Zeno is an automated proof system for Haskell program properties; developed at Imperial College London by William Sonnex, Sophia Drossopoulou and Susan Eisenbach. It aims to solve the general problem of equality between two Haskell terms, for any input value.
Many program verification tools available today are of the model checking variety; able to traverse a very large but finite search space very quickly. These are well suited to problems with a large description, but no recursive datatypes. Zeno on the other hand is designed to inductively prove properties over an infinite search space, but only those with a small and simple specification.
One can try Zeno online at TryZeno, or cabal install zeno to use it from home. You can find the latest paper on Zeno here, though please note that Zeno no longer uses the described proof output syntax but instead outputs proofs as Isabelle theories.
- Outputs proofs and translated Haskell programs to an Isabelle/HOL theory file and will automatically invoke Isabelle to check it (requires isabelle to be visible on the command line).
- Works with full Haskell98 along with any GHC extensions not related to the type system. Unfortunately not all Haskell code is then convertable to Isabelle/HOL, see Zeno#Limitations for details.
- Has a built-in counter-example finder, along the same lines as SmallCheck, but using symbolic evaluation to control search depth.
- Its property language is just a Haskell DSL.
The first thing you need is the Zeno.hs file which contains the definitions for Zeno's property DSL. It should be in your Zeno installation directory but is also given below:
module Zeno ( Bool (..), Equals (..), Prop, prove, proveBool, given, givenBool, ($), otherwise ) where import Prelude ( Bool (..) ) infix 1 :=: infixr 0 $ ($) :: (a -> b) -> a -> b f $ x = f x otherwise :: Bool otherwise = True data Equals = forall a . (:=:) a a data Prop = Given Equals Prop | Prove Equals prove :: Equals -> Prop prove = Prove given :: Equals -> Prop -> Prop given = Given proveBool :: Bool -> Prop proveBool p = Prove (p :=: True) givenBool :: Bool -> Prop -> Prop givenBool p = Given (p :=: True)
Making sure this file is in the same directory we can now start coding:
module Test where import Prelude () import Zeno data Nat = Zero | Succ Nat length :: [a] -> Nat length  = Zero length (x:xs) = Succ (length xs) (++) :: [a] -> [a] -> [a]  ++ ys = ys (x:xs) ++ ys = x : (xs ++ ys) class Num a where (+) :: a -> a -> a instance Num Nat where Zero + y = y Succ x + y = Succ (x + y)
Notice we have stopped any
Prelude functions from being imported, this is important as we have no source code available for them; Zeno can only work with functions for which it can see the definition. The only built-in Haskell types we have are lists and tuples, which are automatically available, and
Bool, which Zeno.hs will import for you.
Now that we have some code we can define a property about this code. Equality is expressed using the
(:=:) constructor defined in Zeno.hs. We then pass this to the
prove function to turn an equality into a property (
The following code will express that the length of two appended lists is the sum of their individual lengths:
prop_length xs ys = prove (length (xs ++ ys) :=: length xs + length ys)
Add this to the above code and save it to Test.hs. We can now check
prop_length by running zeno Test.hs. As a bug/feature this will also check
Zeno.proveBool, a helper function in Zeno.hs, as this looks like a property. To restrict this to just
prop_length we can run zeno -m prop Test.hs, which will only check properties whose name contains the text prop.
Say we want to express arbitrary propositions, we can do an equality check with
True, as in the following code (appended to the code above):
class Eq a where (==) :: a -> a -> Bool instance Eq Nat where Zero == Zero = True Succ x == Succ y = x == y _ == _ = False prop_eq_ref :: Nat -> Prop prop_eq_ref x = prove (x == x :=: True)
We have provided the helper function
proveBool to make this more succinct; an equivalent definition of
prop_eq_ref would be:
prop_eq_ref x = proveBool (x == x)
We can also express implication in our properties, using the
elem :: Eq a => a -> [a] -> Bool elem _  = False elem n (x:xs) | n == x = True | otherwise = elem n xs prop_elem :: Nat -> [Nat] -> [Nat] -> Prop prop_elem n xs ys = givenBool (n `elem` ys) $ proveBool (n `elem` (xs ++ ys))
prop_elem expresses that if
n is an element of
n is an element of
xs ++ ys. Notice that we had to explicitly type everything to be
Nat so as to give Zeno an explicit definition for
While Zeno is able to reason about any valid Haskell definition, not all of these can be converted to Isabelle for checking. There are two main restrictions:
- No internal recursive definitions; don't put recursive functions inside your functions.
- No non-terminating definitions. This also means you cannot use default type-class methods, as GHC transforms these internally to a co-recursive value. Isabelle will check for termination but Zeno will not, unfortunately this means that Zeno could be thrown into an infinite loop with such a definition.
While the above restrictions are founded in Isabelle's input language there are a few which are just laziness on part of Zeno's developers, and on our to-do list:
- No partial definitions; only use total pattern matches.
- No mututally recursive datatypes.
- No tuple types beyond quadruples.
- No name clashes, even across modules. Zeno will automatically strip module names in its output for clarity, and we have not yet implemented a flag to control this.
If you are wondering why a certain bit of code cannot be converted to Isabelle try running Zeno with the --print-core flag, this will output Zeno's internal representation for your code.
Zeno can only reason about inductive datatypes, meaning the only built-in types it can use are lists, tuples and
Bool. Any values of type
Char, etc. Zeno will replace with
Infinite and undefined values
When we said that Zeno proves properties of Haskell programs this wasn't entirely true, it only proves those for which every value is finite and well-defined. For example, Zeno can prove
reverse (reverse xs) = xs, which is not true for infinite lists, as
xs could still be pattern matched upon, whereas evaluating
reverse (reverse xs) starts an infinite loop (
Another example (courtesy of Tillmann Rendel) is
takeWhile p xs ++ dropWhile p xs = xs, which Zeno will prove but in fact is not true when
p = const undefined and
xs /= , as the left hand side of the equality would be
You might ask why this is a Haskell theorem prover, rather than an ML one, if it cannot deal with infinite values, which would be a very valid question. As it stands Zeno is more a base-line for us to start more advanced research into lazy functional program verification, which will include attempting to tackle this issue.