test_ID stringlengths 3 3 | test_file stringlengths 14 119 | ground_truth stringlengths 70 28.7k | hints_removed stringlengths 58 28.7k |
|---|---|---|---|
200 | Dafny_tmp_tmpmvs2dmry_examples2.dfy | method add_by_inc(x: nat, y:nat) returns (z:nat)
ensures z == x+y;
{
z := x;
var i := 0;
while (i < y)
decreases y-i;
invariant 0 <= i <= y;
invariant z == x + i;
{
z := z+1;
i := i+1;
}
assert (z == x+y);
assert (i == y);
}
method Product(m: nat, n:nat) return... | method add_by_inc(x: nat, y:nat) returns (z:nat)
ensures z == x+y;
{
z := x;
var i := 0;
while (i < y)
{
z := z+1;
i := i+1;
}
}
method Product(m: nat, n:nat) returns (res:nat)
ensures res == m*n;
{
var m1: nat := m;
res:=0;
while (m1 != 0)
{
var n1: na... |
201 | Dafny_tmp_tmpmvs2dmry_pancakesort_findmax.dfy | // returns an index of the largest element of array 'a' in the range [0..n)
method findMax (a : array<int>, n : int) returns (r:int)
requires a.Length > 0
requires 0 < n <= a.Length
ensures 0 <= r < n <= a.Length;
ensures forall k :: 0 <= k < n <= a.Length ==> a[r] >= a[k];
ensures multiset(a[..]) == multiset(old(a[..]... | // returns an index of the largest element of array 'a' in the range [0..n)
method findMax (a : array<int>, n : int) returns (r:int)
requires a.Length > 0
requires 0 < n <= a.Length
ensures 0 <= r < n <= a.Length;
ensures forall k :: 0 <= k < n <= a.Length ==> a[r] >= a[k];
ensures multiset(a[..]) == multiset(old(a[..]... |
202 | Dafny_tmp_tmpmvs2dmry_pancakesort_flip.dfy | // flips (i.e., reverses) array elements in the range [0..num]
method flip (a: array<int>, num: int)
requires a.Length > 0;
requires 0 <= num < a.Length;
modifies a;
ensures forall k :: 0 <= k <= num ==> a[k] == old(a[num-k])
ensures forall k :: num < k < a.Length ==> a[k] == old(a[k])
// ensures multiset(a[..]) == old... | // flips (i.e., reverses) array elements in the range [0..num]
method flip (a: array<int>, num: int)
requires a.Length > 0;
requires 0 <= num < a.Length;
modifies a;
ensures forall k :: 0 <= k <= num ==> a[k] == old(a[num-k])
ensures forall k :: num < k < a.Length ==> a[k] == old(a[k])
// ensures multiset(a[..]) == old... |
203 | Dafny_tmp_tmpv_d3qi10_2_min.dfy |
function min(a: int, b: int): int
ensures min(a, b) <= a && min(a, b) <= b
ensures min(a, b) == a || min(a, b) == b
{
if a < b then a else b
}
method minMethod(a: int, b: int) returns (c: int)
ensures c <= a && c <= b
ensures c == a || c == b
// Ou encore:
ensures c == min(a, b)
{
if a... |
function min(a: int, b: int): int
ensures min(a, b) <= a && min(a, b) <= b
ensures min(a, b) == a || min(a, b) == b
{
if a < b then a else b
}
method minMethod(a: int, b: int) returns (c: int)
ensures c <= a && c <= b
ensures c == a || c == b
// Ou encore:
ensures c == min(a, b)
{
if a... |
204 | Dafny_tmp_tmpv_d3qi10_3_cumsum.dfy | function sum(a: array<int>, i: int): int
requires 0 <= i < a.Length
reads a
{
a[i] + if i == 0 then 0 else sum(a, i - 1)
}
method cumsum(a: array<int>, b: array<int>)
requires a.Length == b.Length && a.Length > 0 && a != b
// when you change a , that's not the same object than b .
//requires... | function sum(a: array<int>, i: int): int
requires 0 <= i < a.Length
reads a
{
a[i] + if i == 0 then 0 else sum(a, i - 1)
}
method cumsum(a: array<int>, b: array<int>)
requires a.Length == b.Length && a.Length > 0 && a != b
// when you change a , that's not the same object than b .
//requires... |
205 | FMSE-2022-2023_tmp_tmp6_x_ba46_Lab10_Lab10.dfy | predicate IsOdd(x: int) {
x % 2 == 1
}
newtype Odd = n : int | IsOdd(n) witness 3
trait OddListSpec
{
var s: seq<Odd>
var capacity: nat
predicate Valid()
reads this
{
0 <= |s| <= this.capacity &&
forall i :: 0 <= i < |s| ==> IsOdd(s[i] as int)
}
method insert(inde... | predicate IsOdd(x: int) {
x % 2 == 1
}
newtype Odd = n : int | IsOdd(n) witness 3
trait OddListSpec
{
var s: seq<Odd>
var capacity: nat
predicate Valid()
reads this
{
0 <= |s| <= this.capacity &&
forall i :: 0 <= i < |s| ==> IsOdd(s[i] as int)
}
method insert(inde... |
206 | FMSE-2022-2023_tmp_tmp6_x_ba46_Lab1_Lab1.dfy | /// Types defined as part of Tasks 3, 5 and 9
// Since we have created the IsOddNat predicate we use it to define the new Odd subsort
newtype Odd = n : int | IsOddNat(n) witness 1
// Since we have created the IsEvenNat predicate we use it to define the new Even subsort
newtype Even = n : int | IsEvenNat(n) witness 2
... | /// Types defined as part of Tasks 3, 5 and 9
// Since we have created the IsOddNat predicate we use it to define the new Odd subsort
newtype Odd = n : int | IsOddNat(n) witness 1
// Since we have created the IsEvenNat predicate we use it to define the new Even subsort
newtype Even = n : int | IsEvenNat(n) witness 2
... |
207 | FMSE-2022-2023_tmp_tmp6_x_ba46_Lab2_Lab2.dfy | /*
* Task 2: Define the natural numbers as an algebraic data type
*
* Being an inductive data type, it's required that we have a base case constructor and an inductive one.
*/
datatype Nat = Zero | S(Pred: Nat)
/// Task 2
// Exercise (a'): proving that the successor constructor is injective
/*
* It's known that ... | /*
* Task 2: Define the natural numbers as an algebraic data type
*
* Being an inductive data type, it's required that we have a base case constructor and an inductive one.
*/
datatype Nat = Zero | S(Pred: Nat)
/// Task 2
// Exercise (a'): proving that the successor constructor is injective
/*
* It's known that ... |
208 | FMSE-2022-2023_tmp_tmp6_x_ba46_Lab3_Lab3.dfy | /*
* Task 2: Define in Dafny the conatural numbers as a coinductive datatype
*
* Being a coinductive data type, it's required that we have a base case constructor and an inductive one
* (as is the case with inductive ones as well)
*/
codatatype Conat = Zero | Succ(Pred: Conat)
// Exercise (a): explain why the f... | /*
* Task 2: Define in Dafny the conatural numbers as a coinductive datatype
*
* Being a coinductive data type, it's required that we have a base case constructor and an inductive one
* (as is the case with inductive ones as well)
*/
codatatype Conat = Zero | Succ(Pred: Conat)
// Exercise (a): explain why the f... |
209 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Exercise3_Increment_Array.dfy | method incrementArray(a:array<int>)
requires a.Length > 0
ensures forall i :: 0 <= i < a.Length ==> a[i] == old(a[i]) + 1
modifies a
{
var j : int := 0;
while(j < a.Length)
invariant 0 <= j <= a.Length
invariant forall i :: j <= i < a.Length ==> a[i] == old(a[i])
invariant forall i :: 0 <= i < j =... | method incrementArray(a:array<int>)
requires a.Length > 0
ensures forall i :: 0 <= i < a.Length ==> a[i] == old(a[i]) + 1
modifies a
{
var j : int := 0;
while(j < a.Length)
{
a[j] := a[j] + 1;
j := j+1;
}
}
|
210 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Exercise4_Find_Max.dfy | method findMax(a:array<int>) returns (pos:int, maxVal: int)
requires a.Length > 0;
requires forall i :: 0 <= i < a.Length ==> a[i] >= 0;
ensures forall i :: 0 <= i < a.Length ==> a[i] <= maxVal;
