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