Question

2. (15 marks) Consider the following program: >> Precondition: x and y E Z. Postcondition: Return the sum x + y. add(x, y): 1. if x == 0: 2. return y 3. elif x > 0: 4. return add(x - 1, y) + 1 5. else: 6. return add(x + 1, y) - 1 Prove that this program is correct in terms of its specification.

Answer 1

add (x,y): ly return y 3) diy wo: return add (x-1, y ) + 1 3) else return add (2+1, y) - 1 This is a program for addition of two numbers using recursion Here based on value of , add function is called using different parameters until x = 0 (Ternmiating condition and starts calculation from there a As the functions are called they are there return values are stored in stark When Terminating condition is reached, stack is poped by 2 the and return value to Top da caller - Sumlish

example 1 , 4 = 3 add (2, 3) return 3 I add(0, 3) tik return 4 Ladd (1,3)+1 4+1 = 5 stuck at last will contain the result and again, 1) add (2, 3) 2 0 so return add (2-1,3)+1 to add (1,3)+1 stored in stack radd (1,3) is called 2) add (1, 3) x70 to return add (1-1,3)+1 so add (0, 3) to stored in stack and called sidd 10,3) add (0,3) is Now That is which Now x==0 so return 3 it will return 3 to its is add 60,3) parent coller function add (0, 3) =3 get added by a Now It will Top of ow top of stack - add (0,3)+1 stack = add op og stack = 3+1 = 4 with return

4 is returned to its caller function that is add (1,3) Now It is Now so add (1, 3) = 4 Top of Stack - 4+1 = 5 the last uadue in stark and last value of Stack - result .. so add (2,3) = 5 it is the result enample 2 u=-2, 4-3 add (-2,3) n=0 return 3 add lo , 3) -1 3-1 add (1,3)- e return 3-1 - 3-1-1 = 1 add (-2,3) nho so return It will get stored add (-2+1,3)-1 in stack.

F 2) add (-1, 3) is called Te As a Lo so return It will get stored in stark add (-1+1, 3) -1 =) add (0,3)-1 3 add (0,3) is called Now x = 0 to return 3 that is add (0,3) to its caller function, Now add (0, 3) = 3 4) Top of stack = add 60, 3) - 1 = 3-1=2 return 2 to caller Junction that is add (-1,3) s) Tof of stack = add (-1,3) - 1 - - 2-1 -1 Lest nature Last value in stack & result = 1 add (-2, 3) = 1