var i:=0;
while i<a.Length
invariant 0<=i<=a.Length
invariant forall j:: 0 <= j< i==> a[j] * b[j]==c[j]
{
c[i]:=a[i]*b[i];
i:=i+1;
}
}
| method arrayProduct(a: array<int>, b: array<int>) returns (c: array<int> )
requires a.Length==b.Length
ensures c.Length==a.Length
ensures forall i:: 0 <= i< a.Length==> a[i] * b[i]==c[i]
{
c:= new int[a.Length];
var i:=0;
while i<a.Length
{
c[i]:=a[i]*b[i];
i:=i+1;
}
}
|
026 | Clover_array_sum.dfy | method arraySum(a: array<int>, b: array<int>) returns (c: array<int> )
requires a.Length==b.Length
ensures c.Length==a.Length
ensures forall i:: 0 <= i< a.Length==> a[i] + b[i]==c[i]
{
c:= new int[a.Length];
var i:=0;
while i<a.Length
invariant 0<=i<=a.Length
invariant forall j:: 0 <= j< i==> a[j] + b[j]==c[j]
{
c[i]:=a[i]+b[i];
i:=i+1;
}
}
| method arraySum(a: array<int>, b: array<int>) returns (c: array<int> )
requires a.Length==b.Length
ensures c.Length==a.Length
ensures forall i:: 0 <= i< a.Length==> a[i] + b[i]==c[i]
{
c:= new int[a.Length];
var i:=0;
while i<a.Length
{
c[i]:=a[i]+b[i];
i:=i+1;
}
}
|
027 | Clover_avg.dfy | method ComputeAvg(a: int, b: int) returns (avg:int)
ensures avg == (a+b)/2
{
avg:= (a + b) / 2;
}
| method ComputeAvg(a: int, b: int) returns (avg:int)
ensures avg == (a+b)/2
{
avg:= (a + b) / 2;
}
|
028 | Clover_below_zero.dfy | method below_zero(operations: seq<int>) returns (s:array<int>, result:bool)
ensures s.Length == |operations| + 1
ensures s[0]==0
ensures forall i :: 0 <= i < s.Length-1 ==> s[i+1]==s[i]+operations[i]
ensures result == true ==> (exists i :: 1 <= i <= |operations| && s[i] < 0)
ensures result == false ==> forall i :: 0 <= i < s.Length ==> s[i] >= 0
{
result := false;
s := new int[|operations| + 1];
var i := 0;
s[i] := 0;
while i < s.Length
invariant 0 <= i <= s.Length
invariant s[0]==0
invariant s.Length == |operations| + 1
invariant forall x :: 0 <= x < i-1 ==> s[x+1]==s[x]+operations[x]
{
if i>0{
s[i] := s[i - 1] + operations[i - 1];
}
i := i + 1;
}
i:=0;
while i < s.Length
invariant 0 <= i <= s.Length
invariant forall x :: 0 <= x < i ==> s[x] >= 0
{
if s[i] < 0 {
result := true;
return;
}
i := i + 1;
}
}
| method below_zero(operations: seq<int>) returns (s:array<int>, result:bool)
ensures s.Length == |operations| + 1
ensures s[0]==0
ensures forall i :: 0 <= i < s.Length-1 ==> s[i+1]==s[i]+operations[i]
ensures result == true ==> (exists i :: 1 <= i <= |operations| && s[i] < 0)
ensures result == false ==> forall i :: 0 <= i < s.Length ==> s[i] >= 0
{
result := false;
s := new int[|operations| + 1];
var i := 0;
s[i] := 0;
while i < s.Length
{
if i>0{
s[i] := s[i - 1] + operations[i - 1];
}
i := i + 1;
}
i:=0;
while i < s.Length
{
if s[i] < 0 {
result := true;
return;
}
i := i + 1;
}
}
|
029 | Clover_binary_search.dfy | method BinarySearch(a: array<int>, key: int) returns (n: int)
requires forall i,j :: 0<=i<j<a.Length ==> a[i]<=a[j]
ensures 0<= n <=a.Length
ensures forall i :: 0<= i < n ==> a[i] < key
ensures n == a.Length ==> forall i :: 0 <= i < a.Length ==> a[i] < key
ensures forall i :: n<= i < a.Length ==> a[i]>=key
{
var lo, hi := 0, a.Length;
while lo<hi
invariant 0<= lo <= hi <= a.Length
invariant forall i :: 0<=i<lo ==> a[i] < key
invariant forall i :: hi<=i<a.Length ==> a[i] >= key
{
var mid := (lo + hi) / 2;
if a[mid] < key {
lo := mid + 1;
} else {
hi := mid;
}
}
n:=lo;
}
| method BinarySearch(a: array<int>, key: int) returns (n: int)
requires forall i,j :: 0<=i<j<a.Length ==> a[i]<=a[j]
ensures 0<= n <=a.Length
ensures forall i :: 0<= i < n ==> a[i] < key
ensures n == a.Length ==> forall i :: 0 <= i < a.Length ==> a[i] < key
ensures forall i :: n<= i < a.Length ==> a[i]>=key
{
var lo, hi := 0, a.Length;
while lo<hi
{
var mid := (lo + hi) / 2;
if a[mid] < key {
lo := mid + 1;
} else {
hi := mid;
}
}
n:=lo;
}
|
030 | Clover_bubble_sort.dfy | method BubbleSort(a: array<int>)
modifies a
ensures forall i,j::0<= i < j < a.Length ==> a[i] <= a[j]
ensures multiset(a[..])==multiset(old(a[..]))
{
var i := a.Length - 1;
while (i > 0)
invariant i < 0 ==> a.Length == 0
invariant -1 <= i < a.Length
invariant forall ii,jj::i <= ii< jj <a.Length ==> a[ii] <= a[jj]
invariant forall k,k'::0<=k<=i<k'<a.Length==>a[k]<=a[k']
invariant multiset(a[..])==multiset(old(a[..]))
{
var j := 0;
while (j < i)
invariant 0 < i < a.Length && 0 <= j <= i
invariant forall ii,jj::i<= ii <= jj <a.Length ==> a[ii] <= a[jj]
invariant forall k, k'::0<=k<=i<k'<a.Length==>a[k]<=a[k']
invariant forall k :: 0 <= k <= j ==> a[k] <= a[j]
invariant multiset(a[..])==multiset(old(a[..]))
{
if (a[j] > a[j + 1])
{
a[j], a[j + 1] := a[j + 1], a[j];
}
j := j + 1;
}
i := i - 1;
}
}
| method BubbleSort(a: array<int>)
modifies a
ensures forall i,j::0<= i < j < a.Length ==> a[i] <= a[j]
ensures multiset(a[..])==multiset(old(a[..]))
{
var i := a.Length - 1;
while (i > 0)
{
var j := 0;
while (j < i)
{
if (a[j] > a[j + 1])
{
a[j], a[j + 1] := a[j + 1], a[j];
}
j := j + 1;
}
i := i - 1;
}
}
|
031 | Clover_cal_ans.dfy | method CalDiv() returns (x:int, y:int)
ensures x==191/7
ensures y==191%7
{
x, y := 0, 191;
while 7 <= y
invariant 0 <= y && 7 * x + y == 191
{
x := x+1;
y:=191-7*x;
}
}
| method CalDiv() returns (x:int, y:int)
ensures x==191/7
ensures y==191%7
{
x, y := 0, 191;
while 7 <= y
{
x := x+1;
y:=191-7*x;
}
}
|
032 | Clover_cal_sum.dfy | method Sum(N:int) returns (s:int)
requires N >= 0
ensures s == N * (N + 1) / 2
{
var n := 0;
s := 0;
while n != N
invariant 0 <= n <= N
invariant s == n * (n + 1) / 2
{
n := n + 1;
s := s + n;
}
}
| method Sum(N:int) returns (s:int)
requires N >= 0
ensures s == N * (N + 1) / 2
{
var n := 0;
s := 0;
while n != N
{
n := n + 1;
s := s + n;
}
}
|
033 | Clover_canyon_search.dfy | method CanyonSearch(a: array<int>, b: array<int>) returns (d:nat)
requires a.Length !=0 && b.Length!=0
requires forall i,j :: 0<=i<j<a.Length ==> a[i]<=a[j]
requires forall i,j :: 0<=i<j<b.Length ==> b[i]<=b[j]
ensures exists i,j:: 0<=i<a.Length && 0<=j<b.Length && d==if a[i] < b[j] then (b[j]-a[i]) else (a[i]-b[j])
ensures forall i,j:: 0<=i<a.Length && 0<=j<b.Length ==> d<=if a[i] < b[j] then (b[j]-a[i]) else (a[i]-b[j])
{
var m,n:=0,0;
d:=if a[0] < b[0] then (b[0]-a[0]) else (a[0]-b[0]);
while m<a.Length && n<b.Length
invariant 0<=m<=a.Length && 0<=n<=b.Length
decreases a.Length -m+b.Length-n
invariant exists i,j:: 0<=i<a.Length && 0<=j<b.Length && d==if a[i] < b[j] then (b[j]-a[i]) else (a[i]-b[j])
invariant forall i,j:: 0<=i<a.Length && 0<=j<b.Length ==> d<=(if a[i] < b[j] then (b[j]-a[i]) else (a[i]-b[j]))|| (m<=i&&n<=j)
{
var t := if a[m] < b[n] then (b[n]-a[m]) else (a[m]-b[n]);
d:=if t<d then t else d;
if
case a[m]<=b[n] =>
m:=m+1;
case b[n]<=a[m] =>
n:=n+1;
}
}
| method CanyonSearch(a: array<int>, b: array<int>) returns (d:nat)
requires a.Length !=0 && b.Length!=0
requires forall i,j :: 0<=i<j<a.Length ==> a[i]<=a[j]
requires forall i,j :: 0<=i<j<b.Length ==> b[i]<=b[j]
ensures exists i,j:: 0<=i<a.Length && 0<=j<b.Length && d==if a[i] < b[j] then (b[j]-a[i]) else (a[i]-b[j])
ensures forall i,j:: 0<=i<a.Length && 0<=j<b.Length ==> d<=if a[i] < b[j] then (b[j]-a[i]) else (a[i]-b[j])
{
var m,n:=0,0;
d:=if a[0] < b[0] then (b[0]-a[0]) else (a[0]-b[0]);
while m<a.Length && n<b.Length
{
var t := if a[m] < b[n] then (b[n]-a[m]) else (a[m]-b[n]);
d:=if t<d then t else d;
if
case a[m]<=b[n] =>
m:=m+1;
case b[n]<=a[m] =>
n:=n+1;
}
}
|
034 | Clover_compare.dfy | method Compare<T(==)>(a: T, b: T) returns (eq: bool)
ensures a==b ==> eq==true
ensures a!=b ==> eq==false
{
if a == b { eq := true; } else { eq := false; }
}
| method Compare<T(==)>(a: T, b: T) returns (eq: bool)
ensures a==b ==> eq==true
ensures a!=b ==> eq==false
{
if a == b { eq := true; } else { eq := false; }
}
|
035 | Clover_convert_map_key.dfy | method convert_map_key(inputs: map<nat, bool>, f: nat->nat) returns(r:map<nat, bool>)
requires forall n1: nat, n2: nat :: n1 != n2 ==> f(n1) != f(n2)
ensures forall k :: k in inputs <==> f(k) in r
ensures forall k :: k in inputs ==> r[f(k)] == inputs[k]
{
r:= map k | k in inputs :: f(k) := inputs[k];
}
| method convert_map_key(inputs: map<nat, bool>, f: nat->nat) returns(r:map<nat, bool>)
requires forall n1: nat, n2: nat :: n1 != n2 ==> f(n1) != f(n2)
ensures forall k :: k in inputs <==> f(k) in r
ensures forall k :: k in inputs ==> r[f(k)] == inputs[k]
{
r:= map k | k in inputs :: f(k) := inputs[k];
}
|
036 | Clover_copy_part.dfy | method copy( src: array<int>, sStart: nat, dest: array<int>, dStart: nat, len: nat) returns (r: array<int>)
requires src.Length >= sStart + len
requires dest.Length >= dStart + len
ensures r.Length == dest.Length
ensures r[..dStart] == dest[..dStart]
ensures r[dStart + len..] == dest[dStart + len..]
ensures r[dStart..len+dStart] == src[sStart..len+sStart]
{
if len == 0 { return dest; }
var i: nat := 0;
r := new int[dest.Length];
while (i < r.Length)
invariant i <= r.Length
invariant r[..i] == dest[..i]
{
r[i] := dest[i];
i := i + 1;
}
assert r[..]==dest[..];
i := 0;
while (i < len)
invariant i <= len
invariant r[..dStart] == dest[..dStart]
invariant r[(dStart + len)..] == dest[(dStart + len)..]
invariant r[dStart .. dStart + i] == src[sStart .. sStart + i]
{
assert r[(dStart + len)..] == dest[(dStart + len)..];
r[dStart + i] := src[sStart + i];
i := i + 1;
}
} | method copy( src: array<int>, sStart: nat, dest: array<int>, dStart: nat, len: nat) returns (r: array<int>)
requires src.Length >= sStart + len
requires dest.Length >= dStart + len
ensures r.Length == dest.Length
ensures r[..dStart] == dest[..dStart]
ensures r[dStart + len..] == dest[dStart + len..]
ensures r[dStart..len+dStart] == src[sStart..len+sStart]
{
if len == 0 { return dest; }
var i: nat := 0;
r := new int[dest.Length];
while (i < r.Length)
{
r[i] := dest[i];
i := i + 1;
}
i := 0;
while (i < len)
{
r[dStart + i] := src[sStart + i];
i := i + 1;
}
} |
037 | Clover_count_lessthan.dfy | method CountLessThan(numbers: set<int>, threshold: int) returns (count: int)
ensures count == |set i | i in numbers && i < threshold|
{
count := 0;
var shrink := numbers;
var grow := {};
while |shrink | > 0
decreases shrink
invariant shrink + grow == numbers
invariant grow !! shrink
invariant count == |set i | i in grow && i < threshold|
{
var i: int :| i in shrink;
shrink := shrink - {i};
var grow' := grow+{i};
assert (set i | i in grow' && i < threshold) ==
(set i | i in grow && i < threshold )+ if i < threshold then {i} else {};
grow := grow + {i};
if i < threshold {
count := count + 1;
}
}
}
| method CountLessThan(numbers: set<int>, threshold: int) returns (count: int)
ensures count == |set i | i in numbers && i < threshold|
{
count := 0;
var shrink := numbers;
var grow := {};
while |shrink | > 0
{
var i: int :| i in shrink;
shrink := shrink - {i};
var grow' := grow+{i};
grow := grow + {i};
if i < threshold {
count := count + 1;
}
}
}
|
038 | Clover_double_array_elements.dfy | method double_array_elements(s: array<int>)
modifies s
ensures forall i :: 0 <= i < s.Length ==> s[i] == 2 * old(s[i])
{
var i:= 0;
while (i < s.Length)
invariant 0 <= i <= s.Length
invariant forall x :: i <= x < s.Length ==> s[x] == old(s[x])
invariant forall x :: 0 <= x < i ==> s[x] == 2 * old(s[x])
{
s[i] := 2 * s[i];
i := i + 1;
}
} | method double_array_elements(s: array<int>)
modifies s
ensures forall i :: 0 <= i < s.Length ==> s[i] == 2 * old(s[i])
{
var i:= 0;
while (i < s.Length)
{
s[i] := 2 * s[i];
i := i + 1;
}
} |
039 | Clover_double_quadruple.dfy | method DoubleQuadruple(x: int) returns (a: int, b: int)
ensures a == 2 * x && b == 4 * x
{
a := 2 * x;
b := 2 * a;
}
| method DoubleQuadruple(x: int) returns (a: int, b: int)
ensures a == 2 * x && b == 4 * x
{
a := 2 * x;
b := 2 * a;
}
|
040 | Clover_even_list.dfy | method FindEvenNumbers (arr: array<int>) returns (evenNumbers: array<int>)
ensures forall x {:trigger (x%2) }:: x in arr[..] && (x%2==0)==> x in evenNumbers[..]
