# Need Solution Of Question 2 And 3 Question 2 Doubling Numbers 2835507

Need solution of question 2 and 3
Question 2 Doubling Numbers
The following piece of code is called Half:
x := 0;
y := 0;
while (x
x := x + 2;
y := y + 1;
We wish to use Hoare Logic to show that:
{True} Half {x = 2 * y}
In the questions below (and your answers), we may refer to the loop code as Loop, the body of the
loop (i.e. x:=x+2;y:=y+1;) as Body, and the initialisation assignments (i.e. x:=0;y:=0;) as Init.
1. Given the desired postcondition {x = 2 * y}, what is a suitable invariant for Loop?
(Hint: notice that the postcondition is independent of the value of a.)
2. Prove that your answer to the previous question is indeed a loop invariant. That is, if we call
your invariant P, show that {P} Body {P}. Be sure to properly justify each step of your proof.
3. Using the previous result and some more proof steps show that
{True} Half {x = 2 * y}
Be sure to properly justify each step of your proof.
4. To prove total correctness of the program Half, identify and state a suitable variant for the loop.
Using the same invariant P as above, the variant E should have the following two properties:
• it should be = 0 when the loop is entered, i.e. P ? (x
• it should decrease every time the loop body is executed, i.e. [P ?(x
E
1
You just need to state the variant, and do not need to prove the two bullet points above (yet).
5. For the variant E you have identified above, give a proof of the premise of the while-rule for
total correctness, i.e. give a Hoare-logic proof of [P ? (x
argue that P ? (x
Question 3 Counting Modulo 7 [5 + 20 + 10 + 5 credits]
Consider the following code fragment that we refer to as Count below, and we refer to the body of the
loop (i.e. the two assignments together with the if-statement) as Body.
while (y
y := y + 1;
x := x + 1;
if (x = 7) then x := 0 else x := x
The goal of the exercise is to show that {x
1. Given the desired postcondition, what is a suitable invariant P for the loop? You just need to
state the invariant.
2. Give a Hoare Logic proof of the fact that your invariant above is indeed an invariant, i.e. prove
the Hoare-triple {P}Body{P}.
3. Hence, or otherwise, give a Hoare-logic proof of the triple {x
4. Give an example of a precondition P so that the Hoare-triple {P}Count{x

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