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In the universe of programming, lambda calculations hold a prominent position. Simple yet profound, they build the cornerstone for functional programming languages, and particularly, Haskell. This article explores the fundamental understanding of lambda calculations in Haskell, diving into its problem-solving capabilities, and providing explanatory code examples for clarity. Throughout we shall highlight specific libraries and functions in Haskell that illuminate the functionality and utility of lambda expressions.

Lambda calculations or lambda expressions comprise anonymous functions in programming. These functions delve into the root of functional programming, offering versatility and precision. Especially in Haskell, they rev up the simplicity and usability.

Lambda and Haskell: The Synergistic Combination

Lambda expressions in programming, generated from the mathematical lambda calculus, are functions with no names – ‘anonymous functions’. Initially brought up in the mid-20th century, lambda calculus caught a ride to the programming world with numerous languages. Haskell, a purely functional language, is among the standards where lambda is broadly utilized.

In Haskell, a lambda expression gets defined as a function with no name, that is used to encapsulate and return function behavior. Key benefits are its simplicity, providing us the ability to pass behaviors as parameters, without the need of defining separate functions.

The following code snippet offers a glance at how lambda expressions work in Haskell:

(x -> 2 * x + 1) 2

Unravelling The Problem: Lambda at Work

Let’s say we’re presented with a small problem. We have a list of numbers and we need to transform this list by doubling all its elements. The solutions for this problem can demonstrate how Haskell and lambda complement each other beautifully.

Normally in Haskell, you’d define a function that doubles a number. Then you’d map that function across a list. But with lambda expressions, the whole process becomes much simpler, especially when the function is only used once.

One way to solve the problem is shown below:

map (x -> 2 * x) [1,2,3,4,5]

Haskell Libraries and Functions: The Defining Toolkit

When using lambda expressions in Haskell, certain libraries and functions make the task handy. Two especially worthy of note are the “map” function and the “Control.Monad” library.

In our problem solution, we used the “map” function. It’s a high-order function that takes a function and a list as arguments, applies the function to all elements in the list, and returns a list with the results.

Moreover, the library “Control.Monad” gives us numerous functions for working with monads, that underpin many operations in functional programming. Lambda expressions play a big part here, helping us generate flexible, reusable code pieces.

Whether you’re a seasoned Haskell programmer or a beginner exploring the field, understanding lambda calculus will open doors to efficient functional programming. The power and simplicity of lambda, when harnessed correctly, can greatly streamline your code and make programming in Haskell a delightful experience.

NOTE: This is a simplified explanation. Lambda calculus in Haskell can get quite complex and it’s a fascinating topic loaded with potential for learning and growth in functional programming.

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