## Pigeonhole principle

Pigeonhole principle (aka Dirichlet's drawer principle) proved using a lemma function.

Authors: Jean-Christophe Filliâtre

Topics: Mathematics / Ghost code

Tools: Why3

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Pigeonhole principle (also know as Dirichlet's drawer principle)

Proved using a lemma function.

```module Pigeonhole

use int.Int, set.Fset, ref.Ref

let rec ghost below (n: int) : fset int
requires { 0 <= n }
ensures  { forall i. mem i result <-> 0 <= i < n }
ensures  { cardinal result = n }
variant  { n }
= if n = 0 then empty else add (n-1) (below (n-1))

let lemma pigeonhole (n m: int) (f: int -> int)
requires { 0 <= m < n }
requires { forall i. 0 <= i < n -> 0 <= f i < m }
ensures  { exists i1, i2. 0 <= i1 < i2 < n /\ f i1 = f i2 }
=
let s = ref empty in
for i = 0 to n-1 do
invariant { cardinal !s = i }
invariant { forall x. mem x !s <-> (exists j. 0 <= j < i /\ x = f j) }
if mem (f i) !s then return;
s := add (f i) !s
done;
let b = below m in assert { subset !s b };
absurd

end
```