Give a useful precondition, postcondition, loop invariant, and variant in Dafny.

Question

Give a useful precondition, postcondition, loop invariant, and variant in Dafny.

method f(x0: int,y: int) returns (r: int)
requires true
ensures true
{
var x := x0;
var d := 0;
while x < y
invariant true
{
x := x + 1;
d := d + 1;
}
return d;
}

method g(x: int) returns (r: int)
requires true
ensures true
{
var y := 0;
var i := 0;
while i < x
invariant true
{
y := y + 2;
i := i + 1;
}
return y;
}

method h(x0: int, y0: int) returns (r: int)
requires true
ensures true
{
var x := x0;
var y := y0;
while x > 0
invariant true
{
x := x - 2;
y := y + 1;
}
return y;
}

Explanation / Answer

method f(x0: int,y: int) returns (r: int)
requires true
ensures true
{
var x := x0;
var d := 0;
while x < y
invariant true
{
x := x + 1;
d := d + 1;
}
return d;
}

method g(x: int) returns (r: int)
requires true
ensures true
{
var y := 0;
var i := 0;
while i < x
invariant true
{
y := y + 2;
i := i + 1;
}
return y;
}

method h(x0: int, y0: int) returns (r: int)
requires true
ensures true
{
var x := x0;
var y := y0;
while x > 0
invariant true
{
x := x - 2;
y := y + 1;
}
return y;
}

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