ensures exists i :: 0 <= i < a.Length && a[i] == maxVal;
ensures 0 <= pos < a.Length
ensures a[pos] == maxVal;
{... | method findMax(a:array<int>) returns (pos:int, maxVal: int)
requires a.Length > 0;
requires forall i :: 0 <= i < a.Length ==> a[i] >= 0;
ensures forall i :: 0 <= i < a.Length ==> a[i] <= maxVal;
ensures exists i :: 0 <= i < a.Length && a[i] == maxVal;
ensures 0 <= pos < a.Length
ensures a[pos] == maxVal;
{... |
211 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Exercise6_Binary_Search.dfy | method binarySearch(a:array<int>, val:int) returns (pos:int)
requires a.Length > 0
requires forall i, j :: 0 <= i < j < a.Length ==> a[i] <= a[j]
ensures 0 <= pos < a.Length ==> a[pos] == val
ensures pos < 0 || pos >= a.Length ==> forall i :: 0 <= i < a.Length ==> a[i] != val
{
var left := 0;
var right :... | method binarySearch(a:array<int>, val:int) returns (pos:int)
requires a.Length > 0
requires forall i, j :: 0 <= i < j < a.Length ==> a[i] <= a[j]
ensures 0 <= pos < a.Length ==> a[pos] == val
ensures pos < 0 || pos >= a.Length ==> forall i :: 0 <= i < a.Length ==> a[i] != val
{
var left := 0;
var right :... |
212 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Insertion_Sort_Normal.dfy | predicate sorted (a: array<int>)
reads a
{
sortedA(a, a.Length)
}
predicate sortedA (a: array<int>, i: int)
requires 0 <= i <= a.Length
reads a
{
forall k :: 0 < k < i ==> a[k-1] <= a[k]
}
method lookForMin (a: array<int>, i: int) returns (m: int)
requires 0 <= i < a.Length
ensures i <= m < a.Length
ensure... | predicate sorted (a: array<int>)
reads a
{
sortedA(a, a.Length)
}
predicate sortedA (a: array<int>, i: int)
requires 0 <= i <= a.Length
reads a
{
forall k :: 0 < k < i ==> a[k-1] <= a[k]
}
method lookForMin (a: array<int>, i: int) returns (m: int)
requires 0 <= i < a.Length
ensures i <= m < a.Length
ensure... |
213 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Insertion_Sorted_Standard.dfy | predicate InsertionSorted(Array: array<int>, left: int, right: int)
requires 0 <= left <= right <= Array.Length
reads Array
{
forall i,j :: left <= i < j < right ==> Array[i] <= Array[j]
}
method sorting(Array: array<int>)
requires Array.Length > 1
ensures InsertionSorted(Array, ... | predicate InsertionSorted(Array: array<int>, left: int, right: int)
requires 0 <= left <= right <= Array.Length
reads Array
{
forall i,j :: left <= i < j < right ==> Array[i] <= Array[j]
}
method sorting(Array: array<int>)
requires Array.Length > 1
ensures InsertionSorted(Array, ... |
214 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Merge_Sort.dfy | method mergeSort(a: array<int>)
modifies a
{
sorting(a, 0, a.Length-1);
}
method merging(a: array<int>, low: int, medium: int, high: int)
requires 0 <= low <= medium <= high < a.Length
modifies a
{
var x := 0;
var y := 0;
var z := 0;
var a1: array<int> := new [medium - low + 1];
var a2: array<int> := new [... | method mergeSort(a: array<int>)
modifies a
{
sorting(a, 0, a.Length-1);
}
method merging(a: array<int>, low: int, medium: int, high: int)
requires 0 <= low <= medium <= high < a.Length
modifies a
{
var x := 0;
var y := 0;
var z := 0;
var a1: array<int> := new [medium - low + 1];
var a2: array<int> := new [... |
215 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Quick_Sort.dfy | predicate quickSorted(Seq: seq<int>)
{
forall idx_1, idx_2 :: 0 <= idx_1 < idx_2 < |Seq| ==> Seq[idx_1] <= Seq[idx_2]
}
method threshold(thres:int,Seq:seq<int>) returns (Seq_1:seq<int>,Seq_2:seq<int>)
ensures (forall x | x in Seq_1 :: x <= thres) && (forall x | x in Seq_2 :: x >= thres)
ensures |Seq_1| + |Seq_2|... | predicate quickSorted(Seq: seq<int>)
{
forall idx_1, idx_2 :: 0 <= idx_1 < idx_2 < |Seq| ==> Seq[idx_1] <= Seq[idx_2]
}
method threshold(thres:int,Seq:seq<int>) returns (Seq_1:seq<int>,Seq_2:seq<int>)
ensures (forall x | x in Seq_1 :: x <= thres) && (forall x | x in Seq_2 :: x >= thres)
ensures |Seq_1| + |Seq_2|... |
216 | Final-Project-Dafny_tmp_tmpmcywuqox_Attempts_Selection_Sort_Standard.dfy | method selectionSorted(Array: array<int>)
modifies Array
ensures multiset(old(Array[..])) == multiset(Array[..])
{
var idx := 0;
while (idx < Array.Length)
invariant 0 <= idx <= Array.Length
invariant forall i,j :: 0 <= i < idx <= j < Array.Length ==> Array[i] <= Array[j]
invariant forall i,j :: 0... | method selectionSorted(Array: array<int>)
modifies Array
ensures multiset(old(Array[..])) == multiset(Array[..])
{
var idx := 0;
while (idx < Array.Length)
{
var minIndex := idx;
var idx' := idx + 1;
while (idx' < Array.Length)
{
if (Array[idx'] < Array[minIndex]) {
minIndex := ... |
217 | Final-Project-Dafny_tmp_tmpmcywuqox_Final_Project_3.dfy | method nonZeroReturn(x: int) returns (y: int)
ensures y != 0
{
if x == 0 {
return x + 1;
} else {
return -x;
}
}
method test() {
var input := nonZeroReturn(-1);
assert input != 0;
}
| method nonZeroReturn(x: int) returns (y: int)
ensures y != 0
{
if x == 0 {
return x + 1;
} else {
return -x;
}
}
method test() {
var input := nonZeroReturn(-1);
}
|
218 | FlexWeek_tmp_tmpc_tfdj_3_ex2.dfy | // 2. Given an array of positive and negative integers, it returns an array of the absolute value of all the integers. [-4,1,5,-2,-5]->[4,1,5,2,5]
function abs(a:int):nat
{
if a < 0 then -a else a
}
method aba(a:array<int>)returns (b:array<int>)
ensures a.Length == b.Length // needed for next line
ensures forall ... | // 2. Given an array of positive and negative integers, it returns an array of the absolute value of all the integers. [-4,1,5,-2,-5]->[4,1,5,2,5]
function abs(a:int):nat
{
if a < 0 then -a else a
}
method aba(a:array<int>)returns (b:array<int>)
ensures a.Length == b.Length // needed for next line
ensures forall ... |
219 | FlexWeek_tmp_tmpc_tfdj_3_ex3.dfy | method Max(a:array<nat>)returns(m:int)
ensures a.Length > 0 ==> forall k :: 0<=k<a.Length ==> m >= a[k]// not strong enough
ensures a.Length == 0 ==> m == -1
ensures a.Length > 0 ==> m in a[..] // finally at the top // approach did not work for recusrive function
{
if(a.Length == 0){
return -1;
}
as... | method Max(a:array<nat>)returns(m:int)
ensures a.Length > 0 ==> forall k :: 0<=k<a.Length ==> m >= a[k]// not strong enough
ensures a.Length == 0 ==> m == -1
ensures a.Length > 0 ==> m in a[..] // finally at the top // approach did not work for recusrive function
{
if(a.Length == 0){
return -1;
}
va... |
220 | FlexWeek_tmp_tmpc_tfdj_3_ex4.dfy | method join(a:array<int>,b:array<int>) returns (c:array<int>)
ensures a[..] + b[..] == c[..]
ensures multiset(a[..] + b[..]) == multiset(c[..])
ensures multiset(a[..]) + multiset(b[..]) == multiset(c[..])
ensures a.Length+b.Length == c.Length
// Forall
ensures forall i :: 0<=i<a.Length ==> c[i] == a[i]
ensures forall... | method join(a:array<int>,b:array<int>) returns (c:array<int>)
ensures a[..] + b[..] == c[..]
ensures multiset(a[..] + b[..]) == multiset(c[..])
ensures multiset(a[..]) + multiset(b[..]) == multiset(c[..])
