Please show your steps and reasoning on how you get to your answers:

Prove the following loop invariants about the CountingSort algorithm, below Note that you do not need to prove that this algorithm is correct. Input: data: array of n integers that are between 1 and r Input: n: size of data Input: r: range of data Output: permutation of data such that data[1] data[2] . . . data[n] 1 Algorithm: CountingSort 2 count - Array(r) 3 Initialize count to 0 4 for i -1 to n do 5count[data[i|] - count[data[i] +1 6 end for j 2 to r do count]countcountlj - 1] 9 end 10 output - Array(n 11 for i - 1 to n do 12output[count[data[i|] - data[i] 13count[data[icount[datafi] -1 14 end 15 return output 1. Prove that countlj] equals the number of times j appears in data[1..i], for every j from 1 up to r, after each iteration of the for loop in lines 4-6 2. Prove that count jequals the number of values in data that are less than or equal to j after every iteration of the for loop in lines 7-9 Hint: first, prove that count|j] is the number of times j appears in data after the loop in lines 4-6