ensures forall x :: x !in arr[..] ==> x !in evenNumbers[..]
ensures forall k :: 0 <= k < evenNumbers.Length ==> evenNumbers[k] % 2 == 0
ensures forall k, l :: 0 <= k < l < evenNumbers.Length ==>
exists n, m :: 0 <= n < m < arr.Length && evenNumbers[k] == arr[n] && evenNumbers[l] == arr[m]
{
var evenList: seq<int> := [];
ghost var indices: seq<int> := [];
for i := 0 to arr.Length
invariant 0 <= i <= arr.Length
invariant 0 <= |evenList| <= i
invariant forall x {:trigger (x%2) }:: (x in arr[..i] && (x%2==0) )==> x in evenList[..]
invariant forall k :: 0 <= k < |evenList| ==> evenList[k] % 2 == 0
invariant forall x :: x !in arr[..i] ==> x !in evenList
invariant |evenList| == |indices|
invariant forall k :: 0 <= k < |indices| ==> indices[k] < i
invariant forall k, l :: 0 <= k < l < |indices| ==> indices[k] < indices[l]
invariant forall k :: 0 <= k < |evenList| ==> 0 <= indices[k] < i <= arr.Length && arr[indices[k]] == evenList[k]
{
if arr[i]%2==0
{
evenList := evenList + [arr[i]];
indices := indices + [i];
}
}
evenNumbers := new int[|evenList|](i requires 0 <= i < |evenList| => evenList[i]);
assert evenList == evenNumbers[..];
}
| method FindEvenNumbers (arr: array<int>) returns (evenNumbers: array<int>)
ensures forall x {:trigger (x%2) }:: x in arr[..] && (x%2==0)==> x in evenNumbers[..]
ensures forall x :: x !in arr[..] ==> x !in evenNumbers[..]
ensures forall k :: 0 <= k < evenNumbers.Length ==> evenNumbers[k] % 2 == 0
ensures forall k, l :: 0 <= k < l < evenNumbers.Length ==>
exists n, m :: 0 <= n < m < arr.Length && evenNumbers[k] == arr[n] && evenNumbers[l] == arr[m]
{
var evenList: seq<int> := [];
ghost var indices: seq<int> := [];
for i := 0 to arr.Length
{
if arr[i]%2==0
{
evenList := evenList + [arr[i]];
indices := indices + [i];
}
}
evenNumbers := new int[|evenList|](i requires 0 <= i < |evenList| => evenList[i]);
}
|
041 | Clover_find.dfy | method Find(a: array<int>, key: int) returns (index: int)
ensures -1<=index<a.Length
ensures index!=-1 ==> a[index]==key && (forall i :: 0 <= i < index ==> a[i] != key)
ensures index == -1 ==> (forall i::0 <= i < a.Length ==> a[i] != key)
{
index := 0;
while index < a.Length
invariant 0<=index<=a.Length
invariant (forall i::0 <= i < index==>a[i] != key)
{
if a[index] == key { return; }
index := index + 1;
}
if index >= a.Length {
index := -1;
}
}
| method Find(a: array<int>, key: int) returns (index: int)
ensures -1<=index<a.Length
ensures index!=-1 ==> a[index]==key && (forall i :: 0 <= i < index ==> a[i] != key)
ensures index == -1 ==> (forall i::0 <= i < a.Length ==> a[i] != key)
{
index := 0;
while index < a.Length
{
if a[index] == key { return; }
index := index + 1;
}
if index >= a.Length {
index := -1;
}
}
|
042 | Clover_has_close_elements.dfy | method has_close_elements(numbers: seq<real>, threshold: real) returns (res: bool)
requires threshold >= 0.0
ensures res ==> exists i: int, j: int :: 0 <= i < |numbers| && 0 <= j < |numbers| && i != j && (if numbers[i] - numbers[j] < 0.0 then numbers[j] - numbers[i] else numbers[i] - numbers[j]) < threshold
ensures !res ==> (forall i: int, j: int :: 1 <= i < |numbers| && 0 <= j < i ==> (if numbers[i] - numbers[j] < 0.0 then numbers[j] - numbers[i] else numbers[i] - numbers[j]) >= threshold)
{
res := false;
var idx: int := 0;
while idx < |numbers| && !res
invariant 0 <= idx <= |numbers|
invariant !res
invariant forall i: int, j: int :: 0 <= i < idx && 0 <= j < i ==> (if numbers[i] - numbers[j] < 0.0 then numbers[j] - numbers[i] else numbers[i] - numbers[j]) >= threshold
{
var idx2: int := 0;
while idx2 < idx && !res
invariant 0 <= idx <= |numbers|
invariant 0 <= idx2 <= idx
invariant !res
invariant forall j: int :: 0 <= j < idx2 ==> (if numbers[idx] - numbers[j] < 0.0 then numbers[j] - numbers[idx] else numbers[idx] - numbers[j]) >= threshold
{
var distance := (if numbers[idx2] - numbers[idx] < 0.0 then numbers[idx] - numbers[idx2] else numbers[idx2] - numbers[idx]);
if distance < threshold {
res := true;
return;
}
idx2 := idx2 + 1;
}
idx := idx + 1;
}
}
| method has_close_elements(numbers: seq<real>, threshold: real) returns (res: bool)
requires threshold >= 0.0
ensures res ==> exists i: int, j: int :: 0 <= i < |numbers| && 0 <= j < |numbers| && i != j && (if numbers[i] - numbers[j] < 0.0 then numbers[j] - numbers[i] else numbers[i] - numbers[j]) < threshold
ensures !res ==> (forall i: int, j: int :: 1 <= i < |numbers| && 0 <= j < i ==> (if numbers[i] - numbers[j] < 0.0 then numbers[j] - numbers[i] else numbers[i] - numbers[j]) >= threshold)
{
res := false;
var idx: int := 0;
while idx < |numbers| && !res
{
var idx2: int := 0;
while idx2 < idx && !res
{
var distance := (if numbers[idx2] - numbers[idx] < 0.0 then numbers[idx] - numbers[idx2] else numbers[idx2] - numbers[idx]);
if distance < threshold {
res := true;
return;
}
idx2 := idx2 + 1;
}
idx := idx + 1;
}
}
|
043 | Clover_insert.dfy | method insert(line:array<char>, l:int, nl:array<char>, p:int, at:int)
requires 0 <= l+p <= line.Length
requires 0 <= p <= nl.Length
requires 0 <= at <= l
modifies line
ensures forall i :: (0<=i<p) ==> line[at+i] == nl[i]
ensures forall i :: (0<=i<at) ==> line[i] == old(line[i])
ensures forall i :: (at+p<=i<l+p) ==> line[i] == old(line[i-p])
{
ghost var initialLine := line[..];
var i:int := l;
while(i>at)
invariant line[0..i] == initialLine[0..i]
invariant line[i+p..l+p] == initialLine[i..l]
invariant at<=i<=l
{
i := i - 1;
line[i+p] := line[i];
}
assert line[0..at] == initialLine[0..at];
assert line[at+p..l+p] == initialLine[at..l];
i := 0;
while(i<p)
invariant 0<=i<=p
invariant line[0..at] == initialLine[0..at]
invariant line[at..at+i] == nl[0..i]
invariant line[at+p..l+p] == initialLine[at..l]
{
line[at + i] := nl[i];
i := i + 1;
}
assert line[0..at] == initialLine[0..at];
assert line[at..at+p] == nl[0..p];
assert line[at+p..l+p] == initialLine[at..l];
} | method insert(line:array<char>, l:int, nl:array<char>, p:int, at:int)
requires 0 <= l+p <= line.Length
requires 0 <= p <= nl.Length
requires 0 <= at <= l
modifies line
ensures forall i :: (0<=i<p) ==> line[at+i] == nl[i]
ensures forall i :: (0<=i<at) ==> line[i] == old(line[i])
ensures forall i :: (at+p<=i<l+p) ==> line[i] == old(line[i-p])
{
ghost var initialLine := line[..];
var i:int := l;
while(i>at)
{
i := i - 1;
line[i+p] := line[i];
}
i := 0;
while(i<p)
{
line[at + i] := nl[i];
i := i + 1;
}
} |
044 | Clover_integer_square_root.dfy | method SquareRoot(N:nat) returns (r:nat)
ensures r*r <= N < (r+1)*(r+1)
{
r:=0;
while (r+1)*(r+1)<=N
invariant r*r<=N
{
r:=r+1;
}
}
| method SquareRoot(N:nat) returns (r:nat)
ensures r*r <= N < (r+1)*(r+1)
{
r:=0;
while (r+1)*(r+1)<=N
{
r:=r+1;
}
}
|
045 | Clover_is_even.dfy | method ComputeIsEven(x:int) returns (is_even:bool)
ensures (x % 2 == 0)==is_even
{
is_even:=false;
if x%2==0{
is_even:=true;
}
}
| method ComputeIsEven(x:int) returns (is_even:bool)
ensures (x % 2 == 0)==is_even
{
is_even:=false;
if x%2==0{
is_even:=true;
}
}
|
046 | Clover_is_palindrome.dfy | method IsPalindrome(x: seq<char>) returns (result: bool)
ensures result <==> (forall i :: 0 <= i < |x| ==> x[i] == x[|x| - i - 1])
{
if |x|==0 {
return true;
}
var i := 0;
var j := |x| - 1;
result := true;
while (i < j)
invariant 0<=i<=j+1 && 0<=j < |x|
invariant i+j==|x|-1
invariant (forall k :: 0 <= k < i ==> x[k] == x[|x| - k - 1])
{
if x[i] != x[j] {
result := false;
return;
}
i := i + 1;
j := j - 1;
}
}
| method IsPalindrome(x: seq<char>) returns (result: bool)
ensures result <==> (forall i :: 0 <= i < |x| ==> x[i] == x[|x| - i - 1])
{
if |x|==0 {
return true;
}
var i := 0;
var j := |x| - 1;
result := true;
while (i < j)
{
if x[i] != x[j] {
result := false;
return;
}
i := i + 1;
j := j - 1;
}
}
|
047 | Clover_linear_search1.dfy | method LinearSearch(a: array<int>, e: int) returns (n:int)
ensures 0<=n<=a.Length
ensures n==a.Length || a[n]==e
ensures forall i::0<=i < n ==> e!=a[i]
{
n :=0;
while n!=a.Length
invariant 0<=n<=a.Length
invariant forall i::0<=i<n ==> e!=a[i]
{
if e==a[n]{
return;
}
n:=n+1;
}
}
| method LinearSearch(a: array<int>, e: int) returns (n:int)
ensures 0<=n<=a.Length
ensures n==a.Length || a[n]==e
ensures forall i::0<=i < n ==> e!=a[i]
{
n :=0;
while n!=a.Length
{
if e==a[n]{
return;
}
n:=n+1;
}
}
|
048 | Clover_linear_search2.dfy | method LinearSearch(a: array<int>, e: int) returns (n:int)
requires exists i::0<=i<a.Length && a[i]==e
ensures 0<=n<a.Length && a[n]==e
ensures forall k :: 0 <= k < n ==> a[k]!=e
{
n :=0;
while n!=a.Length
invariant 0<=n<=a.Length
invariant forall i::0<=i<n ==> e!=a[i]
{
if e==a[n]{
return;
}
n:=n+1;
}
}
| method LinearSearch(a: array<int>, e: int) returns (n:int)
requires exists i::0<=i<a.Length && a[i]==e
ensures 0<=n<a.Length && a[n]==e
ensures forall k :: 0 <= k < n ==> a[k]!=e
{
n :=0;
while n!=a.Length
{
if e==a[n]{
return;
}
n:=n+1;
}
}
|
049 | Clover_linear_search3.dfy | method LinearSearch3<T>(a: array<T>, P: T -> bool) returns (n: int)
requires exists i :: 0 <= i < a.Length && P(a[i])
ensures 0 <= n < a.Length && P(a[n])
ensures forall k :: 0 <= k < n ==> !P(a[k])
{
n := 0;
while true
invariant 0 <= n < a.Length
invariant exists i :: n <= i < a.Length && P(a[i])
invariant forall k :: 0 <= k < n ==> !P(a[k])
decreases a.Length - n
{
if P(a[n]) {
return;
}
n := n + 1;
}
}
| method LinearSearch3<T>(a: array<T>, P: T -> bool) returns (n: int)
requires exists i :: 0 <= i < a.Length && P(a[i])
ensures 0 <= n < a.Length && P(a[n])
ensures forall k :: 0 <= k < n ==> !P(a[k])
{
n := 0;
while true
{
if P(a[n]) {
return;
}
n := n + 1;
}
}
|
050 | Clover_longest_prefix.dfy | method LongestCommonPrefix(str1: seq<char>, str2: seq<char>) returns (prefix: seq<char>)
ensures |prefix| <= |str1| && prefix == str1[0..|prefix|]&& |prefix| <= |str2| && prefix == str2[0..|prefix|]
ensures |prefix|==|str1| || |prefix|==|str2| || (str1[|prefix|]!=str2[|prefix|])
{
prefix := [];
var minLength := if |str1| <|str2| then |str1| else |str2|;
for idx:= 0 to minLength
invariant |prefix|==idx <= minLength<=|str1| && minLength<=|str2|
invariant |prefix| <= |str1| && prefix == str1[0..|prefix|]&& |prefix| <= |str2| && prefix == str2[0..|prefix|]
{
if str1[idx] != str2[idx] {
return;
}
prefix := prefix + [str1[idx]];
}
}
| method LongestCommonPrefix(str1: seq<char>, str2: seq<char>) returns (prefix: seq<char>)
ensures |prefix| <= |str1| && prefix == str1[0..|prefix|]&& |prefix| <= |str2| && prefix == str2[0..|prefix|]
ensures |prefix|==|str1| || |prefix|==|str2| || (str1[|prefix|]!=str2[|prefix|])
{
prefix := [];
var minLength := if |str1| <|str2| then |str1| else |str2|;
for idx:= 0 to minLength
{
if str1[idx] != str2[idx] {
return;
}
prefix := prefix + [str1[idx]];
}
}
|
051 | Clover_match.dfy | method Match(s: string, p: string) returns (b: bool)
requires |s| == |p|
ensures b <==> forall n :: 0 <= n < |s| ==> s[n] == p[n] || p[n] == '?'
