(* Floyd's cycle detection, also known as ``tortoise and hare'' algorithm. See The Art of Computer Programming, vol 2, exercise 6 page 7. *) module TortoiseAndHare use import int.Int use int.Iter type t val predicate eq (x y : t) ensures { result <-> x = y } (* let f be a function from t to t *) val function f t : t (* given some initial value x0 *) val constant x0: t (* we can build the infinite sequence defined by x{i+1} = f(xi) *) function x (i: int): t = Iter.iter f i x0 (* let use assume now that the sequence (x_i) has finitely many distinct values (e.g. f is an integer function with values in 0..m) it means that there exists values lambda and mu such that (1) x0, x1, ... x{mu+lambda-1} are all distinct, and (2) x{n+lambda} = x_n when n >= mu. *) (* lambda and mu are skomlemized *) constant mu : int constant lambda : int (* they are axiomatized as follows *) axiom mu_range: 0 <= mu axiom lambda_range: 1 <= lambda axiom distinct: forall i j: int. 0 <= i < mu+lambda -> 0 <= j < mu+lambda -> i <> j -> x i <> x j axiom cycle: forall n: int. mu <= n -> x (n + lambda) = x n lemma cycle_induction: forall n: int. mu <= n -> forall k: int. 0 <= k -> x (n + lambda * k) = x n (* Now comes the code. Two references, tortoise and hare, traverses the sequence (xi) at speed 1 and 2 respectively. We stop whenever they are equal. The challenge is to prove termination. Actually, we even prove that the code runs in time O(lambda + mu). *) use import ref.Ref use import relations.WellFounded (* the minimal distance between x i and x j *) function dist int int : int (* it is defined as soon as i,j >= mu *) axiom dist_def: forall i j: int. mu <= i -> mu <= j -> dist i j >= 0 /\ x (i + dist i j) = x j /\ forall k: int. 0 <= k -> x (i + k) = x j -> dist i j <= k predicate rel (t2 t1: t) = exists i: int. t1 = x i /\ t2 = x (i+1) /\ 1 <= i <= mu + lambda /\ (i >= mu -> dist (2*i+2) (i+1) < dist (2*i) i) axiom wfrel: well_founded rel let tortoise_hare () = let tortoise = ref (f x0) in let hare = ref (f (f x0)) in while not (eq !tortoise !hare) do invariant { exists t: int. 1 <= t <= mu+lambda /\ !tortoise = x t /\ !hare = x (2 * t) /\ forall i: int. 1 <= i < t -> x i <> x (2*i) } variant { !tortoise with rel } tortoise := f !tortoise; hare := f (f !hare) done (* TODO: code to find out lambda and mu. See wikipedia. *) end