ensures a.Length+b.Length == c.Length
// Forall
ensures forall i :: 0<=i<a.Length ==> c[i] == a[i]
ensures forall... |
221 | FlexWeek_tmp_tmpc_tfdj_3_reverse.dfy | // Write an *iterative* Dafny method Reverse with signature:
// method Reverse(a: array<char>) returns (b: array<char>)
// which takes an input array of characters 'a' and outputs array 'b' consisting of
// the elements of the input array in reverse order. The following conditions apply:
// - the input array... | // Write an *iterative* Dafny method Reverse with signature:
// method Reverse(a: array<char>) returns (b: array<char>)
// which takes an input array of characters 'a' and outputs array 'b' consisting of
// the elements of the input array in reverse order. The following conditions apply:
// - the input array... |
222 | Formal-Methods-Project_tmp_tmphh2ar2xv_BubbleSort.dfy | predicate sorted(a: array?<int>, l: int, u: int)
reads a;
requires a != null;
{
forall i, j :: 0 <= l <= i <= j <= u < a.Length ==> a[i] <= a[j]
}
predicate partitioned(a: array?<int>, i: int)
reads a
requires a != null
{
forall k, k' :: 0 <= k <= i < k' < a.Length ==> a[k] <= a[k']
}
method Bu... | predicate sorted(a: array?<int>, l: int, u: int)
reads a;
requires a != null;
{
forall i, j :: 0 <= l <= i <= j <= u < a.Length ==> a[i] <= a[j]
}
predicate partitioned(a: array?<int>, i: int)
reads a
requires a != null
{
forall k, k' :: 0 <= k <= i < k' < a.Length ==> a[k] <= a[k']
}
method Bu... |
223 | Formal-Methods-Project_tmp_tmphh2ar2xv_Factorial.dfy |
method Fact(x: int) returns (y: int)
requires x >= 0;
{
y := 1;
var z := 0;
while(z != x)
decreases x - z;
invariant 0 <= x-z;
{
z := z + 1;
y := y * z;
}
}
method Main() {
var a := Fact(87);
print a;
}
|
method Fact(x: int) returns (y: int)
requires x >= 0;
{
y := 1;
var z := 0;
while(z != x)
{
z := z + 1;
y := y * z;
}
}
method Main() {
var a := Fact(87);
print a;
}
|
224 | Formal-Verification-Project_tmp_tmp9gmwsmyp_strings3.dfy | predicate isPrefixPred(pre:string, str:string)
{
(|pre| <= |str|) &&
pre == str[..|pre|]
}
predicate isNotPrefixPred(pre:string, str:string)
{
(|pre| > |str|) ||
pre != str[..|pre|]
}
lemma PrefixNegationLemma(pre:string, str:string)
ensures isPrefixPred(pre,str) <==> !isNotPrefixPred(pre,str)
ensures !isPre... | predicate isPrefixPred(pre:string, str:string)
{
(|pre| <= |str|) &&
pre == str[..|pre|]
}
predicate isNotPrefixPred(pre:string, str:string)
{
(|pre| > |str|) ||
pre != str[..|pre|]
}
lemma PrefixNegationLemma(pre:string, str:string)
ensures isPrefixPred(pre,str) <==> !isNotPrefixPred(pre,str)
ensures !isPre... |
225 | Formal-Verification_tmp_tmpuyt21wjt_Dafny_strings1.dfy | predicate isPrefixPredicate(pre: string, str:string)
{
|str| >= |pre| && pre <= str
}
method isPrefix(pre: string, str: string) returns (res: bool)
ensures |pre| > |str| ==> !res
ensures res == isPrefixPredicate(pre, str)
{
if |pre| > |str|
{return false;}
var i := 0;
while i < |pre|
decreases |pr... | predicate isPrefixPredicate(pre: string, str:string)
{
|str| >= |pre| && pre <= str
}
method isPrefix(pre: string, str: string) returns (res: bool)
ensures |pre| > |str| ==> !res
ensures res == isPrefixPredicate(pre, str)
{
if |pre| > |str|
{return false;}
var i := 0;
while i < |pre|
{
if pre[i]... |
226 | Formal-Verification_tmp_tmpuyt21wjt_Dafny_strings3.dfy | // We spent 2h each on this assignment
predicate isPrefixPred(pre:string, str:string)
{
(|pre| <= |str|) &&
pre == str[..|pre|]
}
predicate isNotPrefixPred(pre:string, str:string)
{
(|pre| > |str|) ||
pre != str[..|pre|]
}
lemma PrefixNegationLemma(pre:string, str:string)
ensures isPrefixPred(pre,str) <==> !... | // We spent 2h each on this assignment
predicate isPrefixPred(pre:string, str:string)
{
(|pre| <= |str|) &&
pre == str[..|pre|]
}
predicate isNotPrefixPred(pre:string, str:string)
{
(|pre| > |str|) ||
pre != str[..|pre|]
}
lemma PrefixNegationLemma(pre:string, str:string)
ensures isPrefixPred(pre,str) <==> !... |
227 | Formal-methods-of-software-development_tmp_tmppryvbyty_Bloque 1_Lab3.dfy |
method multipleReturns (x:int, y:int) returns (more:int, less:int)
requires y > 0
ensures less < x < more
method multipleReturns2 (x:int, y:int) returns (more:int, less:int)
requires y > 0
ensures more + less == 2*x
// TODO: Hacer en casa
method multipleReturns3 (x:int, y:int) returns (more:int, less:int)
requires ... |
method multipleReturns (x:int, y:int) returns (more:int, less:int)
requires y > 0
ensures less < x < more
method multipleReturns2 (x:int, y:int) returns (more:int, less:int)
requires y > 0
ensures more + less == 2*x
// TODO: Hacer en casa
method multipleReturns3 (x:int, y:int) returns (more:int, less:int)
requires ... |
228 | Formal-methods-of-software-development_tmp_tmppryvbyty_Bloque 2_Lab6.dfy | /*predicate palindrome<T(==)> (s:seq<T>)
{
forall i:: 0<=i<|s| ==> s[i] == s[|s|-i-1]
}
*/
// SUM OF A SEQUENCE OF INTEGERS
function sum(v: seq<int>): int
decreases v
{
if v==[] then 0
else if |v|==1 then v[0]
else v[0]+sum(v[1..])
}
/*
method vector_Sum(v:seq<int>) returns (x:int)
ensures x == sum(v)... | /*predicate palindrome<T(==)> (s:seq<T>)
{
forall i:: 0<=i<|s| ==> s[i] == s[|s|-i-1]
}
*/
// SUM OF A SEQUENCE OF INTEGERS
function sum(v: seq<int>): int
{
if v==[] then 0
else if |v|==1 then v[0]
else v[0]+sum(v[1..])
}
/*
method vector_Sum(v:seq<int>) returns (x:int)
ensures x == sum(v)
{
var ... |
229 | Formal-methods-of-software-development_tmp_tmppryvbyty_Examenes_Beni_Heusel-Benedikt-Ass-1.dfy | // APELLIDOS: Heusel
// NOMBRE: Benedikt
// ESPECIALIDAD: ninguna (Erasmus)
// ESTÁ PERFECTO, NO HAY NINGUN COMENTARIO MAS ABAJO
// EJERCICIO 1
// Demostrar el lemma div10_Lemma por inducción en n
// y luego usarlo para demostrar el lemma div10Forall_Lemma
function exp (x:int,e:nat):int
{
if e == 0 then 1 else x *... | // APELLIDOS: Heusel
// NOMBRE: Benedikt
// ESPECIALIDAD: ninguna (Erasmus)
// ESTÁ PERFECTO, NO HAY NINGUN COMENTARIO MAS ABAJO
// EJERCICIO 1
// Demostrar el lemma div10_Lemma por inducción en n
// y luego usarlo para demostrar el lemma div10Forall_Lemma
function exp (x:int,e:nat):int
{
if e == 0 then 1 else x *... |
230 | FormalMethods_tmp_tmpvda2r3_o_dafny_Invariants_ex1.dfy | method Mult(x:nat, y:nat) returns (r:nat)
ensures r == x * y
{
// Valores passados por parâmetros são imutáveis
var m := x;
var n := y;
r := 0;
// Soma sucessiva para multiplicar dois números.
while m > 0
invariant m*n+r == x*y
invariant m>=0
{
r := r + n;
m := m - 1;... | method Mult(x:nat, y:nat) returns (r:nat)
ensures r == x * y
{
// Valores passados por parâmetros são imutáveis
var m := x;
var n := y;
r := 0;
// Soma sucessiva para multiplicar dois números.
while m > 0
{
r := r + n;
m := m - 1;
}
return r; // NOT(m>0) ^ Inv ==> r =... |
231 | FormalMethods_tmp_tmpvda2r3_o_dafny_Invariants_ex2.dfy | function Potencia(x:nat, y:nat):nat
{
if y == 0
then 1
else x * Potencia(x, y-1)
}
method Pot(x:nat, y:nat) returns (r:nat)
ensures r == Potencia(x,y)
{
r := 1;
var b := x;
var e := y;
while e > 0
invariant Potencia(b,e) * r == Potencia(x,y)
{
r := r * b;
e := e - 1;... | function Potencia(x:nat, y:nat):nat
{
if y == 0
then 1
else x * Potencia(x, y-1)
}
method Pot(x:nat, y:nat) returns (r:nat)
ensures r == Potencia(x,y)
{
r := 1;
var b := x;
var e := y;
while e > 0
{
r := r * b;
e := e - 1;
}
return r;
}
/*
Inv =
Pot(2,3)
Teste ... |
232 | Formal_Verification_With_Dafny_tmp_tmp5j79rq48_Counter.dfy | class Counter {
var value : int ;
constructor init()
ensures value == 0;
{
value := 0 ;
}
method getValue() returns (x:int)
ensures x == value;
{
x := value ;
}
method inc()
modifies this`value
requires value >= 0;
ensures value == old(value) + 1;
{
value := value + ... | class Counter {
var value : int ;
constructor init()
ensures value == 0;
{
value := 0 ;
}
method getValue() returns (x:int)
ensures x == value;
{
x := value ;
}
method inc()
modifies this`value
requires value >= 0;
ensures value == old(value) + 1;
{
value := value + ... |
233 | Formal_Verification_With_Dafny_tmp_tmp5j79rq48_LimitedStack.dfy | // A LIFO queue (aka a stack) with limited capacity.
class LimitedStack{
var capacity : int; // capacity, max number of elements allowed on the stack.
var arr : array<int>; // contents of stack.
var top : int; // The index of the top of the stack, or -1 if the stack is empty
// This predicate... | // A LIFO queue (aka a stack) with limited capacity.
class LimitedStack{
var capacity : int; // capacity, max number of elements allowed on the stack.
var arr : array<int>; // contents of stack.