{
var i := 0;
while i < |s|
invariant 0 <= i <= |s|
invariant forall n :: 0 <= n < i ==> s[n] == p[n] || p[n] == '?'
{
if s[i] != p[i] && p[i] != '?'
{
return false;
}
i := i + 1;
}
return true;
}
| method Match(s: string, p: string) returns (b: bool)
requires |s| == |p|
ensures b <==> forall n :: 0 <= n < |s| ==> s[n] == p[n] || p[n] == '?'
{
var i := 0;
while i < |s|
{
if s[i] != p[i] && p[i] != '?'
{
return false;
}
i := i + 1;
}
return true;
}
|
052 | Clover_max_array.dfy | method maxArray(a: array<int>) returns (m: int)
requires a.Length >= 1
ensures forall k :: 0 <= k < a.Length ==> m >= a[k]
ensures exists k :: 0 <= k < a.Length && m == a[k]
{
m := a[0];
var index := 1;
while (index < a.Length)
invariant 0 <= index <= a.Length
invariant forall k :: 0 <= k < index ==> m >= a[k]
invariant exists k :: 0 <= k < index && m == a[k]
decreases a.Length - index
{
m := if m>a[index] then m else a[index];
index := index + 1;
}
}
| method maxArray(a: array<int>) returns (m: int)
requires a.Length >= 1
ensures forall k :: 0 <= k < a.Length ==> m >= a[k]
ensures exists k :: 0 <= k < a.Length && m == a[k]
{
m := a[0];
var index := 1;
while (index < a.Length)
{
m := if m>a[index] then m else a[index];
index := index + 1;
}
}
|
053 | Clover_min_array.dfy | method minArray(a: array<int>) returns (r:int)
requires a.Length > 0
ensures forall i :: 0 <= i < a.Length ==> r <= a[i]
ensures exists i :: 0 <= i < a.Length && r == a[i]
{
r:=a[0];
var i:=1;
while i<a.Length
invariant 0 <= i <= a.Length
invariant forall x :: 0 <= x < i ==> r <= a[x]
invariant exists x :: 0 <= x < i && r == a[x]
{
if r>a[i]{
r:=a[i];
}
i:=i+1;
}
}
| method minArray(a: array<int>) returns (r:int)
requires a.Length > 0
ensures forall i :: 0 <= i < a.Length ==> r <= a[i]
ensures exists i :: 0 <= i < a.Length && r == a[i]
{
r:=a[0];
var i:=1;
while i<a.Length
{
if r>a[i]{
r:=a[i];
}
i:=i+1;
}
}
|
054 | Clover_min_of_two.dfy | method Min(x: int, y:int) returns (z: int)
ensures x<=y ==> z==x
ensures x>y ==> z==y
{
if x < y {
return x;
} else {
return y;
}
}
| method Min(x: int, y:int) returns (z: int)
ensures x<=y ==> z==x
ensures x>y ==> z==y
{
if x < y {
return x;
} else {
return y;
}
}
|
055 | Clover_modify_2d_array.dfy | method modify_array_element(arr: array<array<nat>>, index1: nat, index2: nat, val: nat)
requires index1 < arr.Length
requires index2 < arr[index1].Length
requires forall i: nat, j:nat :: i < arr.Length && j < arr.Length && i != j ==> arr[i] != arr[j]
modifies arr[index1]
ensures forall i: nat :: 0 <= i < arr.Length ==> arr[i] == old(arr[i])
ensures forall i: nat, j: nat :: 0 <= i < arr.Length && 0 <= j < arr[i].Length && (i != index1 || j != index2) ==> arr[i][j] == old(arr[i][j])
ensures arr[index1][index2] == val
{
arr[index1][index2] := val;
}
| method modify_array_element(arr: array<array<nat>>, index1: nat, index2: nat, val: nat)
requires index1 < arr.Length
requires index2 < arr[index1].Length
requires forall i: nat, j:nat :: i < arr.Length && j < arr.Length && i != j ==> arr[i] != arr[j]
modifies arr[index1]
ensures forall i: nat :: 0 <= i < arr.Length ==> arr[i] == old(arr[i])
ensures forall i: nat, j: nat :: 0 <= i < arr.Length && 0 <= j < arr[i].Length && (i != index1 || j != index2) ==> arr[i][j] == old(arr[i][j])
ensures arr[index1][index2] == val
{
arr[index1][index2] := val;
}
|
056 | Clover_multi_return.dfy | method MultipleReturns(x: int, y: int) returns (more: int, less: int)
ensures more == x+y
ensures less == x-y
{
more := x + y;
less := x - y;
}
| method MultipleReturns(x: int, y: int) returns (more: int, less: int)
ensures more == x+y
ensures less == x-y
{
more := x + y;
less := x - y;
}
|
057 | Clover_online_max.dfy | method onlineMax(a: array<int>, x: int) returns (ghost m:int, p:int)
requires 1<=x<a.Length
requires a.Length!=0
ensures x<=p<a.Length
ensures forall i::0<=i<x==> a[i]<=m
ensures exists i::0<=i<x && a[i]==m
ensures x<=p<a.Length-1 ==> (forall i::0<=i<p ==> a[i]<a[p])
ensures (forall i::x<=i<a.Length && a[i]<=m) ==> p==a.Length-1
{
p:= 0;
var best := a[0];
var i:=1;
while i<x
invariant 0<=i<=x
invariant forall j::0<=j<i==> a[j]<=best
invariant exists j::0<=j<i && a[j]==best
{
if a[i]>best{
best:=a[i];
}
i:=i+1;
}
m:=best;
i:=x;
while i<a.Length
invariant x<=i<=a.Length
invariant forall j::x<=j<i ==> a[j]<=m
{
if a[i]>best{
p:=i;
return;
}
i:=i+1;
}
p:=a.Length-1;
}
| method onlineMax(a: array<int>, x: int) returns (ghost m:int, p:int)
requires 1<=x<a.Length
requires a.Length!=0
ensures x<=p<a.Length
ensures forall i::0<=i<x==> a[i]<=m
ensures exists i::0<=i<x && a[i]==m
ensures x<=p<a.Length-1 ==> (forall i::0<=i<p ==> a[i]<a[p])
ensures (forall i::x<=i<a.Length && a[i]<=m) ==> p==a.Length-1
{
p:= 0;
var best := a[0];
var i:=1;
while i<x
{
if a[i]>best{
best:=a[i];
}
i:=i+1;
}
m:=best;
i:=x;
while i<a.Length
{
if a[i]>best{
p:=i;
return;
}
i:=i+1;
}
p:=a.Length-1;
}
|
058 | Clover_only_once.dfy | method only_once<T(==)>(a: array<T>, key: T) returns (b:bool)
ensures (multiset(a[..])[key] ==1 ) <==> b
{
var i := 0;
b := false;
var keyCount := 0;
while i < a.Length
invariant 0 <= i <= a.Length
invariant multiset(a[..i])[key] == keyCount
invariant b <==> (keyCount == 1)
{
if (a[i] == key)
{
keyCount := keyCount + 1;
}
if (keyCount == 1)
{ b := true; }
else
{ b := false; }
i := i + 1;
}
assert a[..a.Length] == a[..];
}
| method only_once<T(==)>(a: array<T>, key: T) returns (b:bool)
ensures (multiset(a[..])[key] ==1 ) <==> b
{
var i := 0;
b := false;
var keyCount := 0;
while i < a.Length
{
if (a[i] == key)
{
keyCount := keyCount + 1;
}
if (keyCount == 1)
{ b := true; }
else
{ b := false; }
i := i + 1;
}
}
|
059 | Clover_quotient.dfy | method Quotient(x: nat, y:nat) returns (r:int, q:int)
requires y != 0
ensures q * y + r == x && 0 <= r < y && 0 <= q
{
r:=x;
q:=0;
while y<=r
invariant q*y+r==x && r>=0
decreases r
{
r:=r-y;
q:=q+1;
}
}
| method Quotient(x: nat, y:nat) returns (r:int, q:int)
requires y != 0
ensures q * y + r == x && 0 <= r < y && 0 <= q
{
r:=x;
q:=0;
while y<=r
{
r:=r-y;
q:=q+1;
}
}
|
060 | Clover_remove_front.dfy | method remove_front(a:array<int>) returns (c:array<int>)
requires a.Length>0
ensures a[1..] == c[..]
{
c := new int[a.Length-1];
var i:= 1;
while (i < a.Length)
invariant 1 <= i <= a.Length
invariant forall ii::1<=ii<i ==> c[ii-1]==a[ii]
{
c[i-1] := a[i];
i:=i+1;
}
}
| method remove_front(a:array<int>) returns (c:array<int>)
requires a.Length>0
ensures a[1..] == c[..]
{
c := new int[a.Length-1];
var i:= 1;
while (i < a.Length)
{
c[i-1] := a[i];
i:=i+1;
}
}
|
061 | Clover_replace.dfy | method replace(arr: array<int>, k: int)
modifies arr
ensures forall i :: 0 <= i < arr.Length ==> old(arr[i]) > k ==> arr[i] == -1
ensures forall i :: 0 <= i < arr.Length ==> old(arr[i]) <= k ==> arr[i] == old(arr[i])
{
var i := 0;
while i < arr.Length
decreases arr.Length - i
invariant 0 <= i <= arr.Length
invariant forall j :: 0 <= j < i ==> old(arr[j]) > k ==> arr[j] == -1
invariant forall j :: 0 <= j < i ==> old(arr[j]) <= k ==> arr[j] == old(arr[j])
invariant forall j :: i <= j < arr.Length ==> old(arr[j]) == arr[j]
{
if arr[i] > k {
arr[i] := -1;
}
i := i + 1;
}
}
| method replace(arr: array<int>, k: int)
modifies arr
ensures forall i :: 0 <= i < arr.Length ==> old(arr[i]) > k ==> arr[i] == -1
ensures forall i :: 0 <= i < arr.Length ==> old(arr[i]) <= k ==> arr[i] == old(arr[i])
{
var i := 0;
while i < arr.Length
{
if arr[i] > k {
arr[i] := -1;
}
i := i + 1;
}
}
|
062 | Clover_return_seven.dfy | method M(x: int) returns (seven: int)
ensures seven==7
{
seven := 7;
}
| method M(x: int) returns (seven: int)
ensures seven==7
{
seven := 7;
}
|
063 | Clover_reverse.dfy | method reverse(a: array<int>)
modifies a
ensures forall i :: 0 <= i < a.Length ==> a[i] == old(a[a.Length - 1 - i])
{
var i := 0;
while i <a.Length / 2
invariant 0 <= i <= a.Length/2
invariant forall k :: 0 <= k < i || a.Length-1-i < k <= a.Length-1 ==> a[k] == old(a[a.Length-1-k])
invariant forall k :: i <= k <= a.Length-1-i ==> a[k] == old(a[k])
{
a[i], a[a.Length-1-i] := a[a.Length-1-i], a[i];
i := i + 1;
}
}
| method reverse(a: array<int>)
modifies a
ensures forall i :: 0 <= i < a.Length ==> a[i] == old(a[a.Length - 1 - i])
{
var i := 0;
while i <a.Length / 2
{
a[i], a[a.Length-1-i] := a[a.Length-1-i], a[i];
i := i + 1;
}
}
|
064 | Clover_rotate.dfy | method rotate(a: array<int>, offset:int) returns (b: array<int> )
requires 0<=offset
ensures b.Length==a.Length
ensures forall i::0<=i<a.Length ==> b[i]==a[(i+offset)%a.Length]
{
b:= new int[a.Length];
var i:=0;
while i<a.Length
invariant 0<=i<=a.Length
invariant forall j ::0<=j<i ==> b[j]==a[(j+offset)%a.Length]
{
b[i]:=a[(i+offset)%a.Length];
i:=i+1;
}
} | method rotate(a: array<int>, offset:int) returns (b: array<int> )
requires 0<=offset
ensures b.Length==a.Length
ensures forall i::0<=i<a.Length ==> b[i]==a[(i+offset)%a.Length]
{
b:= new int[a.Length];
var i:=0;
while i<a.Length
{
b[i]:=a[(i+offset)%a.Length];
i:=i+1;
}
} |
065 | Clover_selectionsort.dfy | method SelectionSort(a: array<int>)
modifies a
ensures forall i,j :: 0 <= i < j < a.Length ==> a[i] <= a[j]
ensures multiset(a[..]) == old(multiset(a[..]))
{
var n:= 0;
while n != a.Length
invariant 0 <= n <= a.Length
invariant forall i, j :: 0 <= i < n <= j < a.Length ==> a[i] <= a[j]
invariant forall i,j :: 0 <= i < j < n ==> a[i] <= a[j]
invariant multiset(a[..]) == old(multiset(a[..]))
{
var mindex, m := n, n+1;
while m != a.Length
invariant n <= mindex < m <= a.Length
invariant forall i :: n <= i < m ==> a[mindex] <= a[i]
{
if a[m] < a[mindex] {
mindex := m;
}
m := m+1;
}
a[n], a[mindex] := a[mindex], a[n];
n := n+1;
}
}
| method SelectionSort(a: array<int>)
modifies a
ensures forall i,j :: 0 <= i < j < a.Length ==> a[i] <= a[j]
ensures multiset(a[..]) == old(multiset(a[..]))
