Proving Mathematical Identities Using Haskell and Set Arithmetic
Introduction
In this article, we will explore how to prove mathematical identities using Haskell, particularly focusing on the identity of Wall's continued fraction representation of the inverse tangent function. We will discuss the process step-by-step, from understanding the identity to writing a Haskell program to verify it.
Understanding the Identity
The identity we aim to prove is related to the continued fraction representation of the inverse tangent function, as described in the book Analytic Theory of Continued Fractions by H.S. Wall, published in 1948. A key trigonometric identity involved is arcsin(x) 2 arctan(x / (√(1 - x^2) - x)). This identity will play a crucial role in our proof.
Program Development
To prove this identity programmatically, we need to represent the involved sets and their operations using Haskell. We will define a data type for expressions and implement functions to evaluate these expressions under specific conditions.
Defining the Expression Data Type
data Expr Union Expr Expr | Intersect Expr Expr | Complement Expr | Variable String deriving Show
This data type represents basic set operations such as union, intersection, complement, and variables. We will extend this with functions to evaluate the expressions.
Evaluating Expressions
type VariableAssignment String - BoolevalExpr :: VariableAssignment - Expr - BoolevalExpr va Union e1 e2 (evalExpr va e1) (evalExpr va e2)evalExpr va Intersect e1 e2 (evalExpr va e1) (evalExpr va e2)evalExpr va Complement e not $ (evalExpr va e)evalExpr va (Variable v) va v
The above functions evalExpr are designed to evaluate boolean expressions based on the given assignment of variables. Here, VariableAssignment is a mapping from a set of strings (variables) to booleans (true or false).
Checking Expression Equivalence
equivExprs :: Expr - Expr - BoolequivExprs e1 e2 let uniqVars nub $ allVars e1 allVars e2 in all (uncurry (uniqVars where allVars :: Expr - [String] allVars Union e1 e2 allVars e1 allVars e2 allVars Intersect e1 e2 allVars e1 allVars e2 allVars Complement e allVars e allVars Variable v [v]
The function equivExprs determines if two expressions are equivalent by evaluating them under all possible variable assignments. The function subsequences generates all possible combinations of variables, and the function all ensures that for every combination, the two expressions evaluate to the same boolean value.
Parsing Expressions
exprParser :: GenParser Char ExprexprParser do lhs do op oneOf "∪" rhs return if op "∪" then Union else Intersect lhs rhs do return lhsexprParser1 :: GenParser Char ExprexprParser1 do char exprParser char "∪" char exprParser do char char do Variable .: [] oneOf acceptableVarNames Complement char U char exprParser1 where acceptableVarNames :: String - ExprparseExpr s case parse exprParser eof s of Left er - error er Right res - res
The parser uses the Parsec library to parse strings into Expr data types. This allows us to easily input expressions and parse them into a data structure for evaluation.
Running the Program
main putStrLn $ show $ equivExprs >> parseExpr >> parseExpr
The main function runs the expression equivalence check and prints the result. In this case, the program will output True, confirming the identity.
Conclusion
Using Haskell for set arithmetic and programmatic proof of mathematical identities provides a powerful and elegant approach. This method not only verifies the given identity but also offers a general framework for proving other mathematical identities involving set operations.
References
Wall, H.S. (1948). Analytic Theory of Continued Fractions. New York: Chelsea.