var top : int; // The index of the top of the stack, or -1 if the stack is empty
// This predicate... |
234 | HATRA-2022-Paper_tmp_tmp5texxy8l_copilot_verification_Binary Search_binary_search.dfy | // Dafny verification of binary search alogirthm from binary_search.py
// Inspired by: https://ece.uwaterloo.ca/~agurfink/stqam/rise4fun-Dafny/#h211
method BinarySearch(arr: array<int>, target: int) returns (index: int)
requires distinct(arr)
requires sorted(arr)
ensures -1 <= index < arr.Length
ensure... | // Dafny verification of binary search alogirthm from binary_search.py
// Inspired by: https://ece.uwaterloo.ca/~agurfink/stqam/rise4fun-Dafny/#h211
method BinarySearch(arr: array<int>, target: int) returns (index: int)
requires distinct(arr)
requires sorted(arr)
ensures -1 <= index < arr.Length
ensure... |
235 | HATRA-2022-Paper_tmp_tmp5texxy8l_copilot_verification_Largest Sum_largest_sum.dfy | // CoPilot function converted to dafny
method largest_sum(nums: array<int>, k: int) returns (sum: int)
requires nums.Length > 0
ensures max_sum_subarray(nums, sum, 0, nums.Length)
{
var max_sum := 0;
var current_sum := 0;
var i := 0;
while (i < nums.Length)
invariant 0 <= i <= nums... | // CoPilot function converted to dafny
method largest_sum(nums: array<int>, k: int) returns (sum: int)
requires nums.Length > 0
ensures max_sum_subarray(nums, sum, 0, nums.Length)
{
var max_sum := 0;
var current_sum := 0;
var i := 0;
while (i < nums.Length)
{
current_sum := cur... |
236 | HATRA-2022-Paper_tmp_tmp5texxy8l_copilot_verification_Sort Array_sort_array.dfy | method sortArray(arr: array<int>) returns (arr_sorted: array<int>)
// Requires array length to be between 0 and 10000
requires 0 <= arr.Length < 10000
// Ensuring the arry has been sorted
ensures sorted(arr_sorted, 0, arr_sorted.Length)
// Ensuring that we have not modified elements but have only ch... | method sortArray(arr: array<int>) returns (arr_sorted: array<int>)
// Requires array length to be between 0 and 10000
requires 0 <= arr.Length < 10000
// Ensuring the arry has been sorted
ensures sorted(arr_sorted, 0, arr_sorted.Length)
// Ensuring that we have not modified elements but have only ch... |
237 | HATRA-2022-Paper_tmp_tmp5texxy8l_copilot_verification_Two Sum_two_sum.dfy | method twoSum(nums: array<int>, target: int) returns (index1: int, index2: int)
requires 2 <= nums.Length
requires exists i, j :: (0 <= i < j < nums.Length && nums[i] + nums[j] == target)
ensures index1 != index2
ensures 0 <= index1 < nums.Length
ensures 0 <= index2 < nums.Length
ensures nums[in... | method twoSum(nums: array<int>, target: int) returns (index1: int, index2: int)
requires 2 <= nums.Length
requires exists i, j :: (0 <= i < j < nums.Length && nums[i] + nums[j] == target)
ensures index1 != index2
ensures 0 <= index1 < nums.Length
ensures 0 <= index2 < nums.Length
ensures nums[in... |
238 | Invoker_tmp_tmpypx0gs8x_dafny_abstract-interpreter_SimpleVerifier.dfy | module Ints {
const U32_BOUND: nat := 0x1_0000_0000
newtype u32 = x:int | 0 <= x < 0x1_0000_0000
newtype i32 = x: int | -0x8000_0000 <= x < 0x8000_0000
}
module Lang {
import opened Ints
datatype Reg = R0 | R1 | R2 | R3
datatype Expr =
| Const(n: u32)
// overflow during addition is an error
... | module Ints {
const U32_BOUND: nat := 0x1_0000_0000
newtype u32 = x:int | 0 <= x < 0x1_0000_0000
newtype i32 = x: int | -0x8000_0000 <= x < 0x8000_0000
}
module Lang {
import opened Ints
datatype Reg = R0 | R1 | R2 | R3
datatype Expr =
| Const(n: u32)
// overflow during addition is an error
... |
239 | M2_tmp_tmp2laaavvl_Software Verification_Exercices_Exo4-CountAndReturn.dfy | method CountToAndReturnN(n: int) returns (r: int)
requires n >= 0
ensures r == n
{
var i := 0;
while i < n
invariant 0 <= i <= n
{
i := i + 1;
}
r := i;
}
| method CountToAndReturnN(n: int) returns (r: int)
requires n >= 0
ensures r == n
{
var i := 0;
while i < n
{
i := i + 1;
}
r := i;
}
|
240 | M2_tmp_tmp2laaavvl_Software Verification_Exercices_Exo7-ComputeSum.dfy | function Sum(n:nat):nat
{
if n==0 then 0 else n + Sum(n-1)
}
method ComputeSum(n:nat) returns (s:nat)
ensures s ==Sum(n)
{
s := 0;
var i := 0;
while i< n
invariant 0 <= i <= n
invariant s == Sum(i)
{
s := s + i + 1;
i := i+1;
}
}
| function Sum(n:nat):nat
{
if n==0 then 0 else n + Sum(n-1)
}
method ComputeSum(n:nat) returns (s:nat)
ensures s ==Sum(n)
{
s := 0;
var i := 0;
while i< n
{
s := s + i + 1;
i := i+1;
}
}
|
241 | M2_tmp_tmp2laaavvl_Software Verification_Exercices_Exo9-Carre.dfy | method Carre(a: nat) returns (c: nat)
ensures c == a*a
{
var i := 0;
c := 0;
while i != a
invariant 0 <= i <= a
invariant c == i*i
decreases a - i
{
c := c + 2*i +1;
i := i + 1;
}
}
| method Carre(a: nat) returns (c: nat)
ensures c == a*a
{
var i := 0;
c := 0;
while i != a
{
c := c + 2*i +1;
i := i + 1;
}
}
|
242 | MFDS_tmp_tmpvvr5y1t9_Assignments_Ass-1-2020-21-Sol-eGela.dfy | // Ejercicio 1: Demostrar por inducci�n el siguiente lema:
lemma EcCuadDiv2_Lemma (x:int)
requires x >= 1
ensures (x*x + x) % 2 == 0
{
if x != 1 {
EcCuadDiv2_Lemma(x-1);
assert x*x+x == ((x-1)*(x-1) + (x-1)) + 2*x;
}
}
// Ejercicio 2: Demostrar por inducci�n el siguiente lema
// Ind... | // Ejercicio 1: Demostrar por inducci�n el siguiente lema:
lemma EcCuadDiv2_Lemma (x:int)
requires x >= 1
ensures (x*x + x) % 2 == 0
{
if x != 1 {
EcCuadDiv2_Lemma(x-1);
}
}
// Ejercicio 2: Demostrar por inducci�n el siguiente lema
// Indicaciones: (1) Puedes llamar al lema del ejercicio an... |
243 | MFES_2021_tmp_tmpuljn8zd9_Exams_Special_Exam_03_2020_4_CatalanNumbers.dfy | function C(n: nat): nat
decreases n
{
if n == 0 then 1 else (4 * n - 2) * C(n-1) / (n + 1)
}
method calcC(n: nat) returns (res: nat)
ensures res == C(n)
{
var i := 0;
res := 1;
assert res == C(i) && 0 <= i <= n;
while i < n
decreases n - i //a - loop variant
invariant res ... | function C(n: nat): nat
{
if n == 0 then 1 else (4 * n - 2) * C(n-1) / (n + 1)
}
method calcC(n: nat) returns (res: nat)
ensures res == C(n)
{
var i := 0;
res := 1;
while i < n
{
ghost var v0 := n - i;
i := i + 1;
res := (4 * i - 2) * res / (i + 1);
}
}
|
244 | MFES_2021_tmp_tmpuljn8zd9_FCUL_Exercises_10_find.dfy | method find(a: array<int>, key: int) returns(index: int)
requires a.Length > 0;
ensures 0 <= index <= a.Length;
ensures index < a.Length ==> a[index] == key;
{
index := 0;
while index < a.Length && a[index] != key
decreases a.Length - index
invariant 0 <= index <= a.Length
... | method find(a: array<int>, key: int) returns(index: int)
requires a.Length > 0;
ensures 0 <= index <= a.Length;
ensures index < a.Length ==> a[index] == key;
{
index := 0;
while index < a.Length && a[index] != key
{
index := index + 1;
}
}
|
245 | MFES_2021_tmp_tmpuljn8zd9_FCUL_Exercises_8_sum.dfy | function calcSum(n: nat) : nat
{
n * (n - 1) / 2
}
method sum(n: nat) returns(s: nat)
ensures s == calcSum(n + 1)
{
s := 0;
var i := 0;
while i < n
decreases n - i
invariant 0 <= i <= n
invariant s == calcSum(i + 1)
{
i := i + 1;
s := s + i;
}
}
| function calcSum(n: nat) : nat
{
n * (n - 1) / 2
}
method sum(n: nat) returns(s: nat)
ensures s == calcSum(n + 1)
{
s := 0;
var i := 0;
while i < n
{
i := i + 1;
s := s + i;
}
}
|
246 | MFES_2021_tmp_tmpuljn8zd9_PracticalClasses_TP3_2_Insertion_Sort.dfy | // Sorts array 'a' using the insertion sort algorithm.
method insertionSort(a: array<int>)
modifies a
ensures isSorted(a, 0, a.Length)
ensures multiset(a[..]) == multiset(old(a[..]))
{
var i := 0;
while i < a.Length
decreases a.Length - i
invariant 0 <= i <= a.Length
invar... | // Sorts array 'a' using the insertion sort algorithm.
method insertionSort(a: array<int>)
modifies a
ensures isSorted(a, 0, a.Length)
ensures multiset(a[..]) == multiset(old(a[..]))
{
var i := 0;
while i < a.Length
{
var j := i;
while j > 0 && a[j-1] > a[j]
{
... |
247 | MFES_2021_tmp_tmpuljn8zd9_TheoreticalClasses_Power.dfy | /*
* Formal verification of O(n) and O(log n) algorithms to calculate the natural
* power of a real number (x^n), illustrating the usage of lemmas.
* FEUP, MIEIC, MFES, 2020/21.
*/
// Initial specification/definition of x^n, recursive, functional style,
// with time and space complexity O(n).
function power(x: real,... | /*
* Formal verification of O(n) and O(log n) algorithms to calculate the natural
* power of a real number (x^n), illustrating the usage of lemmas.
* FEUP, MIEIC, MFES, 2020/21.
*/
// Initial specification/definition of x^n, recursive, functional style,
// with time and space complexity O(n).
function power(x: real,... |
248 | MFS_tmp_tmpmmnu354t_Praticas_TP9_Power.dfy | /*
* Formal verification of O(n) and O(log n) algorithms to calculate the natural
* power of a real number (x^n), illustrating the usage of lemmas.
* FEUP, M.EIC, MFS, 2021/22.
*/
// Initial specification/definition of x^n, recursive, functional style,
// with time and space complexity O(n).
function power(x: real, ... | /*
* Formal verification of O(n) and O(log n) algorithms to calculate the natural
* power of a real number (x^n), illustrating the usage of lemmas.
* FEUP, M.EIC, MFS, 2021/22.
