sliding window minimum - mock interview and discussion

sliding window minimum

given an integer array, give range size k, find minimum value in each range. 

for example, integer array [1, 2, 3, 4, 5]
k = 3,
min[0] = min{1, 2, 3} = 1
min[1] = min{2, 3, 4} = 2
min[2] = min{3, 4, 5} = 3

The interviewee gave the proposal of solution using O(N) time complexity, preproces the array by two scans, one
is from left to right, second one is from right to left. 

[1,2,3 | 4,5]
                      leftToRight     [1,1,1 | 4,4]
                      rightToLeft     [1,2,3 | 4,5]
                                      [1,2,
                                      [
     [43, 21, 34,    22, 12]
lToR [43, 21, 21,||| 22, 12]
rToL [21, 21, 34     12, 12]
     [
     
       1000 elements, k = 3
       1, 2, 3 |||  4, 5, 6 ||| ....999 ||| 1000
lToR - 1, 1, 1 |||  4, 4, 4 |||       
RtoL - 1   2   3  |||     4   5   6  |||        999 |||       1000   
       
rangeSum[i] = min(righToLeft[i], lefttoRight[i+k-1]);

       i=0, min(1, 1)
       i=1, min(2, 4) =2
       i=2, min(3,4) = 3
       
So Julia as an interviewer tried very hard to understand the algorithm. After more than 3 minutes discussion, 
Julia said that it is better to prove that it is correct as well. 

Julia tried to give a proof about the correctness of the algorithm, since she has math major. 
She likes to prove the algorithm approach will work. 

Given an integer array, given size of k = 3, find minimum number in each sliding window. 

For example, k = 3, there are three possible values. 
i%3 == 0
       0, 1, 2
       
       case 0:
       first test case, rightToLeft[i] cover minimum number of 3 numbers
       
       case 1:
       rightToLeft, rightToLeft[i] covers two numbers
         leftToright -> i + k - 1, next window, first element, 
       cover first one
       in total we have k numbers covers - 3 
       by math, in total there are 3 numbers
       
       case 2:
       rightToLeft, one number
       LeftToright, two number
       in total 3 numbers