Indeed, programming in Haskell involves many intricacies and nuances, wherein understanding how to solve problems is an essential skill. Consequently, one of the basic programming challenges that developers new to the Haskell language typically face is the implementation of the factorial function. This function, which is quite significant in mathematics, particularly in computations involving permutations and combinations, holds the potential to open a myriad of possibilities. Profoundly understanding the factorial function thus forms the basis of more complex Haskell programming.

Nevertheless, despite the factorial function’s seeming complexity, the solution to the problem is however quite straightforward. This function can be implemented in Haskell using the ‘product’ function from the Prelude library — a library that contains a host of commonly used functions. Here’s the code:

factorial :: Integer -> Integer factorial n = product [1..n]

## Decoding the Factorial Function

Typically, in a mathematical context, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. We thus implement the factorial function in Haskell by leveraging the power of lists. As opposed to using traditional loops in imperative languages, we can define our range as a list, such as [1..n], where ‘n’ represents the input to our factorial function.

**The ‘product’ function** then takes this list and multiplies all of its elements together. The resulting value in effect represents the factorial of the integer ‘n’. It’s noteworthy to mention that this function has an Integer type as an argument and returns an Integer, a depiction of the strong static typing in Haskell.

## Prelude Library and Product Function

The **Prelude library** in Haskell is a standard library that is imported by default. It provides a rich variety of functions and data types that are commonly used in Haskell programming, one of which is the **‘product’ function**.

The ‘product’ function is a function that receives a list as an argument and returns the product of all the elements in the list. Consequently, when defining the factorial function earlier, we used ‘[1..n]’ to construct our list, where ‘1’ and ‘n’ formed the start and end of our list’s range respectively.

Thus, the product function multiplied all numbers from 1 to ‘n’, computationally achieving the intended result of a factorial function.

In conclusion, this provides a clear illustration of the elegance and expressiveness of Haskell as a functional programming language. It succinctly solves a complex problem through an efficient implementation of the factorial function.

## Exploration of Haskell Function Definitions and Type Signatures

While understanding the implementation of the factorial function is of utmost importance, being enlightened about **function definitions** and Haskell’s **type system** serves as a bonus.

In Haskell, every expression and function has a type. Consequently, when we define the factorial function: ‘factorial :: Integer -> Integer’, we are essentially stating that this function receives an Integer and also returns an Integer.

This statement is referred to as the **type signature** of the function and serves as an essential component of Haskell’s highly-rated type system. It invariably improves code safety, readability and maintainability by ensuring that data mismanipulation is avoided.

In all, while understanding and leveraging the factorial function, the Prelude library and Haskell’s type system and function definitions could initially seem daunting, upon mastery, their capabilities however become a powerful tool in the arsenal of any Haskell programmer.