*/
// Initial specification/definition of x^n, recursive, functional style,
// with time and space complexity O(n).
function power(x: real, ... |
249 | MFS_tmp_tmpmmnu354t_Testes anteriores_T2_ex5_2020_2.dfy | method leq(a: array<int>, b: array<int>) returns (result: bool)
ensures result <==> (a.Length <= b.Length && a[..] == b[..a.Length]) || (exists k :: 0 <= k < a.Length && k < b.Length && a[..k] == b[..k] && a[k] < b[k])
{
var i := 0;
while i < a.Length && i < b.Length
decreases a.Length - i
... | method leq(a: array<int>, b: array<int>) returns (result: bool)
ensures result <==> (a.Length <= b.Length && a[..] == b[..a.Length]) || (exists k :: 0 <= k < a.Length && k < b.Length && a[..k] == b[..k] && a[k] < b[k])
{
var i := 0;
while i < a.Length && i < b.Length
{
if a[i] < b[i] { return ... |
250 | MIEIC_mfes_tmp_tmpq3ho7nve_TP3_binary_search.dfy | // Checks if array 'a' is sorted.
predicate isSorted(a: array<int>)
reads a
{
forall i, j :: 0 <= i < j < a.Length ==> a[i] <= a[j]
}
// Finds a value 'x' in a sorted array 'a', and returns its index,
// or -1 if not found.
method binarySearch(a: array<int>, x: int) returns (index: int)
requires isSorted(a... | // Checks if array 'a' is sorted.
predicate isSorted(a: array<int>)
reads a
{
forall i, j :: 0 <= i < j < a.Length ==> a[i] <= a[j]
}
// Finds a value 'x' in a sorted array 'a', and returns its index,
// or -1 if not found.
method binarySearch(a: array<int>, x: int) returns (index: int)
requires isSorted(a... |
251 | MIEIC_mfes_tmp_tmpq3ho7nve_exams_appeal_20_p4.dfy | function F(n: nat): nat { if n <= 2 then n else F(n-1) + F(n-3)}
method calcF(n: nat) returns (res: nat)
ensures res == F(n)
{
var a, b, c := 0, 1, 2;
var i := 0;
while i < n
decreases n-i
invariant 0 <= i <= n
invariant a == F(i) && b == F(i+1) && c == F(i+2)
{
a, b, c := b, c, a + c; ... | function F(n: nat): nat { if n <= 2 then n else F(n-1) + F(n-3)}
method calcF(n: nat) returns (res: nat)
ensures res == F(n)
{
var a, b, c := 0, 1, 2;
var i := 0;
while i < n
{
a, b, c := b, c, a + c;
i := i + 1;
}
res := a;
}
|
252 | MIEIC_mfes_tmp_tmpq3ho7nve_exams_mt2_19_p4.dfy | function R(n: nat): nat {
if n == 0 then 0 else if R(n-1) > n then R(n-1) - n else R(n-1) + n
}
method calcR(n: nat) returns (r: nat)
ensures r == R(n)
{
r := 0;
var i := 0;
while i < n
decreases n-i
invariant 0 <= i <= n
invariant r == R(i)
{
i := i + 1;
... | function R(n: nat): nat {
if n == 0 then 0 else if R(n-1) > n then R(n-1) - n else R(n-1) + n
}
method calcR(n: nat) returns (r: nat)
ensures r == R(n)
{
r := 0;
var i := 0;
while i < n
{
i := i + 1;
if r > i {
r := r - i;
}
else {
r := r +... |
253 | MIEIC_mfes_tmp_tmpq3ho7nve_exams_mt2_19_p5.dfy | type T = int // example
// Partitions a nonempty array 'a', by reordering the elements in the array,
// so that elements smaller than a chosen pivot are placed to the left of the
// pivot, and values greater or equal than the pivot are placed to the right of
// the pivot. Returns the pivot position.
method partition... | type T = int // example
// Partitions a nonempty array 'a', by reordering the elements in the array,
// so that elements smaller than a chosen pivot are placed to the left of the
// pivot, and values greater or equal than the pivot are placed to the right of
// the pivot. Returns the pivot position.
method partition... |
254 | MIEIC_mfes_tmp_tmpq3ho7nve_exams_special_20_p5.dfy | type T = int // for demo purposes, but could be another type
predicate sorted(a: array<T>, n: nat)
requires n <= a.Length
reads a
{
forall i,j :: 0 <= i < j < n ==> a[i] <= a[j]
}
// Use binary search to find an appropriate position to insert a value 'x'
// in a sorted array 'a', so that it remains sorted... | type T = int // for demo purposes, but could be another type
predicate sorted(a: array<T>, n: nat)
requires n <= a.Length
reads a
{
forall i,j :: 0 <= i < j < n ==> a[i] <= a[j]
}
// Use binary search to find an appropriate position to insert a value 'x'
// in a sorted array 'a', so that it remains sorted... |
255 | Metodos_Formais_tmp_tmpbez22nnn_Aula_2_ex1.dfy | method Mult(x:nat, y:nat) returns (r: nat)
ensures r == x * y
{
var m := x;
var n := y;
r:=0;
while m > 0
invariant m >= 0
invariant m*n+r == x*y
{
r := r + n;
m := m - 1;
}
return r;
}
| method Mult(x:nat, y:nat) returns (r: nat)
ensures r == x * y
{
var m := x;
var n := y;
r:=0;
while m > 0
{
r := r + n;
m := m - 1;
}
return r;
}
|
256 | Metodos_Formais_tmp_tmpbez22nnn_Aula_2_ex2.dfy | function Potencia(x: nat, y: nat): nat
{
if y == 0
then 1
else x * Potencia(x, y-1)
}
method Pot(x: nat, y: nat) returns (r: nat)
ensures r == Potencia(x,y)
{
var b := x;
var e := y;
r := 1;
while e > 0
invariant Potencia(b, e) * r == Potencia(x,y)
{
r := b * r;
e ... | function Potencia(x: nat, y: nat): nat
{
if y == 0
then 1
else x * Potencia(x, y-1)
}
method Pot(x: nat, y: nat) returns (r: nat)
ensures r == Potencia(x,y)
{
var b := x;
var e := y;
r := 1;
while e > 0
{
r := b * r;
e := e - 1;
}
return r;
}
|
257 | Metodos_Formais_tmp_tmpbez22nnn_Aula_4_ex1.dfy | predicate Par(n:int)
{
n % 2 == 0
}
method FazAlgo (a:int, b:int) returns (x:int, y:int)
requires a >= b && Par (a-b)
{
x := a;
y := b;
while x != y
invariant x >= y
invariant Par(x-y)
decreases x-y
{
x := x - 1;
y := y + 1;
}
}
| predicate Par(n:int)
{
n % 2 == 0
}
method FazAlgo (a:int, b:int) returns (x:int, y:int)
requires a >= b && Par (a-b)
{
x := a;
y := b;
while x != y
{
x := x - 1;
y := y + 1;
}
}
|
258 | Metodos_Formais_tmp_tmpbez22nnn_Aula_4_ex3.dfy | function Fib(n:nat):nat
{
if n < 2
then n
else Fib(n-2) + Fib(n-1)
}
method ComputeFib(n:nat) returns (x:nat)
ensures x == Fib(n)
{
var i := 0;
x := 0;
var y := 1;
while i < n
decreases n - i
invariant 0 <= i <= n
invariant x == Fib(i)
invariant y == Fib(i+1)
{
x... | function Fib(n:nat):nat
{
if n < 2
then n
else Fib(n-2) + Fib(n-1)
}
method ComputeFib(n:nat) returns (x:nat)
ensures x == Fib(n)
{
var i := 0;
x := 0;
var y := 1;
while i < n
{
x, y := y, x + y;
i := i + 1;
}
}
method Teste()
{
var n := 3;
var f := ComputeF... |
259 | Metodos_Formais_tmp_tmpql2hwcsh_Arrays_explicacao.dfy | // Array<T> = visualização de um array
// Uma busca ordenada em um array
// Buscar: Array<Z>xZ -> Z (Z é inteiro)
// Pré: True (pré-condição é sempre verdadeira)
// Pos: R < 0 => Para todo i pertencente aos naturais(0 <= i < A.length => A[i] != X) e
// 0 <= R < A.length => A[R] = x
//
// método em qualquer linguagem:
... | // Array<T> = visualização de um array
// Uma busca ordenada em um array
// Buscar: Array<Z>xZ -> Z (Z é inteiro)
// Pré: True (pré-condição é sempre verdadeira)
// Pos: R < 0 => Para todo i pertencente aos naturais(0 <= i < A.length => A[i] != X) e
// 0 <= R < A.length => A[R] = x
//
// método em qualquer linguagem:
... |
260 | Metodos_Formais_tmp_tmpql2hwcsh_Arrays_somatorioArray.dfy | // Deve ser criado uma função explicando o que é um somatório
// Somatorio: Array<N> -> N
// Pre: True
// Pos: Somatorio(A) = somatório de i = 0 até |A|-1 os valores das posições do array pelo i
//
// function é uma fórmula matemática, ele não possui variaveis globais
// Soma: Array<N>xN -> N
// { Soma(A,0) = A[0]
// ... | // Deve ser criado uma função explicando o que é um somatório
// Somatorio: Array<N> -> N
// Pre: True
// Pos: Somatorio(A) = somatório de i = 0 até |A|-1 os valores das posições do array pelo i
//
// function é uma fórmula matemática, ele não possui variaveis globais
// Soma: Array<N>xN -> N
// { Soma(A,0) = A[0]
// ... |
261 | Metodos_Formais_tmp_tmpql2hwcsh_Invariantes_fatorial2.dfy | function Fat(n:nat):nat
{
if n == 0 then 1 else n*Fat(n-1)
}
method Fatorial(n:nat) returns (f:nat)
ensures f == Fat(n)
{
f := 1;