{
var n:= 0;
while n != a.Length
{
var mindex, m := n, n+1;
while m != a.Length
{
if a[m] < a[mindex] {
mindex := m;
}
m := m+1;
}
a[n], a[mindex] := a[mindex], a[n];
n := n+1;
}
}
|
066 | Clover_seq_to_array.dfy | method ToArray<T>(xs: seq<T>) returns (a: array<T>)
ensures fresh(a)
ensures a.Length == |xs|
ensures forall i :: 0 <= i < |xs| ==> a[i] == xs[i]
{
a := new T[|xs|](i requires 0 <= i < |xs| => xs[i]);
}
| method ToArray<T>(xs: seq<T>) returns (a: array<T>)
ensures fresh(a)
ensures a.Length == |xs|
ensures forall i :: 0 <= i < |xs| ==> a[i] == xs[i]
{
a := new T[|xs|](i requires 0 <= i < |xs| => xs[i]);
}
|
067 | Clover_set_to_seq.dfy | method SetToSeq<T>(s: set<T>) returns (xs: seq<T>)
ensures multiset(s) == multiset(xs)
{
xs := [];
var left: set<T> := s;
while left != {}
invariant multiset(left) + multiset(xs) == multiset(s)
{
var x :| x in left;
left := left - {x};
xs := xs + [x];
}
}
| method SetToSeq<T>(s: set<T>) returns (xs: seq<T>)
ensures multiset(s) == multiset(xs)
{
xs := [];
var left: set<T> := s;
while left != {}
{
var x :| x in left;
left := left - {x};
xs := xs + [x];
}
}
|
068 | Clover_slope_search.dfy | method SlopeSearch(a: array2<int>, key: int) returns (m:int, n:int)
requires forall i,j,j'::0<=i<a.Length0 && 0<=j<j'<a.Length1 ==> a[i,j]<=a[i,j']
requires forall i,i',j::0<=i<i'<a.Length0 && 0<=j<a.Length1 ==> a[i,j]<=a[i',j]
requires exists i,j :: 0<=i<a.Length0 && 0<=j<a.Length1 && a[i,j]==key
ensures 0<=m<a.Length0 && 0<=n<a.Length1
ensures a[m,n]==key
{
m,n:=0, a.Length1-1;
while a[m,n] !=key
invariant 0<=m<a.Length0 && 0<=n<a.Length1
invariant exists i,j :: m<=i<a.Length0 && 0<=j<=n && a[i,j]==key
decreases a.Length0-m+n
{
if a[m,n] < key {
m:=m+1;
}else{
n:=n-1;
}
}
}
| method SlopeSearch(a: array2<int>, key: int) returns (m:int, n:int)
requires forall i,j,j'::0<=i<a.Length0 && 0<=j<j'<a.Length1 ==> a[i,j]<=a[i,j']
requires forall i,i',j::0<=i<i'<a.Length0 && 0<=j<a.Length1 ==> a[i,j]<=a[i',j]
requires exists i,j :: 0<=i<a.Length0 && 0<=j<a.Length1 && a[i,j]==key
ensures 0<=m<a.Length0 && 0<=n<a.Length1
ensures a[m,n]==key
{
m,n:=0, a.Length1-1;
while a[m,n] !=key
{
if a[m,n] < key {
m:=m+1;
}else{
n:=n-1;
}
}
}
|
069 | Clover_swap.dfy | method Swap(X: int, Y: int) returns(x: int, y: int)
ensures x==Y
ensures y==X
{
x, y := X, Y;
var tmp := x;
x := y;
y := tmp;
assert x == Y && y == X;
}
| method Swap(X: int, Y: int) returns(x: int, y: int)
ensures x==Y
ensures y==X
{
x, y := X, Y;
var tmp := x;
x := y;
y := tmp;
}
|
070 | Clover_swap_arith.dfy | method SwapArithmetic(X: int, Y: int) returns(x: int, y: int)
ensures x==Y
ensures y==X
{
x, y := X, Y;
x := y - x;
y := y - x;
x := y + x;
}
| method SwapArithmetic(X: int, Y: int) returns(x: int, y: int)
ensures x==Y
ensures y==X
{
x, y := X, Y;
x := y - x;
y := y - x;
x := y + x;
}
|
071 | Clover_swap_bitvector.dfy | method SwapBitvectors(X: bv8, Y: bv8) returns(x: bv8, y: bv8)
ensures x==Y
ensures y==X
{
x, y := X, Y;
x := x ^ y;
y := x ^ y;
x := x ^ y;
}
| method SwapBitvectors(X: bv8, Y: bv8) returns(x: bv8, y: bv8)
ensures x==Y
ensures y==X
{
x, y := X, Y;
x := x ^ y;
y := x ^ y;
x := x ^ y;
}
|
072 | Clover_swap_in_array.dfy | method swap(arr: array<int>, i: int, j: int)
requires 0 <= i < arr.Length && 0 <= j < arr.Length
modifies arr
ensures arr[i] == old(arr[j]) && arr[j] == old(arr[i])
ensures forall k :: 0 <= k < arr.Length && k != i && k != j ==> arr[k] == old(arr[k])
{
var tmp := arr[i];
arr[i] := arr[j];
arr[j] := tmp;
}
| method swap(arr: array<int>, i: int, j: int)
requires 0 <= i < arr.Length && 0 <= j < arr.Length
modifies arr
ensures arr[i] == old(arr[j]) && arr[j] == old(arr[i])
ensures forall k :: 0 <= k < arr.Length && k != i && k != j ==> arr[k] == old(arr[k])
{
var tmp := arr[i];
arr[i] := arr[j];
arr[j] := tmp;
}
|
073 | Clover_swap_sim.dfy | method SwapSimultaneous(X: int, Y: int) returns(x: int, y: int)
ensures x==Y
ensures y==X
{
x, y := X, Y;
x, y := y, x;
}
| method SwapSimultaneous(X: int, Y: int) returns(x: int, y: int)
ensures x==Y
ensures y==X
{
x, y := X, Y;
x, y := y, x;
}
|
074 | Clover_test_array.dfy | method TestArrayElements(a:array<int>, j: nat)
requires 0<=j < a.Length
modifies a
ensures a[j] == 60
ensures forall k :: 0 <= k < a.Length && k != j ==> a[k] == old(a[k])
{
a[j] := 60;
}
| method TestArrayElements(a:array<int>, j: nat)
requires 0<=j < a.Length
modifies a
ensures a[j] == 60
ensures forall k :: 0 <= k < a.Length && k != j ==> a[k] == old(a[k])
{
a[j] := 60;
}
|
075 | Clover_triple.dfy | method Triple (x:int) returns (r:int)
ensures r==3*x
{
r:= x*3;
}
| method Triple (x:int) returns (r:int)
ensures r==3*x
{
r:= x*3;
}
|
076 | Clover_triple2.dfy | method Triple (x:int) returns (r:int)
ensures r==3*x
{
if {
case x<18 =>
var a,b := 2*x, 4*x;
r:=(a+b)/2;
case 0<=x =>
var y:=2*x;
r:= x+y;
}
}
| method Triple (x:int) returns (r:int)
ensures r==3*x
{
if {
case x<18 =>
var a,b := 2*x, 4*x;
r:=(a+b)/2;
case 0<=x =>
var y:=2*x;
r:= x+y;
}
}
|
077 | Clover_triple3.dfy | method Triple (x:int) returns (r:int)
ensures r==3*x
{
if x==0 {
r:=0;
}
else{
var y:=2*x;
r:= x+y;
}
}
| method Triple (x:int) returns (r:int)
ensures r==3*x
{
if x==0 {
r:=0;
}
else{
var y:=2*x;
r:= x+y;
}
}
|
078 | Clover_triple4.dfy | method Triple (x:int) returns (r:int)
ensures r==3*x
{
var y:= x*2;
r := y+x;
}
| method Triple (x:int) returns (r:int)
ensures r==3*x
{
var y:= x*2;
r := y+x;
}
|
079 | Clover_two_sum.dfy | method twoSum(nums: array<int>, target: int) returns (i: int, j: int)
requires nums.Length > 1
requires exists i,j::0 <= i < j < nums.Length && nums[i] + nums[j] == target
ensures 0 <= i < j < nums.Length && nums[i] + nums[j] == target
ensures forall ii,jj:: (0 <= ii < i && ii < jj < nums.Length) ==> nums[ii] + nums[jj] != target
ensures forall jj:: i < jj < j ==> nums[i] + nums[jj] != target
{
var n := nums.Length;
i := 0;
j := 1;
while i < n - 1
invariant 0 <= i < j <= n
invariant forall ii, jj :: 0 <= ii < i && ii < jj < n ==> nums[ii] + nums[jj] != target
{
j := i + 1;
while j < n
invariant 0 <= i < j <= n
invariant forall jj :: i < jj < j ==> nums[i] + nums[jj] != target
{
if nums[i] + nums[j] == target {
return;
}
j := j + 1;
}
i := i + 1;
}
}
| method twoSum(nums: array<int>, target: int) returns (i: int, j: int)
requires nums.Length > 1
requires exists i,j::0 <= i < j < nums.Length && nums[i] + nums[j] == target
ensures 0 <= i < j < nums.Length && nums[i] + nums[j] == target
ensures forall ii,jj:: (0 <= ii < i && ii < jj < nums.Length) ==> nums[ii] + nums[jj] != target
ensures forall jj:: i < jj < j ==> nums[i] + nums[jj] != target
{
var n := nums.Length;
i := 0;
j := 1;
while i < n - 1
{
j := i + 1;
while j < n
{
if nums[i] + nums[j] == target {
return;
}
j := j + 1;
}
i := i + 1;
}
}
|
080 | Clover_update_array.dfy | method UpdateElements(a: array<int>)
requires a.Length >= 8
modifies a
ensures old(a[4]) +3 == a[4]
ensures a[7]==516
ensures forall i::0 <= i<a.Length ==> i != 7 && i != 4 ==> a[i] == old(a[i])
{
a[4] := a[4] + 3;
a[7] := 516;
}
| method UpdateElements(a: array<int>)
requires a.Length >= 8
modifies a
ensures old(a[4]) +3 == a[4]
ensures a[7]==516
ensures forall i::0 <= i<a.Length ==> i != 7 && i != 4 ==> a[i] == old(a[i])
{
a[4] := a[4] + 3;
a[7] := 516;
}
|
081 | Clover_update_map.dfy | method update_map<K(!new), V>(m1: map<K, V>, m2: map<K, V>) returns (r: map<K, V>)
ensures (forall k :: k in m2 ==> k in r)
ensures (forall k :: k in m1 ==> k in r)
ensures (forall k :: k in m2 ==> r[k] == m2[k])
ensures (forall k :: !(k in m2) && k in m1 ==> r[k] == m1[k])
ensures (forall k :: !(k in m2) && !(k in m1) ==> !(k in r))
{
r:= map k | k in (m1.Keys + m2.Keys) :: if k in m2 then m2[k] else m1[k];
}
| method update_map<K(!new), V>(m1: map<K, V>, m2: map<K, V>) returns (r: map<K, V>)
ensures (forall k :: k in m2 ==> k in r)
ensures (forall k :: k in m1 ==> k in r)
ensures (forall k :: k in m2 ==> r[k] == m2[k])
ensures (forall k :: !(k in m2) && k in m1 ==> r[k] == m1[k])
ensures (forall k :: !(k in m2) && !(k in m1) ==> !(k in r))
{
r:= map k | k in (m1.Keys + m2.Keys) :: if k in m2 then m2[k] else m1[k];
}
|
082 | Correctness_tmp_tmpwqvg5q_4_HoareLogic_exam.dfy | // Redo for exam
function gcd(a: nat, b: nat): nat
lemma r1(a: nat)
ensures gcd(a, 0) == a
lemma r2(a:nat)
ensures gcd(a, a) == a
lemma r3(a: nat, b: nat)
ensures gcd(a, b) == gcd(b, a)
lemma r4 (a: nat, b: nat)
ensures b > 0 ==> gcd(a, b) == gcd(b, a % b)
method GCD1(a: int, b: int) returns (r: int)
requires a > 0 && b > 0
ensures gcd(a,b) == r
decreases b
{
if a < b {
r3(a,b);
r := GCD1(b, a);
} else if (a % b == 0) {
r4(a,b);
assert b > 0;
assert gcd(a, b) == gcd(b, a % b);
assert a % b == 0;
assert gcd(a, b) == gcd(b, 0);
r1(b);
assert gcd(a, b) == b;
r := b;
assert gcd(a,b) == r;
} else {
r4(a,b);
r := GCD1(b, a % b);
assert gcd(a,b) == r;
}
assert gcd(a,b) == r;
}
method GCD2(a: int, b: int) returns (r: int)
requires a > 0 && b >= 0
decreases b
ensures gcd(a,b) == r
{
r1(a);
r4(a,b);
assert
( b != 0 || (a > 0 && b >= 0 && gcd(a,b) == a) )
&&
( (b < 0 || b == 0) || (b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b))) );
assert
b != 0 || (a > 0 && b >= 0 && gcd(a,b) == a);
assert
b == 0 ==> a > 0 && b >= 0 && gcd(a,b) == a;
assert
(b < 0 || b == 0) || (b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b)));
assert
b >= 0 && b != 0 ==> b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b));
if b == 0 {
r1(a);
assert
gcd(a,b) == a;
r := a;
assert
gcd(a,b) == r;
} else {
r4(a,b);
// Method call rule
assert
b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b));
// assert
// gcd(a,b) == GCD2(b, a % b);
r := GCD2(b, a % b);
assert
gcd(a,b) == r;
}
assert
gcd(a,b) == r;
}
| // Redo for exam
function gcd(a: nat, b: nat): nat
lemma r1(a: nat)
ensures gcd(a, 0) == a
lemma r2(a:nat)
ensures gcd(a, a) == a
lemma r3(a: nat, b: nat)
ensures gcd(a, b) == gcd(b, a)
lemma r4 (a: nat, b: nat)
ensures b > 0 ==> gcd(a, b) == gcd(b, a % b)
method GCD1(a: int, b: int) returns (r: int)
requires a > 0 && b > 0
ensures gcd(a,b) == r
{
if a < b {
r3(a,b);
r := GCD1(b, a);
} else if (a % b == 0) {
r4(a,b);
r1(b);
r := b;
} else {
r4(a,b);
r := GCD1(b, a % b);
}
}
method GCD2(a: int, b: int) returns (r: int)
requires a > 0 && b >= 0
ensures gcd(a,b) == r
{
r1(a);
r4(a,b);
( b != 0 || (a > 0 && b >= 0 && gcd(a,b) == a) )
&&
( (b < 0 || b == 0) || (b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b))) );
b != 0 || (a > 0 && b >= 0 && gcd(a,b) == a);
b == 0 ==> a > 0 && b >= 0 && gcd(a,b) == a;
(b < 0 || b == 0) || (b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b)));
b >= 0 && b != 0 ==> b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b));
if b == 0 {
r1(a);
gcd(a,b) == a;
r := a;
gcd(a,b) == r;
} else {
r4(a,b);
// Method call rule
b > 0 && (a % b) >= 0 ==> gcd(a,b) == gcd(b,(a % b));
// assert
// gcd(a,b) == GCD2(b, a % b);
r := GCD2(b, a % b);
gcd(a,b) == r;
}
gcd(a,b) == r;
}
|
083 | Correctness_tmp_tmpwqvg5q_4_MethodCalls_q1.dfy | /**
(a) Verify whether or not the following program
satisfies total correctness.
