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A small mathematical puzzle from a Dijkstra paper

Authors: Jean-Christophe Filliâtre

Topics: Mathematics

Tools: Why3

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(* Let f be a function over natural numbers such that, for all n
     f(f(n)) < f(n+1)
   Show that f(n)=n for all n.

   Inspired by a Dafny example (see
   Original reference is

    Edsger W. Dijkstra: Heuristics for a Calculational Proof.
    Inf. Process. Lett. (IPL) 53(3):141-143 (1995)

theory Puzzle

  use export int.Int

  function f int: int

  axiom H1: forall n: int. 0 <= n -> 0 <= f n
  axiom H2: forall n: int. 0 <= n -> f (f n) < f (n+1)


theory Step1 (* k <= f(n+k) by induction over k *)

  use import Puzzle

  predicate p (k: int) = forall n: int. 0 <= n -> k <= f (n+k)
  clone int.SimpleInduction as I1
    with predicate p = p, lemma base, lemma induction_step


theory Solution

  use import Puzzle
  use import Step1

  lemma L3: forall n: int. 0 <= n -> n <= f n && f n <= f (f n)
  lemma L4: forall n: int. 0 <= n -> f n < f (n+1)

  (* so f is increasing *)
  predicate p' (k: int) = forall n m: int. 0 <= n <= m <= k -> f n <= f m
  clone int.SimpleInduction as I2
    with predicate p = p', lemma base, lemma induction_step

  lemma L5: forall n m: int. 0 <= n <= m -> f n <= f m
  lemma L6: forall n: int. 0 <= n -> f n < n+1

  goal G: forall n: int. 0 <= n -> f n = n


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