Question

To prove the correctness of the following:

# pre: int >= 1

fac = 1
i = n
while i>0:
     fac = fac*i
     i = i-1

# post: fac = n!

which loop invariant do we need?

(Hint, you will need two statements that are both true. One about fac and one about i)

2. To prove correctness of the following: # pre: int n>=1 fac 1 while i>0: fac = fac치 i = i-1 # post: fac n! which loop invariant do we need? (Hint, you will need two statements that are both true. One about fac and one about i )

Please provide both full statements as well as which loop invariant is needed in a full proof.

Answer 1

# pre: int >= 1

fac = 1
i = n
while i>0:
fac = fac*i
i = i-1

# post: fac = n!

statement 1:loop invariant would be fac*(i!) = n! where n is the initial value of i that is passed to the function. This statement is always true, no matter how many iterations of the loop have already been done.

statement 2: in each iteration value of i reduced by 1 till it reaches 0
proof:
Precondition :
i = n ,fac=1
fac*(i!) = 1*(n!) = n!

Maintenance :
in each iteration fac = fac*i
and i=i-1;
(fac*i)*((i-1)!) = n!

Termination :
fac = 1*2*3*4*.........*(n-1)*n = n!