You should use weakest precondition reasoning
and may extend the loop invariant if required.
You will need to add a decreases clause to prove termination
(a) Weakest precondition proof (without termination) (6 marks)
Termination proof (2marks)
*/
function fusc(n: int): nat
lemma rule1()
ensures fusc(0) == 0
lemma rule2()
ensures fusc(1) == 1
lemma rule3(n:nat)
ensures fusc(2*n) == fusc(n)
lemma rule4(n:nat)
ensures fusc(2*n+1) == fusc(n) + fusc(n+1)
method ComputeFusc(N: int) returns (b: int)
requires N >= 0
ensures b == fusc(N)
{
b := 0;
var n, a := N, 1;
assert 0 <= n <= N;
assert fusc(N) == a * fusc(n) + b * fusc(n + 1);
while (n != 0)
invariant 0 <= n <= N // J
invariant fusc(N) == a * fusc(n) + b * fusc(n + 1) // J
decreases n // D
{
ghost var d := n; // termination metric
assert fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert n != 0;
assert (n % 2 != 0 && n % 2 == 0) || fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert (n % 2 != 0 || n % 2 == 0) ==> fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert n % 2 != 0 || fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert n % 2 == 0 || fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert n % 2 == 0 ==> fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert n % 2 != 0 ==> fusc(N) == a * fusc(n) + b * fusc(n + 1);
if (n % 2 == 0)
{
rule4(n/2);
assert fusc((n/2) + 1) == fusc(n + 1) - fusc(n/2);
rule3(n/2);
assert fusc(n/2) == fusc(n);
assert fusc(N) == (a + b) * fusc(n/2) + b * fusc((n/2) + 1);
a := a + b;
assert fusc(N) == a * fusc(n/2) + b * fusc((n/2) + 1);
n := n / 2;
assert fusc(N) == a * fusc(n) + b * fusc(n + 1);
} else {
rule4((n-1)/2);
assert fusc(n) - fusc((n-1)/2) == fusc(((n-1)/2)+1);
rule3((n-1)/2);
assert fusc((n-1)/2) == fusc(n-1);
assert fusc(((n-1)/2)+1) == fusc((n+1)/2);
rule3((n+1)/2);
assert fusc((n+1)/2) == fusc(n+1);
assert fusc(N) == a * fusc(n) + b * fusc(n + 1);
assert fusc(N) == b * fusc(((n-1)/2)+1) + a * fusc(n);
assert fusc(N) ==
b * fusc(n) - b * fusc(n) + b * fusc(((n-1)/2)+1) + a * fusc(n);
assert fusc(N) ==
b * fusc(n) - b * (fusc(n) - fusc(((n-1)/2)+1)) + a * fusc(n);
assert fusc(N) == b * fusc(n) - b * fusc((n-1)/2) + a * fusc(n);
assert fusc(N) == b * fusc(n) - b * fusc(n-1) + a * fusc(n);
assert fusc(N) == b * fusc(n) - b * fusc(n-1) + a * fusc(n);
assert fusc(N) ==
a * fusc(n - 1) + b * fusc(n) - b * fusc(n-1) + a * fusc(n) - a * fusc(n-1);
assert fusc(N) == a * fusc(n - 1) + (b + a) * (fusc(n) - fusc(n-1));
assert fusc(N) == a * fusc((n - 1)) + (b + a) * (fusc(n) - fusc((n-1)/2));
assert fusc(N) == a * fusc((n - 1) / 2) + (b + a) * fusc(((n - 1) / 2) + 1);
b := b + a;
assert fusc(N) == a * fusc((n - 1) / 2) + b * fusc(((n - 1) / 2) + 1);
n := (n - 1) / 2;
assert fusc(N) == a * fusc(n) + b * fusc(n + 1);
}
assert n < d; // termination metric
assert fusc(N) == a * fusc(n) + b * fusc(n + 1); // J
}
assert n == 0; // !B
rule1();
assert fusc(0) == 0;
rule2();
assert fusc(1) == 1;
assert fusc(N) == a * fusc(0) + b * fusc(0 + 1); // J
assert fusc(N) == a * 0 + b * 1; // J
assert b == fusc(N);
}
| /**
(a) Verify whether or not the following program
satisfies total correctness.
You should use weakest precondition reasoning
and may extend the loop invariant if required.
You will need to add a decreases clause to prove termination
(a) Weakest precondition proof (without termination) (6 marks)
Termination proof (2marks)
*/
function fusc(n: int): nat
lemma rule1()
ensures fusc(0) == 0
lemma rule2()
ensures fusc(1) == 1
lemma rule3(n:nat)
ensures fusc(2*n) == fusc(n)
lemma rule4(n:nat)
ensures fusc(2*n+1) == fusc(n) + fusc(n+1)
method ComputeFusc(N: int) returns (b: int)
requires N >= 0
ensures b == fusc(N)
{
b := 0;
var n, a := N, 1;
while (n != 0)
{
ghost var d := n; // termination metric
if (n % 2 == 0)
{
rule4(n/2);
rule3(n/2);
a := a + b;
n := n / 2;
} else {
rule4((n-1)/2);
rule3((n-1)/2);
rule3((n+1)/2);
b * fusc(n) - b * fusc(n) + b * fusc(((n-1)/2)+1) + a * fusc(n);
b * fusc(n) - b * (fusc(n) - fusc(((n-1)/2)+1)) + a * fusc(n);
a * fusc(n - 1) + b * fusc(n) - b * fusc(n-1) + a * fusc(n) - a * fusc(n-1);
b := b + a;
n := (n - 1) / 2;
}
}
rule1();
rule2();
}
|
084 | Correctness_tmp_tmpwqvg5q_4_Sorting_Tangent.dfy | /**
Ather, Mohammad Faiz (s4648481/3)
CSSE3100
Assignemnt 3
The University of Queensland
*/
// Question 1
method Tangent(r: array<int>, x: array<int>)
returns (found: bool)
requires forall i:: 1 <= i < x.Length ==>
x[i-1] < x[i]
requires forall i, j ::
0 <= i < j < x.Length ==>
x[i] < x[j]
ensures !found ==>
forall i,j ::
0 <= i < r.Length &&
0 <= j < x.Length ==>
r[i] != x[j]
ensures found ==>
exists i,j ::
0 <= i < r.Length &&
0 <= j < x.Length &&
r[i] == x[j]
{
found := false;
var n, f := 0, x.Length;
while n != r.Length && !found
invariant 0 <= n <= r.Length
invariant !found ==>
forall i,j ::
0 <= i < n &&
0 <= j < x.Length ==>
r[i] != x[j]
invariant found ==>
exists i,j ::
0 <= i < r.Length &&
0 <= j < x.Length &&
n == i && f == j &&
r[i] == x[j]
decreases r.Length - n, !found
{
f := BinarySearch(x, r[n]);
/*
not iterate over either array
once a tangent has been found
*/ // see if
if (f != x.Length && r[n] == x[f]) {
found := true;
} else {
n := n + 1;
}
}
assert
(!found && n == r.Length) ||
( found && n != r.Length && r[n] == x[f]);
assert
!false; // sanity check
}
// Author: Leino, Title: Program Proofs
method BinarySearch(a: array<int>, circle: int)
returns (n: int)
requires forall i ::
1 <= i < a.Length
==> a[i-1] < a[i]
requires forall i, j ::
0 <= i < j < a.Length ==>
a[i] < a[j]
ensures 0 <= n <= a.Length
ensures forall i ::
0 <= i < n ==>
a[i] < circle
ensures forall i ::
n <= i < a.Length ==>
circle <= a[i]
{
var lo, hi := 0, a.Length;
while lo < hi
invariant 0 <= lo <= hi <= a.Length
invariant forall i ::
0 <= i < lo ==>
a[i] < circle
invariant forall i ::
hi <= i < a.Length ==>
a[i] >= circle
decreases hi - lo
{
var mid := (lo + hi) / 2;
calc {
lo;
==
(lo + lo) / 2;
<= { assert lo <= hi; }
(lo + hi) / 2;
< { assert lo < hi; }
(hi + hi) / 2;
==
hi;
}
/*
for a given circle in r,
should not iterate over array x
once it can be deduced that
no tangent will be found for that circle.
*/ // see if and 1st else if
if (a[lo] > circle) {
hi := lo;
} else if (a[hi-1] < circle) {
lo := hi;
} else if (a[mid] < circle) {
lo := mid + 1;
} else {
hi := mid;
}
}
n := lo;
assert
!false; // sanity check
}
| /**
Ather, Mohammad Faiz (s4648481/3)
CSSE3100
Assignemnt 3
The University of Queensland
*/
// Question 1
method Tangent(r: array<int>, x: array<int>)
returns (found: bool)
requires forall i:: 1 <= i < x.Length ==>
x[i-1] < x[i]
requires forall i, j ::
0 <= i < j < x.Length ==>
x[i] < x[j]
ensures !found ==>
forall i,j ::
0 <= i < r.Length &&
0 <= j < x.Length ==>
r[i] != x[j]
ensures found ==>
exists i,j ::
0 <= i < r.Length &&
0 <= j < x.Length &&
r[i] == x[j]
{
found := false;
var n, f := 0, x.Length;
while n != r.Length && !found
forall i,j ::
0 <= i < n &&
0 <= j < x.Length ==>
r[i] != x[j]
exists i,j ::
0 <= i < r.Length &&
0 <= j < x.Length &&
n == i && f == j &&
r[i] == x[j]
{
f := BinarySearch(x, r[n]);
/*
not iterate over either array
once a tangent has been found
*/ // see if
if (f != x.Length && r[n] == x[f]) {
found := true;
} else {
n := n + 1;
}
}
(!found && n == r.Length) ||
( found && n != r.Length && r[n] == x[f]);
!false; // sanity check
}
// Author: Leino, Title: Program Proofs
method BinarySearch(a: array<int>, circle: int)
returns (n: int)
requires forall i ::
1 <= i < a.Length
==> a[i-1] < a[i]
requires forall i, j ::
0 <= i < j < a.Length ==>
a[i] < a[j]
ensures 0 <= n <= a.Length
ensures forall i ::
0 <= i < n ==>
a[i] < circle
ensures forall i ::
n <= i < a.Length ==>
circle <= a[i]
{
var lo, hi := 0, a.Length;
while lo < hi
0 <= i < lo ==>
a[i] < circle
hi <= i < a.Length ==>
a[i] >= circle
{
var mid := (lo + hi) / 2;
calc {
lo;
==
(lo + lo) / 2;
<= { assert lo <= hi; }
(lo + hi) / 2;
< { assert lo < hi; }
(hi + hi) / 2;
==
hi;
}
/*
for a given circle in r,
should not iterate over array x
once it can be deduced that
no tangent will be found for that circle.
*/ // see if and 1st else if
if (a[lo] > circle) {
hi := lo;
} else if (a[hi-1] < circle) {
lo := hi;
} else if (a[mid] < circle) {
lo := mid + 1;
} else {
hi := mid;
}
}
n := lo;
!false; // sanity check
}
|
085 | Dafny-Exercises_tmp_tmpjm75muf__Session10Exercises_ExerciseBarrier.dfy |
//Method barrier below receives an array and an integer p
//and returns a boolean b which is true if and only if
//all the positions to the left of p and including also position p contain elements
//that are strictly smaller than all the elements contained in the positions to the right of p
//Examples:
// If v=[7,2,5,8] and p=0 or p=1 then the method must return false,
// but for p=2 the method should return true
//1.Specify the method
//2.Implement an O(v.size()) method
//3.Verify the method
method barrier(v:array<int>,p:int) returns (b:bool)
//Give the precondition
//Give the postcondition
//{Implement and verify}
requires v.Length > 0
requires 0<=p<v.Length
ensures b==forall k,l::0<=k<=p && p<l<v.Length ==> v[k]<v[l]
{
var i:=1;
var max:=0;
while(i<=p)
decreases p-i
invariant 0<=i<=p+1
invariant 0<=max<i
invariant forall k::0<=k<i ==> v[max] >= v[k]
{
if(v[i]>v[max]){
max:=i;
}
i:=i+1;
}
while(i<v.Length && v[i]>v[max])
decreases v.Length - i
invariant 0<=i<=v.Length
invariant forall k::0<=k<=p ==> v[k]<=v[max]
invariant forall k::p<k<i ==> v[k] > v[max]
{
i:=i+1;
}
b:=i==v.Length;
}
|
//Method barrier below receives an array and an integer p
//and returns a boolean b which is true if and only if
//all the positions to the left of p and including also position p contain elements
//that are strictly smaller than all the elements contained in the positions to the right of p
//Examples:
// If v=[7,2,5,8] and p=0 or p=1 then the method must return false,
// but for p=2 the method should return true
//1.Specify the method
//2.Implement an O(v.size()) method
//3.Verify the method
method barrier(v:array<int>,p:int) returns (b:bool)
//Give the precondition
//Give the postcondition
//{Implement and verify}
requires v.Length > 0
requires 0<=p<v.Length
ensures b==forall k,l::0<=k<=p && p<l<v.Length ==> v[k]<v[l]
{
var i:=1;
var max:=0;
while(i<=p)
{
if(v[i]>v[max]){
max:=i;
}
i:=i+1;
}
while(i<v.Length && v[i]>v[max])
{
i:=i+1;
}
b:=i==v.Length;
}
|
086 | Dafny-Exercises_tmp_tmpjm75muf__Session2Exercises_ExerciseExp.dfy | function exp(x:int, e:int):int
decreases e
requires e >= 0
ensures x > 0 ==> exp(x,e) > 0
{
if e == 0 then 1 else x*exp(x,e-1)
}
lemma exp3_Lemma(n:int)
decreases n
requires n >= 1
ensures (exp(3,n)-1)%2 == 0
{}
lemma mult8_Lemma(n:int)
decreases n
requires n >= 1
ensures (exp(3,2*n) - 1)%8 == 0
{
if(n==1){
}
else{
calc =={
(exp(3,2*n) -1) % 8;
(exp(3, 2*(n-1)) *8 + exp(3,2*(n-1)) - 1) % 8;
{
mult8_Lemma(n-1);
assert exp(3,2*(n-1)) * 8 %8==0;
}
0;
}
}
}
| function exp(x:int, e:int):int
requires e >= 0
ensures x > 0 ==> exp(x,e) > 0
{
if e == 0 then 1 else x*exp(x,e-1)
}
lemma exp3_Lemma(n:int)
requires n >= 1
ensures (exp(3,n)-1)%2 == 0
{}
lemma mult8_Lemma(n:int)
requires n >= 1
ensures (exp(3,2*n) - 1)%8 == 0
{
if(n==1){
}
else{
calc =={
(exp(3,2*n) -1) % 8;
(exp(3, 2*(n-1)) *8 + exp(3,2*(n-1)) - 1) % 8;
{
mult8_Lemma(n-1);
}
0;
}
}
}
|
087 | Dafny-Exercises_tmp_tmpjm75muf__Session2Exercises_ExerciseFibonacci.dfy | function fib(n: nat): nat
decreases n
{
if n == 0 then 0 else
if n == 1 then 1 else
fib(n - 1) + fib(n - 2)
}
method fibonacci1(n:nat) returns (f:nat)
ensures f==fib(n)
{
var i := 0;
f := 0;
var fsig := 1;
while i < n
decreases n - i//write the bound
invariant f==fib(i) && fsig==fib(i+1)//write the invariant
invariant i<=n
{
f, fsig := fsig, f + fsig;
i := i + 1;
}
}
method fibonacci2(n:nat) returns (f:nat)
ensures f==fib(n)
{
if (n==0) {f:=0;}
else{
var i := 1;
var fant := 0;
f := 1;
while i < n
decreases n-i//write the bound
invariant fant==fib(i-1) && f==fib(i)//write the invariant
invariant i<=n
{
fant, f := f, fant + f;
i := i + 1;
}
}
}
method fibonacci3(n:nat) returns (f:nat)
ensures f==fib(n)
{
{
var i: int := 0;
var a := 1;
f := 0;
while i < n
decreases n-i//write the bound
invariant 0<=i<=n
invariant if i ==0 then a==fib(i+1) && f==fib(i)//write the invariant
else a==fib(i-1) && f==fib(i)
{
a, f := f, a + f;
i := i + 1;
}
}
}
| function fib(n: nat): nat
{
if n == 0 then 0 else
if n == 1 then 1 else
fib(n - 1) + fib(n - 2)
}
method fibonacci1(n:nat) returns (f:nat)
ensures f==fib(n)
{
var i := 0;
f := 0;
var fsig := 1;
while i < n
{
f, fsig := fsig, f + fsig;
i := i + 1;
}
}
method fibonacci2(n:nat) returns (f:nat)
ensures f==fib(n)
{
if (n==0) {f:=0;}
else{
var i := 1;
var fant := 0;
f := 1;
while i < n
{
fant, f := f, fant + f;
i := i + 1;
}
}
}
method fibonacci3(n:nat) returns (f:nat)
ensures f==fib(n)
{
{
var i: int := 0;
var a := 1;
f := 0;
while i < n
else a==fib(i-1) && f==fib(i)
{
a, f := f, a + f;
i := i + 1;
}
}
}
|
088 | Dafny-Exercises_tmp_tmpjm75muf__Session2Exercises_ExercisePositive.dfy | predicate positive(s:seq<int>)
{forall u::0<=u<|s| ==> s[u]>=0}
method mpositive(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=0;
//1. assert positive(v[..0])
while i<v.Length && v[i]>=0
decreases v.Length - i
invariant 0<=i<=v.Length
invariant positive(v[..i])
{
//2. assert 0<=i<v.Length && positive(v[..i]);
i:=i+1;
//2. assert 0<=i<v.Length && positive(v[..i]);
}
//3. assert i==v.Length ==> positive(v[..]);
//3. assert i<v.Length => v[i]<0;
b := i==v.Length;
}
method mpositive3(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=0; b:=true;
while(i<v.Length && b)
decreases v.Length - i
invariant 0 <= i <= v.Length
invariant b==positive(v[0..i])
invariant !b ==> !positive(v[0..v.Length])
{
b:=v[i]>=0;
i:=i+1;
}
}
method mpositive4(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=0; b:=true;
while(i<v.Length && b)
decreases v.Length - i
invariant 0 <= i <= v.Length
invariant b==positive(v[0..i])
invariant !b ==> !positive(v[0..v.Length])
{
b:=v[i]>=0;
i:=i+1;
}
}
method mpositivertl(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=v.Length-1;
while(i>=0 && v[i]>=0)
decreases i
invariant -1 <= i < v.Length
invariant positive(v[i+1..])