var i := 1;
while i <= n
decreases n-i //variante
invariant 1 <= i <= n+1 //invariante
invariant f == Fat(i-1) //invariante
{
f := f * i;... | function Fat(n:nat):nat
{
if n == 0 then 1 else n*Fat(n-1)
}
method Fatorial(n:nat) returns (f:nat)
ensures f == Fat(n)
{
f := 1;
var i := 1;
while i <= n
{
f := f * i;
i := i + 1;
}
return f;
}
// i | n | variante
// 1 | 3 | 2
// 2 | 3 | 1
// 3 | 3 | 0
// 4 | 3 | -1
// var... |
262 | Metodos_Formais_tmp_tmpql2hwcsh_Invariantes_fibonacci.dfy | // Provando fibonacci
function Fib(n:nat):nat
{
if n < 2
then n
else Fib(n-2) + Fib(n-1)
}
method ComputeFib(n:nat) returns (x:nat)
ensures x == Fib(n)
{
var i := 0;
x := 0;
var y := 1;
while i < n
decreases n-i
invariant 0 <= i <= n
invariant x == Fib(i)
invariant y == Fib(... | // Provando fibonacci
function Fib(n:nat):nat
{
if n < 2
then n
else Fib(n-2) + Fib(n-1)
}
method ComputeFib(n:nat) returns (x:nat)
ensures x == Fib(n)
{
var i := 0;
x := 0;
var y := 1;
while i < n
{
x, y := y, x + y; //multiplas atribuições
i := i + 1;
}
}
// Fibon... |
263 | Metodos_Formais_tmp_tmpql2hwcsh_Invariantes_multiplicador.dfy | // Exemplo de invariantes
// Invariante significa que o valor não muda desde a pré-condição até a pós-condição
method Mult(x:nat, y:nat) returns (r:nat)
ensures r == x * y
{
// parâmetros de entrada são imutáveis, por isso
// é preciso a atribuir a variáveis locais para usar em blocos de códigos para mudar
... | // Exemplo de invariantes
// Invariante significa que o valor não muda desde a pré-condição até a pós-condição
method Mult(x:nat, y:nat) returns (r:nat)
ensures r == x * y
{
// parâmetros de entrada são imutáveis, por isso
// é preciso a atribuir a variáveis locais para usar em blocos de códigos para mudar
... |
264 | Metodos_Formais_tmp_tmpql2hwcsh_Invariantes_potencia.dfy | // Potência
// deve ser especificado a potência, porque ele não existe n dafny
// Função recursiva da potência
function Potencia(x:nat, y:nat):nat
{
if y == 0
then 1
else x * Potencia(x,y-1)
}
// Quero agora implementar como uma função não recursiva
method Pot(x:nat, y:nat) returns (r:nat)
ensures r == P... | // Potência
// deve ser especificado a potência, porque ele não existe n dafny
// Função recursiva da potência
function Potencia(x:nat, y:nat):nat
{
if y == 0
then 1
else x * Potencia(x,y-1)
}
// Quero agora implementar como uma função não recursiva
method Pot(x:nat, y:nat) returns (r:nat)
ensures r == P... |
265 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_extra_mod.dfy | ghost function f(n: nat): nat {
if n == 0 then 1
else if n%2 == 0 then 1 + 2*f(n/2)
else 2*f(n/2)
}
method mod(n:nat) returns (a:nat)
ensures a == f(n)
{
var x:nat := 0;
var y:nat := 1;
var k:nat := n;
while k > 0
invariant f(n) == x + y*f(k)
invariant 0 <= k <= n
decreases k... | ghost function f(n: nat): nat {
if n == 0 then 1
else if n%2 == 0 then 1 + 2*f(n/2)
else 2*f(n/2)
}
method mod(n:nat) returns (a:nat)
ensures a == f(n)
{
var x:nat := 0;
var y:nat := 1;
var k:nat := n;
while k > 0
{
if (k%2 == 0) {
x := x + y;
} else {
... |
266 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_extra_mod2.dfy |
ghost function f2(n: nat): nat {
if n == 0 then 0
else 5*f2(n/3) + n%4
}
method mod2(n:nat) returns (a:nat)
ensures a == f2(n)
{
var x:nat := 1;
var y:nat := 0;
var k:nat := n;
while k > 0
invariant f2(n) == x*f2(k) + y
invariant 0 <= k <= n
decreases k
{
assert f2(n)... |
ghost function f2(n: nat): nat {
if n == 0 then 0
else 5*f2(n/3) + n%4
}
method mod2(n:nat) returns (a:nat)
ensures a == f2(n)
{
var x:nat := 1;
var y:nat := 0;
var k:nat := n;
while k > 0
{
y := x*(k%4) + y;
x := 5*x;
k := k/3;
}
a := y;
}
|
267 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_extra_pow.dfy | ghost function pow(a: int, e: nat): int {
if e == 0 then 1 else a*pow(a, e-1)
}
method Pow(a: nat, n: nat) returns (y: nat)
ensures y == pow(a, n)
{
var x:nat := 1;
var k:nat := 0;
while k < n
invariant x == pow(a, k)
invariant 0 <= k <= n
decreases n-k
{
assert x == pow(a, k);... | ghost function pow(a: int, e: nat): int {
if e == 0 then 1 else a*pow(a, e-1)
}
method Pow(a: nat, n: nat) returns (y: nat)
ensures y == pow(a, n)
{
var x:nat := 1;
var k:nat := 0;
while k < n
{
x := a*x;
k := k + 1;
}
y := x;
}
|
268 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_extra_sum.dfy |
ghost function sum(n: nat): int
{
if n == 0 then 0 else n + sum(n - 1)
}
method Sum(n: nat) returns (s: int)
ensures s == sum(n)
{
var x:nat := 0;
var y:nat := 1;
var k:nat := n;
while k > 0
invariant sum(n) == x + y*sum(k)
invariant 0 <= k <= n
decreases k
{
assert sum(n... |
ghost function sum(n: nat): int
{
if n == 0 then 0 else n + sum(n - 1)
}
method Sum(n: nat) returns (s: int)
ensures s == sum(n)
{
var x:nat := 0;
var y:nat := 1;
var k:nat := n;
while k > 0
{
x := x + y*k;
k := k-1;
}
s := x;
}
|
269 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_mockExam2_p2.dfy | // problem 2:
// name: Gabriele Berardi
// s-number: s4878728
// table: XXX
method problem2(p:int, q:int, X:int, Y:int) returns (r:int, s:int)
requires p == 2*X + Y && q == X + 3
ensures r == X && s == Y
{
assert p == 2*X + Y && q == X + 3;
r, s := p, q;
assert r == 2*X + Y && s == X + 3;
r :=... | // problem 2:
// name: Gabriele Berardi
// s-number: s4878728
// table: XXX
method problem2(p:int, q:int, X:int, Y:int) returns (r:int, s:int)
requires p == 2*X + Y && q == X + 3
ensures r == X && s == Y
{
r, s := p, q;
r := r - 2*s + 6;
s := s - 3;
r,s := s, r;
}
|
270 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_mockExam2_p3.dfy | // problem 3:
// name: ....... (fill in your name)
// s-number: s....... (fill in your student number)
// table: ....... (fill in your table number)
method problem3(m:int, X:int) returns (r:int)
requires X >= 0 && (2*m == 1 - X || m == X + 3)
ensures r == X
{
assert X >= 0 && (X == 1 - 2*m || m-3 == X);
... | // problem 3:
// name: ....... (fill in your name)
// s-number: s....... (fill in your student number)
// table: ....... (fill in your table number)
method problem3(m:int, X:int) returns (r:int)
requires X >= 0 && (2*m == 1 - X || m == X + 3)
ensures r == X
{
r := m;
if (1-2*r >= 0) {
r := 2*... |
271 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_mockExam2_p5.dfy | // problem 5:
// name: Gabriele Berardi
// s-number: s4878728
// table: XXXX
ghost function f(n: int): int {
if n < 0 then 0 else 3*f(n-5) + n
}
method problem5(n:nat) returns (x: int)
ensures x == f(n)
{
var a := 1;
var b := 0;
var k := n;
while k >= 0
invariant f(n) == a*f(k) + b
... | // problem 5:
// name: Gabriele Berardi
// s-number: s4878728
// table: XXXX
ghost function f(n: int): int {
if n < 0 then 0 else 3*f(n-5) + n
}
method problem5(n:nat) returns (x: int)
ensures x == f(n)
{
var a := 1;
var b := 0;
var k := n;
while k >= 0
{
b := a*k + b;
... |
272 | Prog-Fun-Solutions_tmp_tmp7_gmnz5f_mockExam2_p6.dfy | // problem 6:
// name: Gabriele Berardi
// s-number: s4878728
// table: XXXXX
ghost function f(n: int): int {
if n <= 0 then 1 else n + f(n-1)*f(n-2)
}
ghost function fSum(n: nat): int {
// give the body of this function
// it should return Sum(i: 0<=i < n: f(i))
if n <= 0 then 0 else f(n-1) + fSum(n-... | // problem 6:
// name: Gabriele Berardi
// s-number: s4878728
// table: XXXXX
ghost function f(n: int): int {
if n <= 0 then 1 else n + f(n-1)*f(n-2)
}
ghost function fSum(n: nat): int {
// give the body of this function
// it should return Sum(i: 0<=i < n: f(i))
if n <= 0 then 0 else f(n-1) + fSum(n-... |
273 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_ArrayMap.dfy | // RUN: /print:log.bpl
method ArrayMap<A>(f: int -> A, a: array<A>)
requires a != null
requires forall j :: 0 <= j < a.Length ==> f.requires(j)
