On the Mizar Encoding of the Fibonacci Numbers

Extended Abstract Fibonacci-type sequences are probably most famous for their recursive definition. The classical one, as F ib(n) =      0, if n = 0, 1, if n = 1, F ib(n − 1) + F ib(n − 2), otherwise, (1) however very simple formulated in natural language, caused some troubles during its formalization in [1]. definition let n be Nat; func Fib (n)-> Nat means :: PRE_FF:def 1 ex fib being Function of NAT, [:NAT, NAT:] st it = (fib.n)'1 & fib.0 = [0,1] & for n being Nat holds fib.(n+1) = [ (fib.n)'2, (fib.n)'1 + (fib.n)'2 ]; end; We will propose a slight modification of this definition based on schemes for recursive definitions available in the Mizar Mathematical Library to simplify things even more. In order to make the definition clearer and closer to the mathematical standards , first we define a function which maps the set of all natural numbers into the set of corresponding Fibonacci numbers. definition func Fib-> Function of NAT, NAT means it.0 = 0 & it.1 = 1 & for k being Nat holds it.(k+2) = it.(k+1) + it.k; end; The work briefly described here is a subset of the second author's B.Sc. thesis in mathematics defended in July 2004.