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Jul 21st, 2018
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  1. { x = n , n > 0 }
  2. { x = n , n > 0 , 0 = 0 }
  3. z := 0;
  4. { x = n , n > z , z = 0 }
  5. { x = n , n > z , z = 0 , 1 = 1 }
  6. odd := 1;
  7. { x = n , n > z , z = 0 , odd = 1 }
  8. { x = n , n > z , z = 0 , odd = 1 , 1 = 1}
  9. sum := 1;
  10. { x = n , n > z , z = 0 , odd = sum , sum = 1 }
  11. { x = n , n > z , z = 0 , odd = sum - 0 , sum = 1 }
  12. { x = n , n > z , z = 0 , odd = sum - (z ^ 2) , sum = 1 }
  13. { x = n , n > z , z = 0 , sum - odd = (z ^ 2) , sum = (z + 1) ^ 2 , sum <= x } = INV
  14. while sum <= x
  15. { sum <= x ^ INV }
  16. z := z + 1;
  17. odd := odd + 2;
  18. sum := sum + odd;
  19. { ¬(sum <= x) , INV }
  20. { sum > x , INV }
  21.  
  22. { }
  23. { (z ^ 2) <= n , (z + 1) ^ 2 > n }
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