requires forall j :: 0 <= j < a.Length ==> a !in f.reads(j)
modifies a
ensures forall j :: 0 <= j < a.Length ==> a[j] == f(j)
{
var i := 0;
while i < a.Length
... | // RUN: /print:log.bpl
method ArrayMap<A>(f: int -> A, a: array<A>)
requires a != null
requires forall j :: 0 <= j < a.Length ==> f.requires(j)
requires forall j :: 0 <= j < a.Length ==> a !in f.reads(j)
modifies a
ensures forall j :: 0 <= j < a.Length ==> a[j] == f(j)
{
var i := 0;
while i < a.Length
... |
274 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_EvenPredicate.dfy | // RUN: /compile:0 /nologo
function IsEven(a : int) : bool
requires a >= 0
{
if a == 0 then true
else if a == 1 then false
else IsEven(a - 2)
}
lemma EvenSquare(a : int)
requires a >= 0
ensures IsEven(a) ==> IsEven(a * a)
{
if a >= 2 && IsEven(a) {
EvenSquare(a - 2);
... | // RUN: /compile:0 /nologo
function IsEven(a : int) : bool
requires a >= 0
{
if a == 0 then true
else if a == 1 then false
else IsEven(a - 2)
}
lemma EvenSquare(a : int)
requires a >= 0
ensures IsEven(a) ==> IsEven(a * a)
{
if a >= 2 && IsEven(a) {
EvenSquare(a - 2);
... |
275 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_GenericMax.dfy | method GenericMax<A>(cmp: (A, A) -> bool, a: array<A>) returns (max: A)
requires a != null && a.Length > 0
requires forall x, y :: cmp.requires(x, y)
requires forall x, y :: cmp(x, y) || cmp(y, x);
requires forall x, y, z :: cmp(x, y) && cmp(y, z) ==> cmp(x, z);
ensures forall x :: 0 <= x < a.Length ==>
... | method GenericMax<A>(cmp: (A, A) -> bool, a: array<A>) returns (max: A)
requires a != null && a.Length > 0
requires forall x, y :: cmp.requires(x, y)
requires forall x, y :: cmp(x, y) || cmp(y, x);
requires forall x, y, z :: cmp(x, y) && cmp(y, z) ==> cmp(x, z);
ensures forall x :: 0 <= x < a.Length ==>
... |
276 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_InsertionSort.dfy | predicate sorted (a:array<int>, start:int, end:int) // all "before" end are sorted
requires a!=null
requires 0<=start<=end<=a.Length
reads a
{
forall j,k:: start<=j<k<end ==> a[j]<=a[k]
}
method InsertionSort (a:array<int>)
requires a!=null && a.Length>1
ensures sorted(... | predicate sorted (a:array<int>, start:int, end:int) // all "before" end are sorted
requires a!=null
requires 0<=start<=end<=a.Length
reads a
{
forall j,k:: start<=j<k<end ==> a[j]<=a[k]
}
method InsertionSort (a:array<int>)
requires a!=null && a.Length>1
ensures sorted(... |
277 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_MatrixMultiplication.dfy | function RowColumnProduct(m1: array2<int>, m2: array2<int>, row: nat, column: nat): int
reads m1
reads m2
requires m1 != null && m2 != null && m1.Length1 == m2.Length0
requires row < m1.Length0 && column < m2.Length1
{
RowColumnProductFrom(m1, m2, row, column, 0)
}
function RowColumnProductFrom(m1:... | function RowColumnProduct(m1: array2<int>, m2: array2<int>, row: nat, column: nat): int
reads m1
reads m2
requires m1 != null && m2 != null && m1.Length1 == m2.Length0
requires row < m1.Length0 && column < m2.Length1
{
RowColumnProductFrom(m1, m2, row, column, 0)
}
function RowColumnProductFrom(m1:... |
278 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_Modules.dfy | // RUN: /compile:1
abstract module Interface {
type T
function F(): T
predicate P(x: T)
lemma FP()
ensures P(F())
}
module Implementation refines Interface {
predicate P(x: T) {
false
}
}
abstract module User {
import I : Interface
lemma Main()
ensures I.P(I.... | // RUN: /compile:1
abstract module Interface {
type T
function F(): T
predicate P(x: T)
lemma FP()
ensures P(F())
}
module Implementation refines Interface {
predicate P(x: T) {
false
}
}
abstract module User {
import I : Interface
lemma Main()
ensures I.P(I.... |
279 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_OneHundredPrisonersAndALightbulb.dfy | // RUN: /compile:0 /nologo
method CardinalitySubsetLt<T>(A: set<T>, B: set<T>)
requires A < B
ensures |A| < |B|
decreases B
{
var b :| b in B && b !in A;
var B' := B - {b};
assert |B| == |B'| + 1;
if A < B' {
CardinalitySubsetLt(A, B');
} else {
assert A == B';
}
}
method strategy<T>(P: set<... | // RUN: /compile:0 /nologo
method CardinalitySubsetLt<T>(A: set<T>, B: set<T>)
requires A < B
ensures |A| < |B|
{
var b :| b in B && b !in A;
var B' := B - {b};
if A < B' {
CardinalitySubsetLt(A, B');
} else {
}
}
method strategy<T>(P: set<T>, Special: T) returns (count: int)
requires |P| > 1 && S... |
280 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_Percentile.dfy | // Sum of elements of A from indices 0 to end.
// end is inclusive! (not James's normal way of thinking!!)
function SumUpto(A: array<real>, end: int): real
requires -1 <= end < A.Length
reads A
{
if end == -1 then
0.0
else
A[end] + SumUpto(A, end-1)
}
function Sum(A: array<real>): real
reads A
{
Su... | // Sum of elements of A from indices 0 to end.
// end is inclusive! (not James's normal way of thinking!!)
function SumUpto(A: array<real>, end: int): real
requires -1 <= end < A.Length
reads A
{
if end == -1 then
0.0
else
A[end] + SumUpto(A, end-1)
}
function Sum(A: array<real>): real
reads A
{
Su... |
281 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_Refinement.dfy | // RUN: /nologo /rlimit:10000000 /noNLarith
abstract module Interface {
function addSome(n: nat): nat
ensures addSome(n) > n
}
abstract module Mod {
import A : Interface
method m() {
assert 6 <= A.addSome(5);
print "Test\n";
}
}
module Implementation refines Interface {
fu... | // RUN: /nologo /rlimit:10000000 /noNLarith
abstract module Interface {
function addSome(n: nat): nat
ensures addSome(n) > n
}
abstract module Mod {
import A : Interface
method m() {
print "Test\n";
}
}
module Implementation refines Interface {
function addSome(n: nat): nat
... |
282 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_ReverseString.dfy | // RUN: /compile:0
predicate reversed (arr : array<char>, outarr: array<char>)
requires arr != null && outarr != null
//requires 0<=k<=arr.Length-1
requires arr.Length == outarr.Length
reads arr, outarr
{
forall k :: 0<=k<=arr.Length-1 ==> outarr[k] == arr[(arr.Length-1-k)]
}
method yarra(arr : array<char>) returns... | // RUN: /compile:0
predicate reversed (arr : array<char>, outarr: array<char>)
requires arr != null && outarr != null
//requires 0<=k<=arr.Length-1
requires arr.Length == outarr.Length
reads arr, outarr
{
forall k :: 0<=k<=arr.Length-1 ==> outarr[k] == arr[(arr.Length-1-k)]
}
method yarra(arr : array<char>) returns... |
283 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_advanced examples_demo.dfy | method Partition(a: array<int>) returns (lo: int, hi: int)
modifies a
ensures 0 <= lo <= hi <= a.Length
ensures forall x | 0 <= x < lo :: a[x] < 0
ensures forall x | lo <= x < hi :: a[x] == 0
ensures forall x | hi <= x < a.Length :: a[x] > 0
{
var i := 0;
var j := a.Length;
var k := a.Length;
while i... | method Partition(a: array<int>) returns (lo: int, hi: int)
modifies a
ensures 0 <= lo <= hi <= a.Length
ensures forall x | 0 <= x < lo :: a[x] < 0
ensures forall x | lo <= x < hi :: a[x] == 0
ensures forall x | hi <= x < a.Length :: a[x] > 0
{
var i := 0;
var j := a.Length;
var k := a.Length;
while i... |
284 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_ProgramProofs_ch15.dfy | predicate SplitPoint(a: array<int>, n: int)
reads a
requires 0 <= n <= n
{
forall i,j :: 0 <= i < n <= j < a.Length ==> a[i] <= a[j]
}
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[..]))
{
... | predicate SplitPoint(a: array<int>, n: int)
reads a
requires 0 <= n <= n
{
forall i,j :: 0 <= i < n <= j < a.Length ==> a[i] <= a[j]
}
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[..]))
