Fun With Functions

This is a collection of cute puzzles of the form “Show that if a function satisfies the following constraints, it must be . . . ” Please add further examples to this collection! Apart from the one about factorial, they all come from the delightful booklet by Terence Tao [1] but go back to Math Olympiads and similar events. Please add further examples of this kind, either directly or by sending them to me. Let us make this a growing body of fun! theory FunWithFunctions imports Complex-Main begin See [1]. Was first brought to our attention by Herbert Ehler who provided a similar proof. theorem identity1 : fixes f :: nat ⇒ nat assumes fff : ∧ n. f (f (n)) < f (Suc(n)) shows f (n) = n 〈proof 〉 See [1]. Possible extension: Should also hold if the range of f is the reals! lemma identity2 : fixes f :: nat ⇒ nat assumes f (k) = k and k ≥ 2 and f-times: ∧ m n. f (m∗n) = f (m)∗f (n) and f-mono: ∧ m n. m<n =⇒ f m < f n shows f (n) = n 〈proof 〉 One more from Tao’s booklet. If f is also assumed to be continuous, f x = x + 1 holds for all reals, not only rationals. Extend the proof! theorem plus1 : fixes f :: real ⇒ real assumes 0 : f 0 = 1 and f-add : ∧ x y . f (x+y+1 ) = f x + f y assumes r : Q shows f (r) = r + 1 〈proof 〉