{
i:=i-1;
}
b:= i==-1;
}
| predicate positive(s:seq<int>)
{forall u::0<=u<|s| ==> s[u]>=0}
method mpositive(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=0;
//1. assert positive(v[..0])
while i<v.Length && v[i]>=0
{
//2. assert 0<=i<v.Length && positive(v[..i]);
i:=i+1;
//2. assert 0<=i<v.Length && positive(v[..i]);
}
//3. assert i==v.Length ==> positive(v[..]);
//3. assert i<v.Length => v[i]<0;
b := i==v.Length;
}
method mpositive3(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=0; b:=true;
while(i<v.Length && b)
{
b:=v[i]>=0;
i:=i+1;
}
}
method mpositive4(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=0; b:=true;
while(i<v.Length && b)
{
b:=v[i]>=0;
i:=i+1;
}
}
method mpositivertl(v:array<int>) returns (b:bool)
ensures b==positive(v[0..v.Length])
{
var i:=v.Length-1;
while(i>=0 && v[i]>=0)
{
i:=i-1;
}
b:= i==-1;
}
|
089 | Dafny-Exercises_tmp_tmpjm75muf__Session2Exercises_ExerciseSquare_root.dfy | method mroot1(n:int) returns (r:int) //Cost O(root n)
requires n>=0
ensures r>=0 && r*r <= n <(r+1)*(r+1)
{
r:=0;
while (r+1)*(r+1) <=n
invariant r>=0 && r*r <=n
decreases n-r*r
{
r:=r+1;
}
}
method mroot2(n:int) returns (r:int) //Cost O(n)
requires n>=0
ensures r>=0 && r*r <= n <(r+1)*(r+1)
{
r:=n;
while n<r*r
invariant 0<=r<=n && n<(r+1)*(r+1)//write the invariant
invariant r*r<=n ==> n<(r+1)*(r+1)
decreases r//write the bound
{
r:=r-1;
}
}
method mroot3(n:int) returns (r:int) //Cost O(log n)
requires n>=0
ensures r>=0 && r*r <= n <(r+1)*(r+1)
{ var y:int;
var h:int;
r:=0;
y:=n+1;
//Search in interval [0,n+1)
while (y!=r+1) //[r,y]
invariant r>=0 && r*r<=n<y*y && y>=r+1// write the invariant
decreases y-r//write the bound
{
h:=(r+y)/2;
if (h*h<=n)
{r:=h;}
else
{y:=h;}
}
}
| method mroot1(n:int) returns (r:int) //Cost O(root n)
requires n>=0
ensures r>=0 && r*r <= n <(r+1)*(r+1)
{
r:=0;
while (r+1)*(r+1) <=n
{
r:=r+1;
}
}
method mroot2(n:int) returns (r:int) //Cost O(n)
requires n>=0
ensures r>=0 && r*r <= n <(r+1)*(r+1)
{
r:=n;
while n<r*r
{
r:=r-1;
}
}
method mroot3(n:int) returns (r:int) //Cost O(log n)
requires n>=0
ensures r>=0 && r*r <= n <(r+1)*(r+1)
{ var y:int;
var h:int;
r:=0;
y:=n+1;
//Search in interval [0,n+1)
while (y!=r+1) //[r,y]
{
h:=(r+y)/2;
if (h*h<=n)
{r:=h;}
else
{y:=h;}
}
}
|
090 | Dafny-Exercises_tmp_tmpjm75muf__Session3Exercises_ExerciseMaximum.dfy | //Algorithm 1: From left to right return the first
method mmaximum1(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
{
var j:=1; i:=0;
while(j<v.Length)
decreases v.Length - j
invariant 0<=j<=v.Length
invariant 0<=i<j
invariant forall k:: 0<=k<j ==> v[i] >= v[k]
{
if(v[j] > v[i]){i:=j;}
j:=j+1;
}
}
//Algorithm 2: From right to left return the last
method mmaximum2(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
{
var j:=v.Length-2; i:=v.Length - 1;
while(j>=0)
decreases j
invariant 0<=i<v.Length
invariant -1<=j<v.Length-1
invariant forall k :: v.Length>k>j ==> v[k]<=v[i]
{
if(v[j] > v[i]){i:=j;}
j:=j-1;
}
}
method mfirstMaximum(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
ensures forall l:: 0<=l<i ==> v[i]>v[l]
//Algorithm: from left to right
{
var j:=1; i:=0;
while(j<v.Length)
decreases v.Length - j
invariant 0<=j<=v.Length
invariant 0<=i<j
invariant forall k:: 0<=k<j ==> v[i] >= v[k]
invariant forall k:: 0<=k<i ==> v[i] > v[k]
{
if(v[j] > v[i]){i:=j;}
j:=j+1;
}
}
method mlastMaximum(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
ensures forall l:: i<l<v.Length ==> v[i]>v[l]
{
var j:=v.Length-2;
i := v.Length-1;
while(j>=0)
decreases j
invariant -1<=j<v.Length-1
invariant 0<=i<v.Length
invariant forall k :: v.Length>k>j ==> v[k]<=v[i]
invariant forall k :: v.Length>k>i ==> v[k]<v[i]
{
if(v[j] > v[i]){i:=j;}
j:=j-1;
}
}
//Algorithm : from left to right
//Algorithm : from right to left
method mmaxvalue1(v:array<int>) returns (m:int)
requires v.Length>0
ensures m in v[..]
ensures forall k::0<=k<v.Length ==> m>=v[k]
{
var i:=mmaximum1(v);
m:=v[i];
}
method mmaxvalue2(v:array<int>) returns (m:int)
requires v.Length>0
ensures m in v[..]
ensures forall k::0<=k<v.Length ==> m>=v[k]
{
var i:=mmaximum2(v);
m:=v[i];
}
| //Algorithm 1: From left to right return the first
method mmaximum1(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
{
var j:=1; i:=0;
while(j<v.Length)
{
if(v[j] > v[i]){i:=j;}
j:=j+1;
}
}
//Algorithm 2: From right to left return the last
method mmaximum2(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
{
var j:=v.Length-2; i:=v.Length - 1;
while(j>=0)
{
if(v[j] > v[i]){i:=j;}
j:=j-1;
}
}
method mfirstMaximum(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
ensures forall l:: 0<=l<i ==> v[i]>v[l]
//Algorithm: from left to right
{
var j:=1; i:=0;
while(j<v.Length)
{
if(v[j] > v[i]){i:=j;}
j:=j+1;
}
}
method mlastMaximum(v:array<int>) returns (i:int)
requires v.Length>0
ensures 0<=i<v.Length
ensures forall k:: 0<=k<v.Length ==> v[i]>=v[k]
ensures forall l:: i<l<v.Length ==> v[i]>v[l]
{
var j:=v.Length-2;
i := v.Length-1;
while(j>=0)
{
if(v[j] > v[i]){i:=j;}
j:=j-1;
}
}
//Algorithm : from left to right
//Algorithm : from right to left
method mmaxvalue1(v:array<int>) returns (m:int)
requires v.Length>0
ensures m in v[..]
ensures forall k::0<=k<v.Length ==> m>=v[k]
{
var i:=mmaximum1(v);
m:=v[i];
}
method mmaxvalue2(v:array<int>) returns (m:int)
requires v.Length>0
ensures m in v[..]
ensures forall k::0<=k<v.Length ==> m>=v[k]
{
var i:=mmaximum2(v);
m:=v[i];
}
|
091 | Dafny-Exercises_tmp_tmpjm75muf__Session4Exercises_ExerciseAllEqual.dfy | predicate allEqual(s:seq<int>)
{forall i,j::0<=i<|s| && 0<=j<|s| ==> s[i]==s[j] }
//{forall i,j::0<=i<=j<|s| ==> s[i]==s[j] }
//{forall i::0<i<|s| ==> s[i-1]==s[i]}
//{forall i::0<=i<|s|-1 ==> s[i]==s[i+1]}
//Ordered indexes
lemma equivalenceNoOrder(s:seq<int>)
ensures allEqual(s) <==> forall i,j::0<=i<=j<|s| ==> s[i]==s[j]
{}
//All equal to first
lemma equivalenceEqualtoFirst(s:seq<int>)
requires s!=[]
ensures allEqual(s) <==> (forall i::0<=i<|s| ==> s[0]==s[i])
{}
lemma equivalenceContiguous(s:seq<int>)
ensures (allEqual(s) ==> forall i::0<=i<|s|-1 ==> s[i]==s[i+1])
ensures (allEqual(s) <== forall i::0<=i<|s|-1 ==> s[i]==s[i+1])
{
assert allEqual(s) ==> forall i::0<=i<|s|-1 ==> s[i]==s[i+1];
if(|s|==0 || |s|==1){
}
else{
calc {
forall i::0<=i<|s|-1 ==> s[i]==s[i+1];
==>
{
equivalenceContiguous(s[..|s|-1]);
assert s[|s|-2] == s[|s|-1];
}
allEqual(s);
}
}
}
method mallEqual1(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i := 0;
b := true;
while (i < v.Length && b)
invariant 0 <= i <= v.Length
invariant b==allEqual(v[..i])
decreases v.Length - i
{
b:=(v[i]==v[0]);
i := i+1;
}
}
method mallEqual2(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i:int;
b:=true;
i:=0;
while (i < v.Length && v[i] == v[0])
invariant 0 <= i <= v.Length
invariant forall k:: 0 <= k < i ==> v[k] == v[0]
decreases v.Length - i
{
i:=i+1;
}
b:=(i==v.Length);
}
method mallEqual3(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
equivalenceContiguous(v[..]);
var i:int;
b:=true;
if (v.Length >0){
i:=0;
while (i<v.Length-1 && v[i]==v[i+1])
invariant 0<=i<=v.Length -1
invariant b==allEqual(v[..i+1])
decreases v.Length - i
{
i:=i+1;
}
b:=(i==v.Length-1);
}
}
method mallEqual4(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i:int;
b:=true;
if (v.Length>0){
i:=0;
while (i < v.Length-1 && b)
invariant 0 <= i < v.Length
invariant b==allEqual(v[..i+1])
decreases v.Length - i
{
b:=(v[i]==v[i+1]);
i:=i+1;
}
}
}
method mallEqual5(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i := 0;
b := true;
while (i < v.Length && b)
invariant 0<=i<=v.Length//
invariant b ==> forall k::0<=k<i ==> v[k] == v[0]
invariant !b ==> exists k:: 0<=k<v.Length && v[k]!=v[0]
decreases v.Length - i - (if b then 0 else 1)//
{
if (v[i] != v[0]) { b := false; }
else { i := i+1;}
}
}
| predicate allEqual(s:seq<int>)
{forall i,j::0<=i<|s| && 0<=j<|s| ==> s[i]==s[j] }
//{forall i,j::0<=i<=j<|s| ==> s[i]==s[j] }
//{forall i::0<i<|s| ==> s[i-1]==s[i]}
//{forall i::0<=i<|s|-1 ==> s[i]==s[i+1]}
//Ordered indexes
lemma equivalenceNoOrder(s:seq<int>)
ensures allEqual(s) <==> forall i,j::0<=i<=j<|s| ==> s[i]==s[j]
{}
//All equal to first
lemma equivalenceEqualtoFirst(s:seq<int>)
requires s!=[]
ensures allEqual(s) <==> (forall i::0<=i<|s| ==> s[0]==s[i])
{}
lemma equivalenceContiguous(s:seq<int>)
ensures (allEqual(s) ==> forall i::0<=i<|s|-1 ==> s[i]==s[i+1])
ensures (allEqual(s) <== forall i::0<=i<|s|-1 ==> s[i]==s[i+1])
{
if(|s|==0 || |s|==1){
}
else{
calc {
forall i::0<=i<|s|-1 ==> s[i]==s[i+1];
==>
{
equivalenceContiguous(s[..|s|-1]);
}
allEqual(s);
}
}
}
method mallEqual1(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i := 0;
b := true;
while (i < v.Length && b)
{
b:=(v[i]==v[0]);
i := i+1;
}
}
method mallEqual2(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i:int;
b:=true;
i:=0;
while (i < v.Length && v[i] == v[0])
{
i:=i+1;
}
b:=(i==v.Length);
}
method mallEqual3(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
equivalenceContiguous(v[..]);
var i:int;
b:=true;
if (v.Length >0){
i:=0;
while (i<v.Length-1 && v[i]==v[i+1])
{
i:=i+1;
}
b:=(i==v.Length-1);
}
}
method mallEqual4(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i:int;
b:=true;
if (v.Length>0){
i:=0;
while (i < v.Length-1 && b)
{
b:=(v[i]==v[i+1]);
i:=i+1;
}
}
}
method mallEqual5(v:array<int>) returns (b:bool)
ensures b==allEqual(v[0..v.Length])
{
var i := 0;
b := true;
while (i < v.Length && b)
{
if (v[i] != v[0]) { b := false; }
else { i := i+1;}
}
}
|
092 | Dafny-Exercises_tmp_tmpjm75muf__Session4Exercises_ExerciseContained.dfy |
predicate strictSorted(s : seq<int>) {
forall u, w :: 0 <= u < w < |s| ==> s[u] < s[w]
}
method mcontained(v:array<int>,w:array<int>,n:int,m:int) returns (b:bool)
//Specify and implement an O(m+n) algorithm that returns b
//v and w are strictly increasing ordered arrays
//b is true iff the first n elements of v are contained in the first m elements of w
requires n<=m && n>=0
requires strictSorted(v[..])
