




def 	_a      (						SCREAMING_SNAKE_CASE_							:				list[int]      ,      SCREAMING_SNAKE_CASE_							:				int     ):
					__lowerCAmelCase				   =       len(SCREAMING_SNAKE_CASE_     )
					__lowerCAmelCase				   =       [[False] * (required_sum + 1) for _ in range(arr_len + 1     )]

					# for each arr value, a sum of zero(0) can be formed by not taking any element
					# hence True/1
					for i in range(arr_len + 1     ):
										__lowerCAmelCase				   =       True

					# sum is not zero and set is empty then false
					for i in range(1      ,      required_sum + 1     ):
										__lowerCAmelCase				   =       False

					for i in range(1      ,      arr_len + 1     ):
										for j in range(1      ,      required_sum + 1     ):
															if arr[i - 1] > j:
																				__lowerCAmelCase				   =       subset[i - 1][j]
															if arr[i - 1] <= j:
																				__lowerCAmelCase				   =       subset[i - 1][j] or subset[i - 1][j - arr[i - 1]]

					return subset[arr_len][required_sum]


if __name__ == "__main__":
					import doctest

					doctest.testmod()





"""simple docstring"""






def SCREAMING_SNAKE_CASE__			(   __UpperCAmelCase       ,    __UpperCAmelCase )  ->  List[str]:
   lowercase__:   Optional[Any]					=      0
   lowercase__:   List[Any]					=      len(__UpperCAmelCase ) - 1

   while left <= right:
      # avoid divided by 0 during interpolation
      if sorted_collection[left] == sorted_collection[right]:
         if sorted_collection[left] == item:
            return left
         else:
            return None

      lowercase__:   Tuple					=      left + ((item - sorted_collection[left]) * (right - left)) // (
          sorted_collection[right] - sorted_collection[left]
      )

      # out of range check
      if point < 0 or point >= len(__UpperCAmelCase ):
         return None

      lowercase__:   Optional[int]					=      sorted_collection[point]
      if current_item == item:
         return point
      else:
         if point < left:
            lowercase__:   List[Any]					=      left
            lowercase__:   int					=      point
         elif point > right:
            lowercase__:   Dict					=      right
            lowercase__:   List[str]					=      point
         else:
            if item < current_item:
               lowercase__:   int					=      point - 1
            else:
               lowercase__:   int					=      point + 1
   return None



def SCREAMING_SNAKE_CASE__			(   __UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase )  ->  List[str]:

   # avoid divided by 0 during interpolation
   if sorted_collection[left] == sorted_collection[right]:
      if sorted_collection[left] == item:
         return left
      else:
         return None

   lowercase__:   Optional[int]					=      left + ((item - sorted_collection[left]) * (right - left)) // (
       sorted_collection[right] - sorted_collection[left]
   )

   # out of range check
   if point < 0 or point >= len(__UpperCAmelCase ):
      return None

   if sorted_collection[point] == item:
      return point
   elif point < left:
      return interpolation_search_by_recursion(__UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase )
   elif point > right:
      return interpolation_search_by_recursion(__UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase )
   else:
      if sorted_collection[point] > item:
         return interpolation_search_by_recursion(
             __UpperCAmelCase       ,    __UpperCAmelCase       ,    __UpperCAmelCase       ,    point - 1 )
      else:
         return interpolation_search_by_recursion(
             __UpperCAmelCase       ,    __UpperCAmelCase       ,    point + 1       ,    __UpperCAmelCase )



def SCREAMING_SNAKE_CASE__			(   __UpperCAmelCase )  ->  List[Any]:
   if collection != sorted(__UpperCAmelCase ):
      raise ValueError('''Collection must be ascending sorted''' )
   return True


if __name__ == "__main__":
       import sys
       __A			   = 0
       if debug == 1:
              __A			   = [1_0, 3_0, 4_0, 4_5, 5_0, 6_6, 7_7, 9_3]
              try:
                     __assert_sorted(collection)
              except ValueError:
                     sys.exit("Sequence must be ascending sorted to apply interpolation search")
              __A			   = 6_7

       __A			   = interpolation_search(collection, target)
       if result is not None:
              print(f'''{target} found at positions: {result}''')
       else:
              print("Not found")
