# Algorithm Implementation

In this section, we explore the practical applications of Type-C by diving into algorithm implementation. Implementing algorithms will help demonstrate the syntax of Type-C and give you a glimpse of the language's features.

## Fibonacci Sequence

The Fibonacci sequence is one of the most famous formulas in mathematics and computer science. It starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. Mathematically, the sequence is defined as ( F(n) = F(n-1) + F(n-2) ) with ( F(0) = 0 ) and ( F(1) = 1 ).

In this code snippet, we define a function fib that takes an unsigned 32-bit integer ( x ) and returns an unsigned 32-bit integer. We use pattern matching to define the function recursively:

- If ( x = 0 ), the function returns 0.
- If ( x = 1 ), the function returns 1.
- For all other values of ( x ), the function returns fib(x-1) + fib(x-2). This is captured using the wild card value _.

This implementation illustrates how straightforward it is to define a recursive function in Type-C, while also showcasing the language's pattern matching capabilities.

## Quick Sort

Quick Sort is a divide-and-conquer algorithm known for its impressive sorting speed. It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot.

In this implementation, quicksort is the main function that takes the array, and two indices, low and high, which define the segment to be sorted. The partition function takes the same arguments and rearranges the elements so that all elements greater than the pivot are to its right, and all smaller elements are to its left. swap is a utility function that swaps two elements in the array.