requires strictSorted(w[..])
requires v.Length >= n && w.Length >= m
ensures b==forall k:: 0<= k< n ==> v[k] in w[..m]//exists j :: 0 <= j < m && v[k] == w[j]
{
var i:=0;
var j:=0;
while(i<n && j<m && (v[i] >= w[j])) //&& b)
invariant 0<=i<=n
invariant 0<=j<=m
invariant strictSorted(v[..])
invariant strictSorted(w[..])
invariant forall k::0<=k<i ==> v[k] in w[..j]
invariant i<n ==> !(v[i] in w[..j])
decreases w.Length-j
decreases v.Length-i
{
if(v[i] == w[j]){
i:=i+1;
}
j:=j+1;
}
assert i<n ==> !(v[i] in w[..m]);
b := i==n;
}
|
predicate strictSorted(s : seq<int>) {
forall u, w :: 0 <= u < w < |s| ==> s[u] < s[w]
}
method mcontained(v:array<int>,w:array<int>,n:int,m:int) returns (b:bool)
//Specify and implement an O(m+n) algorithm that returns b
//v and w are strictly increasing ordered arrays
//b is true iff the first n elements of v are contained in the first m elements of w
requires n<=m && n>=0
requires strictSorted(v[..])
requires strictSorted(w[..])
requires v.Length >= n && w.Length >= m
ensures b==forall k:: 0<= k< n ==> v[k] in w[..m]//exists j :: 0 <= j < m && v[k] == w[j]
{
var i:=0;
var j:=0;
while(i<n && j<m && (v[i] >= w[j])) //&& b)
{
if(v[i] == w[j]){
i:=i+1;
}
j:=j+1;
}
b := i==n;
}
|
093 | Dafny-Exercises_tmp_tmpjm75muf__Session4Exercises_ExerciseFirstNegative.dfy | predicate positive(s:seq<int>)
{forall u::0<=u<|s| ==> s[u]>=0}
method mfirstNegative(v:array<int>) returns (b:bool, i:int)
ensures b <==> exists k::0<=k<v.Length && v[k]<0
ensures b ==> 0<=i<v.Length && v[i]<0 && positive(v[0..i])
{
i:=0;
b:=false;
while (i<v.Length && !b)
invariant 0<=i<=v.Length
invariant b <==> exists k::0<=k<i && v[k]<0
invariant b ==> v[i-1]<0 && positive(v[0..i-1])
decreases v.Length - i
{
b:=(v[i]<0);
i:=i+1;
}
if (b){i:=i-1;}
}
method mfirstNegative2(v:array<int>) returns (b:bool, i:int)
ensures b <==> exists k::0<=k<v.Length && v[k]<0
ensures b ==> 0<=i<v.Length && v[i]<0 && positive(v[0..i])
{
i:=0;b:=false;
while (i<v.Length && !b)
invariant 0<=i<=v.Length
invariant b ==> i<v.Length && v[i]<0 && !(exists k::0<=k<i && v[k]<0)
invariant b <== exists k::0<=k<i && v[k]<0
decreases v.Length - i - (if b then 1 else 0)
{
b:=(v[i]<0);
if (!b) {i:=i+1;}
}
}
| predicate positive(s:seq<int>)
{forall u::0<=u<|s| ==> s[u]>=0}
method mfirstNegative(v:array<int>) returns (b:bool, i:int)
ensures b <==> exists k::0<=k<v.Length && v[k]<0
ensures b ==> 0<=i<v.Length && v[i]<0 && positive(v[0..i])
{
i:=0;
b:=false;
while (i<v.Length && !b)
{
b:=(v[i]<0);
i:=i+1;
}
if (b){i:=i-1;}
}
method mfirstNegative2(v:array<int>) returns (b:bool, i:int)
ensures b <==> exists k::0<=k<v.Length && v[k]<0
ensures b ==> 0<=i<v.Length && v[i]<0 && positive(v[0..i])
{
i:=0;b:=false;
while (i<v.Length && !b)
{
b:=(v[i]<0);
if (!b) {i:=i+1;}
}
}
|
094 | Dafny-Exercises_tmp_tmpjm75muf__Session4Exercises_ExercisefirstZero.dfy |
method mfirstCero(v:array<int>) returns (i:int)
ensures 0 <=i<=v.Length
ensures forall j:: 0<=j<i ==> v[j]!=0
ensures i!=v.Length ==> v[i]==0
{
i:=0;
while (i<v.Length && v[i]!=0)
invariant 0<=i<=v.Length
invariant forall j:: 0<=j<i ==> v[j]!=0
decreases v.Length -i
{i:=i+1;}
}
|
method mfirstCero(v:array<int>) returns (i:int)
ensures 0 <=i<=v.Length
ensures forall j:: 0<=j<i ==> v[j]!=0
ensures i!=v.Length ==> v[i]==0
{
i:=0;
while (i<v.Length && v[i]!=0)
{i:=i+1;}
}
|
095 | Dafny-Exercises_tmp_tmpjm75muf__Session5Exercises_ExerciseSumElems.dfy | function SumR(s:seq<int>):int
decreases s
{
if (s==[]) then 0
else SumR(s[..|s|-1])+s[|s|-1]
}
function SumL(s:seq<int>):int
decreases s
{
if (s==[]) then 0
else s[0]+SumL(s[1..])
}
lemma concatLast(s:seq<int>,t:seq<int>)
requires t!=[]
ensures (s+t)[..|s+t|-1] == s+(t[..|t|-1])
{}
lemma concatFirst(s:seq<int>,t:seq<int>)
requires s!=[]
ensures (s+t)[1..] == s[1..]+t
{}
lemma {:induction s,t} SumByPartsR(s:seq<int>,t:seq<int>)
decreases s,t
ensures SumR(s+t) == SumR(s)+SumR(t)
{ if (t==[])
{assert s+t == s;}
else if (s==[])
{assert s+t==t;}
else
{
calc =={
SumR(s+t);
SumR((s+t)[..|s+t|-1])+(s+t)[|s+t|-1];
SumR((s+t)[..|s+t|-1])+t[|t|-1];
{concatLast(s,t);}
SumR(s+t[..|t|-1])+t[|t|-1];
{SumByPartsR(s,t[..|t|-1]);}
SumR(s)+SumR(t[..|t|-1])+t[|t|-1];
SumR(s)+SumR(t);
}
}
}
lemma {:induction s,t} SumByPartsL(s:seq<int>,t:seq<int>)
decreases s,t
ensures SumL(s+t) == SumL(s)+SumL(t)
//Prove this
{
if(t==[]){
assert s+t==s;
}
else if(s==[]){
assert s+t==t;
}
else{
calc == {
SumL(s+t);
(s+t)[0] + SumL((s+t)[1..]);
s[0] + SumL((s+t)[1..]);
{concatFirst(s,t);}
s[0] + SumL(s[1..] + t);
{SumByPartsL(s[1..], t);}
s[0] + SumL(s[1..]) + SumL(t);
}
}
}
lemma {:induction s,i,j} equalSumR(s:seq<int>,i:int,j:int)
decreases j-i
requires 0<=i<=j<=|s|
ensures SumR(s[i..j])==SumL(s[i..j])
//Prove this
{
if(s==[]){
assert SumR(s) == SumL(s);
}else{
if(i==j){
assert SumR(s[i..j]) == SumL(s[i..j]);
}
else{
calc == {
SumR(s[i..j]);
{
assert s[i..j] == s[i..j-1] + [s[j-1]];
assert SumR(s[i..j]) == SumR(s[i..j-1]) + s[j-1];
}
SumR(s[i..j-1]) + s[j-1];
{equalSumR(s, i, j-1);}
SumL(s[i..j-1]) + s[j-1];
{assert s[j-1] == SumL([s[j-1]]);}
SumL(s[i..j-1]) + SumL([s[j-1]]);
{SumByPartsL(s[i..j-1], [s[j-1]]);}
SumL(s[i..j-1] + [s[j-1]]);
{
assert s[i..j] == s[i..j-1] + [s[j-1]];
}
SumL(s[i..j]);
/*SumR(s[i..j-1])+SumR(s[j..j]);
SumR(s[i..j-1]) + s[j..j];
SumL(s);*/
}
}
}
}
lemma equalSumsV()
ensures forall v:array<int>,i,j | 0<=i<=j<=v.Length :: SumR(v[i..j])==SumL(v[i..j])
//proving the forall
{ forall v:array<int>,i,j | 0<=i<=j<=v.Length
ensures SumR(v[i..j])==SumL(v[i..j])
{equalSumR(v[..],i,j);}
}
function SumV(v:array<int>,c:int,f:int):int
requires 0<=c<=f<=v.Length
reads v
{SumR(v[c..f])}
lemma ArrayFacts<T>()
ensures forall v : array<T> :: v[..v.Length] == v[..];
ensures forall v : array<T> :: v[0..] == v[..];
ensures forall v : array<T> :: v[0..v.Length] == v[..];
ensures forall v : array<T> ::|v[0..v.Length]|==v.Length;
ensures forall v : array<T> | v.Length>=1 ::|v[1..v.Length]|==v.Length-1;
ensures forall v : array<T> ::forall k : nat | k < v.Length :: v[..k+1][..k] == v[..k]
// ensures forall v:array<int>,i,j | 0<=i<=j<=v.Length :: SumR(v[i..j])==SumL(v[i..j])
{equalSumsV();}
method sumElems(v:array<int>) returns (sum:int)
//ensures sum==SumL(v[0..v.Length])
ensures sum==SumR(v[..])
//ensures sum==SumV(v,0,v.Length)
{ArrayFacts<int>();
sum:=0;
var i:int;
i:=0;
while (i<v.Length)
decreases v.Length - i//write
invariant 0<=i<=v.Length && sum == SumR(v[..i])//write
{
sum:=sum+v[i];
i:=i+1;
}
}
method sumElemsB(v:array<int>) returns (sum:int)
//ensures sum==SumL(v[0..v.Length])
ensures sum==SumR(v[0..v.Length])
{
ArrayFacts<int>();
sum:=0;
var i:int;
i:=v.Length;
equalSumsV();
while(i>0)
decreases i//write
invariant 0<=i<=v.Length
invariant sum == SumL(v[i..]) == SumR(v[i..])
{
sum:=sum+v[i-1];
i:=i-1;
}
}
| function SumR(s:seq<int>):int
{
if (s==[]) then 0
else SumR(s[..|s|-1])+s[|s|-1]
}
function SumL(s:seq<int>):int
{
if (s==[]) then 0
else s[0]+SumL(s[1..])