{
... |
285 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_examples_bubblesort.dfy | //https://stackoverflow.com/questions/69364687/how-to-prove-time-complexity-of-bubble-sort-using-dafny
function NChoose2(n: int): int
{
n * (n - 1) / 2
}
// sum of all integers in the range [lo, hi)
// (inclusive of lo, exclusive of hi)
function SumRange(lo: int, hi: int): int
decreases hi - lo
{
if lo >= hi the... | //https://stackoverflow.com/questions/69364687/how-to-prove-time-complexity-of-bubble-sort-using-dafny
function NChoose2(n: int): int
{
n * (n - 1) / 2
}
// sum of all integers in the range [lo, hi)
// (inclusive of lo, exclusive of hi)
function SumRange(lo: int, hi: int): int
{
if lo >= hi then 0
else SumRange(... |
286 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_examples_relativeOrder.dfy |
predicate IsEven (n: int)
{
n % 2 == 0
}
method FindEvenNumbers (arr: array<int>)
returns (evenNumbers: array<int>)
ensures forall x :: x in arr[..] && IsEven(x) ==> x in evenNumbers[..];
ensures forall x :: x !in arr[..] ==> x !in evenNumbers[..]
ensures forall k, l :: 0 <= k < l < evenNumbers.Length ==>... |
predicate IsEven (n: int)
{
n % 2 == 0
}
method FindEvenNumbers (arr: array<int>)
returns (evenNumbers: array<int>)
ensures forall x :: x in arr[..] && IsEven(x) ==> x in evenNumbers[..];
ensures forall x :: x !in arr[..] ==> x !in evenNumbers[..]
ensures forall k, l :: 0 <= k < l < evenNumbers.Length ==>... |
287 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_examples_simpleMultiplication.dfy |
method Foo(y: int, x: int) returns (z: int)
requires 0 <= y
ensures z == x*y
{
var a: int := 0;
z := 0;
while a != y
invariant 0 <= a <= y
invariant z == a*x
decreases y-a
{
z := z + x;
a := a + 1;
}
return z;
}
function stringToSet(s: string): (r: set<char>)
ensures forall x :: 0 <... |
method Foo(y: int, x: int) returns (z: int)
requires 0 <= y
ensures z == x*y
{
var a: int := 0;
z := 0;
while a != y
{
z := z + x;
a := a + 1;
}
return z;
}
function stringToSet(s: string): (r: set<char>)
ensures forall x :: 0 <= x < |s| ==> s[x] in r
{
set x | 0 <= x < |s| :: s[x]
}
//ensu... |
288 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_heap2.dfy | class Heap {
var arr: array<int>
constructor Heap (input: array<int>)
ensures this.arr == input {
this.arr := input;
}
function parent(idx: int): int
{
if idx == 0 then -1
else if idx % 2 == 0 then (idx-2)/2
else (idx-1)/2
}
predicate IsMaxHeap(input: seq<int>)
{
forall i :: 0... | class Heap {
var arr: array<int>
constructor Heap (input: array<int>)
ensures this.arr == input {
this.arr := input;
}
function parent(idx: int): int
{
if idx == 0 then -1
else if idx % 2 == 0 then (idx-2)/2
else (idx-1)/2
}
predicate IsMaxHeap(input: seq<int>)
{
forall i :: 0... |
289 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_BoatsToSavePeople.dfy | /*
function numRescueBoats(people: number[], limit: number): number {
//people.sort((a,b) => a-b);
binsort(people, limit);
let boats = 0;
let lower = 0, upper = people.length-1;
while(lower <= upper) {
if(people[upper] == limit || people[upper]+people[lower] > limit) {
boats++
... | /*
function numRescueBoats(people: number[], limit: number): number {
//people.sort((a,b) => a-b);
binsort(people, limit);
let boats = 0;
let lower = 0, upper = people.length-1;
while(lower <= upper) {
if(people[upper] == limit || people[upper]+people[lower] > limit) {
boats++
... |
290 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_FindPivotIndex.dfy | /*
https://leetcode.com/problems/find-pivot-index/description/
Given an array of integers nums, calculate the pivot index of this array.
The pivot index is the index where the sum of all the numbers strictly to the left of the index is equal to the sum of all the numbers strictly to the index's right.
If the index is... | /*
https://leetcode.com/problems/find-pivot-index/description/
Given an array of integers nums, calculate the pivot index of this array.
The pivot index is the index where the sum of all the numbers strictly to the left of the index is equal to the sum of all the numbers strictly to the index's right.
If the index is... |
291 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_ReverseLinkedList.dfy | /*
https://leetcode.com/problems/reverse-linked-list/description/
* class ListNode {
* val: number
* next: ListNode | null
* constructor(val?: number, next?: ListNode | null) {
* this.val = (val===undefined ? 0 : val)
* this.next = (next===undefined ? null : next)
* }
* }
functi... | /*
https://leetcode.com/problems/reverse-linked-list/description/
* class ListNode {
* val: number
* next: ListNode | null
* constructor(val?: number, next?: ListNode | null) {
* this.val = (val===undefined ? 0 : val)
* this.next = (next===undefined ? null : next)
* }
* }
functi... |
292 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_lc-remove-element.dfy | //https://leetcode.com/problems/remove-element/
method removeElement(nums: array<int>, val: int) returns (i: int)
ensures forall k :: 0 < k < i < nums.Length ==> nums[k] != val
modifies nums
{
i := 0;
var end := nums.Length - 1;
while i <= end
invariant 0 <= i <= nums.Length
invaria... | //https://leetcode.com/problems/remove-element/
method removeElement(nums: array<int>, val: int) returns (i: int)
ensures forall k :: 0 < k < i < nums.Length ==> nums[k] != val
modifies nums
{
i := 0;
var end := nums.Length - 1;
while i <= end
{
if(nums[i] == val) {
if(nums[... |
293 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_pathSum.dfy | //https://leetcode.com/problems/path-sum
/**
function hasPathSum(root: TreeNode | null, targetSum: number): boolean {
if(root == null) {
return false;
}
if(root.val-targetSum == 0 && root.left == null && root.right == null) {
return true;
}
return hasPathSum(root.left, targetSum-root... | //https://leetcode.com/problems/path-sum
/**
function hasPathSum(root: TreeNode | null, targetSum: number): boolean {
if(root == null) {
return false;
}
if(root.val-targetSum == 0 && root.left == null && root.right == null) {
return true;
}
return hasPathSum(root.left, targetSum-root... |
294 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_stairClimbing.dfy | /*
You are climbing a staircase. It takes n steps to reach the top.
Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?
function climbStairs(n: number): number {
let steps = new Array(n+1);
steps[0] = 0;
steps[1] = 1;
steps[2] = 2;
for(let i = 3; i <= n;... | /*
You are climbing a staircase. It takes n steps to reach the top.
Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?
function climbStairs(n: number): number {
let steps = new Array(n+1);
steps[0] = 0;
steps[1] = 1;
steps[2] = 2;
for(let i = 3; i <= n;... |
295 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_leetcode_validAnagram.dfy |
method toMultiset(s: string) returns (mset: multiset<char>)
ensures multiset(s) == mset
{
mset := multiset{};
for i := 0 to |s|
invariant mset == multiset(s[0..i])
{
assert s == s[0..i] + [s[i]] + s[(i+1)..];
// assert multiset(s) == multiset(s[0..i])+multiset{s[i]}+multiset(s[... |
method toMultiset(s: string) returns (mset: multiset<char>)
ensures multiset(s) == mset
{
mset := multiset{};
for i := 0 to |s|
{
// assert multiset(s) == multiset(s[0..i])+multiset{s[i]}+multiset(s[(i+1)..]);
mset := mset + multiset{s[i]};
}
// assert mset == multiset(s[0..|s|... |
296 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_lib_seq.dfy |
module Seq {
export reveals *
function ToSet<A>(xs: seq<A>): set<A>
ensures forall x :: x in ToSet(xs) ==> x in xs
ensures forall x :: x !in ToSet(xs) ==> x !in xs
{
if xs == [] then {} else {xs[0]}+ToSet(xs[1..])
}
predicate substring1<A(==)>(sub: seq<A>, super: seq<A>) {
... |
module Seq {
export reveals *
function ToSet<A>(xs: seq<A>): set<A>
ensures forall x :: x in ToSet(xs) ==> x in xs
ensures forall x :: x !in ToSet(xs) ==> x !in xs
{
if xs == [] then {} else {xs[0]}+ToSet(xs[1..])
}
predicate substring1<A(==)>(sub: seq<A>, super: seq<A>) {
... |
297 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_algorithms and leetcode_math_pearson.dfy |
function eight(x: nat):nat {
9 * x + 5
}
predicate isOdd(x: nat) {
x % 2 == 1
}
predicate isEven(x: nat) {
x % 2 == 0
}
lemma eightL(x: nat)
requires isOdd(x)
ensures isEven(eight(x))
{
}
function nineteenf(x: nat): nat {
7*x+4
}
function nineteens(x: nat): nat {
3*x+11
}
lemma ninet... |
function eight(x: nat):nat {
9 * x + 5
}
predicate isOdd(x: nat) {
x % 2 == 1
}
predicate isEven(x: nat) {
x % 2 == 0
}
lemma eightL(x: nat)
requires isOdd(x)
ensures isEven(eight(x))
{
}
function nineteenf(x: nat): nat {
7*x+4
}
function nineteens(x: nat): nat {
3*x+11
}
lemma ninet... |
298 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_basic examples_BubbleSort.dfy | predicate sorted(a:array<int>, from:int, to:int)
requires a != null;
reads a;
requires 0 <= from <= to <= a.Length;
{
forall u, v :: from <= u < v < to ==> a[u] <= a[v]
}
predicate pivot(a:array<int>, to:int, pvt:int)
requires a != null;
reads a;
requires 0 <= pvt < to <= a.Length;
{
forall u, v :: 0 <... | predicate sorted(a:array<int>, from:int, to:int)
requires a != null;
reads a;
requires 0 <= from <= to <= a.Length;
{
forall u, v :: from <= u < v < to ==> a[u] <= a[v]
}
predicate pivot(a:array<int>, to:int, pvt:int)
requires a != null;
reads a;
requires 0 <= pvt < to <= a.Length;
{
forall u, v :: 0 <... |
299 | Program-Verification-Dataset_tmp_tmpgbdrlnu__Dafny_basic examples_BubbleSort_sol.dfy | predicate sorted_between (a:array<int>, from:nat, to:nat)
reads a;
requires a != null;
requires from <= to;
requires to <= a.Length;
{
forall i,j :: from <= i < j < to && 0 <= i < j < a.Length ==> a[i] <= a[j]
}
predicate sorted (a:array<int>)
reads a;
requires a!=null;
{
sorted_between (a, 0, a.Leng... | predicate sorted_between (a:array<int>, from:nat, to:nat)
reads a;
requires a != null;
requires from <= to;
requires to <= a.Length;
{
forall i,j :: from <= i < j < to && 0 <= i < j < a.Length ==> a[i] <= a[j]
}
predicate sorted (a:array<int>)
reads a;
requires a!=null;
{
sorted_between (a, 0, a.Leng... |
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