}
lemma concatLast(s:seq<int>,t:seq<int>)
requires t!=[]
ensures (s+t)[..|s+t|-1] == s+(t[..|t|-1])
{}
lemma concatFirst(s:seq<int>,t:seq<int>)
requires s!=[]
ensures (s+t)[1..] == s[1..]+t
{}
lemma {:induction s,t} SumByPartsR(s:seq<int>,t:seq<int>)
ensures SumR(s+t) == SumR(s)+SumR(t)
{ if (t==[])
{assert s+t == s;}
else if (s==[])
{assert s+t==t;}
else
{
calc =={
SumR(s+t);
SumR((s+t)[..|s+t|-1])+(s+t)[|s+t|-1];
SumR((s+t)[..|s+t|-1])+t[|t|-1];
{concatLast(s,t);}
SumR(s+t[..|t|-1])+t[|t|-1];
{SumByPartsR(s,t[..|t|-1]);}
SumR(s)+SumR(t[..|t|-1])+t[|t|-1];
SumR(s)+SumR(t);
}
}
}
lemma {:induction s,t} SumByPartsL(s:seq<int>,t:seq<int>)
ensures SumL(s+t) == SumL(s)+SumL(t)
//Prove this
{
if(t==[]){
}
else if(s==[]){
}
else{
calc == {
SumL(s+t);
(s+t)[0] + SumL((s+t)[1..]);
s[0] + SumL((s+t)[1..]);
{concatFirst(s,t);}
s[0] + SumL(s[1..] + t);
{SumByPartsL(s[1..], t);}
s[0] + SumL(s[1..]) + SumL(t);
}
}
}
lemma {:induction s,i,j} equalSumR(s:seq<int>,i:int,j:int)
requires 0<=i<=j<=|s|
ensures SumR(s[i..j])==SumL(s[i..j])
//Prove this
{
if(s==[]){
}else{
if(i==j){
}
else{
calc == {
SumR(s[i..j]);
{
}
SumR(s[i..j-1]) + s[j-1];
{equalSumR(s, i, j-1);}
SumL(s[i..j-1]) + s[j-1];
{assert s[j-1] == SumL([s[j-1]]);}
SumL(s[i..j-1]) + SumL([s[j-1]]);
{SumByPartsL(s[i..j-1], [s[j-1]]);}
SumL(s[i..j-1] + [s[j-1]]);
{
}
SumL(s[i..j]);
/*SumR(s[i..j-1])+SumR(s[j..j]);
SumR(s[i..j-1]) + s[j..j];
SumL(s);*/
}
}
}
}
lemma equalSumsV()
ensures forall v:array<int>,i,j | 0<=i<=j<=v.Length :: SumR(v[i..j])==SumL(v[i..j])
//proving the forall
{ forall v:array<int>,i,j | 0<=i<=j<=v.Length
ensures SumR(v[i..j])==SumL(v[i..j])
{equalSumR(v[..],i,j);}
}
function SumV(v:array<int>,c:int,f:int):int
requires 0<=c<=f<=v.Length
reads v
{SumR(v[c..f])}
lemma ArrayFacts<T>()
ensures forall v : array<T> :: v[..v.Length] == v[..];
ensures forall v : array<T> :: v[0..] == v[..];
ensures forall v : array<T> :: v[0..v.Length] == v[..];
ensures forall v : array<T> ::|v[0..v.Length]|==v.Length;
ensures forall v : array<T> | v.Length>=1 ::|v[1..v.Length]|==v.Length-1;
ensures forall v : array<T> ::forall k : nat | k < v.Length :: v[..k+1][..k] == v[..k]
// ensures forall v:array<int>,i,j | 0<=i<=j<=v.Length :: SumR(v[i..j])==SumL(v[i..j])
{equalSumsV();}
method sumElems(v:array<int>) returns (sum:int)
//ensures sum==SumL(v[0..v.Length])
ensures sum==SumR(v[..])
//ensures sum==SumV(v,0,v.Length)
{ArrayFacts<int>();
sum:=0;
var i:int;
i:=0;
while (i<v.Length)
{
sum:=sum+v[i];
i:=i+1;
}
}
method sumElemsB(v:array<int>) returns (sum:int)
//ensures sum==SumL(v[0..v.Length])
ensures sum==SumR(v[0..v.Length])
{
ArrayFacts<int>();
sum:=0;
var i:int;
i:=v.Length;
equalSumsV();
while(i>0)
{
sum:=sum+v[i-1];
i:=i-1;
}
}
|
096 | Dafny-Exercises_tmp_tmpjm75muf__Session6Exercises_ExerciseCountEven.dfy |
predicate positive(s:seq<int>)
{forall u::0<=u<|s| ==> s[u]>=0}
predicate isEven(i:int)
requires i>=0
{i%2==0}
function CountEven(s:seq<int>):int
decreases s
requires positive(s)
{if s==[] then 0
else (if (s[|s|-1]%2==0) then 1 else 0)+CountEven(s[..|s|-1])
}
lemma ArrayFacts<T>()
ensures forall v : array<T> :: v[..v.Length] == v[..];
ensures forall v : array<T> :: v[0..] == v[..];
ensures forall v : array<T> :: v[0..v.Length] == v[..];
ensures forall v : array<T> ::|v[0..v.Length]|==v.Length;
ensures forall v : array<T> | v.Length>=1 ::|v[1..v.Length]|==v.Length-1;
ensures forall v : array<T> ::forall k : nat | k < v.Length :: v[..k+1][..k] == v[..k]
{}
method mcountEven(v:array<int>) returns (n:int)
requires positive(v[..])
ensures n==CountEven(v[..])
{ ArrayFacts<int>();
n:=0;
var i:int;
i:=0;
while (i<v.Length)
decreases v.Length - i//write
invariant 0<=i<=v.Length//write
invariant n==CountEven(v[..i])
{
if (v[i]%2==0) {n:=n+1;}
i:=i+1;
}
}
|
predicate positive(s:seq<int>)
{forall u::0<=u<|s| ==> s[u]>=0}
predicate isEven(i:int)
requires i>=0
{i%2==0}
function CountEven(s:seq<int>):int
requires positive(s)
{if s==[] then 0
else (if (s[|s|-1]%2==0) then 1 else 0)+CountEven(s[..|s|-1])
}
lemma ArrayFacts<T>()
ensures forall v : array<T> :: v[..v.Length] == v[..];
ensures forall v : array<T> :: v[0..] == v[..];
ensures forall v : array<T> :: v[0..v.Length] == v[..];
ensures forall v : array<T> ::|v[0..v.Length]|==v.Length;
ensures forall v : array<T> | v.Length>=1 ::|v[1..v.Length]|==v.Length-1;
ensures forall v : array<T> ::forall k : nat | k < v.Length :: v[..k+1][..k] == v[..k]
{}
method mcountEven(v:array<int>) returns (n:int)
requires positive(v[..])
ensures n==CountEven(v[..])
{ ArrayFacts<int>();
n:=0;
var i:int;
i:=0;
while (i<v.Length)
{
if (v[i]%2==0) {n:=n+1;}
i:=i+1;
}
}
|
097 | Dafny-Exercises_tmp_tmpjm75muf__Session6Exercises_ExerciseCountMin.dfy | function min(v:array<int>,i:int):int
decreases i
reads v
requires 1<=i<=v.Length
ensures forall k::0<=k<i==> v[k]>=min(v,i)
{if (i==1) then v[0]
else if (v[i-1]<=min(v,i-1)) then v[i-1]
else min(v,i-1)
}
function countMin(v:array<int>,x:int, i:int):int
decreases i
reads v
requires 0<=i<=v.Length
ensures !(x in v[0..i]) ==> countMin(v,x,i)==0
{
if (i==0) then 0
else if (v[i-1]==x) then 1+countMin(v,x,i-1)
else countMin(v,x,i-1)
}
method mCountMin(v:array<int>) returns (c:int)
requires v.Length>0
ensures c==countMin(v,min(v,v.Length),v.Length)
//Implement and verify an O(v.Length) algorithm
{
var i:=1;
c:=1;
var mini:=v[0];
while(i<v.Length)
decreases v.Length -i
invariant 0<i<=v.Length
invariant mini==min(v,i)
invariant c==countMin(v, mini, i)
{
if(v[i]==mini){
c:=c + 1;
}
else if(v[i]<mini){
c:=1;
mini:=v[i];
}
i:=i+1;
}
}
| function min(v:array<int>,i:int):int
reads v
requires 1<=i<=v.Length
ensures forall k::0<=k<i==> v[k]>=min(v,i)
{if (i==1) then v[0]
else if (v[i-1]<=min(v,i-1)) then v[i-1]
else min(v,i-1)
}
function countMin(v:array<int>,x:int, i:int):int
reads v
requires 0<=i<=v.Length
ensures !(x in v[0..i]) ==> countMin(v,x,i)==0
{
if (i==0) then 0
else if (v[i-1]==x) then 1+countMin(v,x,i-1)
else countMin(v,x,i-1)
}
method mCountMin(v:array<int>) returns (c:int)
requires v.Length>0
ensures c==countMin(v,min(v,v.Length),v.Length)
//Implement and verify an O(v.Length) algorithm
{
var i:=1;
c:=1;
var mini:=v[0];
while(i<v.Length)
{
if(v[i]==mini){
c:=c + 1;
}
else if(v[i]<mini){
c:=1;
mini:=v[i];
}
i:=i+1;
}
}
|
098 | Dafny-Exercises_tmp_tmpjm75muf__Session6Exercises_ExercisePeekSum.dfy |
predicate isPeek(v:array<int>,i:int)
reads v
requires 0<=i<v.Length
{forall k::0<=k<i ==> v[i]>=v[k]}
function peekSum(v:array<int>,i:int):int
decreases i
reads v
requires 0<=i<=v.Length
{
if (i==0) then 0
else if isPeek(v,i-1) then v[i-1]+peekSum(v,i-1)
else peekSum(v,i-1)
}
method mPeekSum(v:array<int>) returns (sum:int)
requires v.Length>0
ensures sum==peekSum(v,v.Length)
//Implement and verify an O(v.Length) algorithm to solve this problem
{
var i:=1;
sum:=v[0];
var lmax:=v[0];
while(i<v.Length)
decreases v.Length -i
invariant 0<i<=v.Length
invariant lmax in v[..i]
invariant forall k::0<=k<i ==> lmax>=v[k];
invariant sum==peekSum(v, i)
{
if(v[i]>=lmax){
sum:=sum + v[i];
lmax:=v[i];
}
i:=i+1;
}
}
|
predicate isPeek(v:array<int>,i:int)
reads v
requires 0<=i<v.Length
{forall k::0<=k<i ==> v[i]>=v[k]}
function peekSum(v:array<int>,i:int):int
reads v
requires 0<=i<=v.Length
{
if (i==0) then 0
else if isPeek(v,i-1) then v[i-1]+peekSum(v,i-1)
else peekSum(v,i-1)
}
method mPeekSum(v:array<int>) returns (sum:int)
requires v.Length>0
ensures sum==peekSum(v,v.Length)
//Implement and verify an O(v.Length) algorithm to solve this problem
{
var i:=1;
sum:=v[0];
var lmax:=v[0];
while(i<v.Length)
{
if(v[i]>=lmax){
sum:=sum + v[i];
lmax:=v[i];
}
i:=i+1;
}
}
|
099 | Dafny-Exercises_tmp_tmpjm75muf__Session7Exercises_ExerciseBinarySearch.dfy | predicate sorted(s : seq<int>) {
forall u, w :: 0 <= u < w < |s| ==> s[u] <= s[w]
}
method binarySearch(v:array<int>, elem:int) returns (p:int)
requires sorted(v[0..v.Length])
ensures -1<=p<v.Length
ensures (forall u::0<=u<=p ==> v[u]<=elem) && (forall w::p<w<v.Length ==> v[w]>elem)
{
var c,f:=0,v.Length-1;
while (c<=f)
decreases f-c
invariant 0<=c<=v.Length && -1<=f<=v.Length-1 && c<=f+1
invariant (forall u::0<=u<c ==> v[u]<=elem) &&
(forall w::f<w<v.Length ==> v[w]>elem)
{
var m:=(c+f)/2;
if (v[m]<=elem)
{c:=m+1;}
else {f:=m-1;}
}
p:=c-1;
}
method search(v:array<int>,elem:int) returns (b:bool)
requires sorted(v[0..v.Length])
ensures b==(elem in v[0..v.Length])
//Implement by calling binary search function
{
var p:=binarySearch(v, elem);
if(p==-1){
b:= false;
}
else{
b:=v[p] == elem;
}
}
//Recursive binary search
method {:tailrecursion false} binarySearchRec(v:array<int>, elem:int, c:int, f:int) returns (p:int)
requires sorted(v[0..v.Length])
requires 0<=c<=f+1<=v.Length//0<=c<=v.Length && -1<=f<v.Length && c<=f+1
requires forall k::0<=k<c ==> v[k]<=elem
requires forall k::f<k<v.Length ==> v[k]>elem
decreases f-c
ensures -1<=p<v.Length
ensures (forall u::0<=u<=p ==> v[u]<=elem) && (forall w::p<w<v.Length ==> v[w]>elem)
{
if (c==f+1)
{p:=c-1;} //empty case: c-1 contains the last element less or equal than elem
else
{
var m:=(c+f)/2;
if (v[m]<=elem)
{p:=binarySearchRec(v,elem,m+1,f);}
else
{p:=binarySearchRec(v,elem,c,m-1);}
}
}
method otherbSearch(v:array<int>, elem:int) returns (b:bool,p:int)
requires sorted(v[0..v.Length])
ensures 0<=p<=v.Length
ensures b == (elem in v[0..v.Length])
ensures b ==> p<v.Length && v[p]==elem
ensures !b ==> (forall u::0<=u<p ==> v[u]<elem) &&
(forall w::p<=w<v.Length ==> v[w]>elem)
//Implement and verify
{
p:=binarySearch(v, elem);
if(p==-1){
b:= false;
p:=p+1;
}
else{
b:=v[p] == elem;
p:=p + if b then 0 else 1;
}
}
| predicate sorted(s : seq<int>) {
forall u, w :: 0 <= u < w < |s| ==> s[u] <= s[w]
}
method binarySearch(v:array<int>, elem:int) returns (p:int)
requires sorted(v[0..v.Length])
ensures -1<=p<v.Length
ensures (forall u::0<=u<=p ==> v[u]<=elem) && (forall w::p<w<v.Length ==> v[w]>elem)
{
var c,f:=0,v.Length-1;
while (c<=f)
(forall w::f<w<v.Length ==> v[w]>elem)
{
var m:=(c+f)/2;
if (v[m]<=elem)
{c:=m+1;}
else {f:=m-1;}
}
p:=c-1;
}
method search(v:array<int>,elem:int) returns (b:bool)
requires sorted(v[0..v.Length])
ensures b==(elem in v[0..v.Length])
//Implement by calling binary search function
{
var p:=binarySearch(v, elem);
if(p==-1){
b:= false;
}
else{
b:=v[p] == elem;
}
}
//Recursive binary search
method {:tailrecursion false} binarySearchRec(v:array<int>, elem:int, c:int, f:int) returns (p:int)
requires sorted(v[0..v.Length])
requires 0<=c<=f+1<=v.Length//0<=c<=v.Length && -1<=f<v.Length && c<=f+1
requires forall k::0<=k<c ==> v[k]<=elem
requires forall k::f<k<v.Length ==> v[k]>elem
ensures -1<=p<v.Length
ensures (forall u::0<=u<=p ==> v[u]<=elem) && (forall w::p<w<v.Length ==> v[w]>elem)
{
if (c==f+1)
{p:=c-1;} //empty case: c-1 contains the last element less or equal than elem
else
{
var m:=(c+f)/2;
if (v[m]<=elem)
{p:=binarySearchRec(v,elem,m+1,f);}
else
{p:=binarySearchRec(v,elem,c,m-1);}
}
}
method otherbSearch(v:array<int>, elem:int) returns (b:bool,p:int)
requires sorted(v[0..v.Length])
ensures 0<=p<=v.Length
ensures b == (elem in v[0..v.Length])
ensures b ==> p<v.Length && v[p]==elem
ensures !b ==> (forall u::0<=u<p ==> v[u]<elem) &&
(forall w::p<=w<v.Length ==> v[w]>elem)
//Implement and verify
{
p:=binarySearch(v, elem);
if(p==-1){
b:= false;
p:=p+1;
}
else{
b:=v[p] == elem;
p:=p + if b then 0 else 1;
}
}
|