Testing vs Proving Testing uses a set of “typical” examples, symbolic testing, may find errors, but cannot show absents of errors, “easy” to do. Proving correctness establishes properties of programs by a mathematical proof, failure  error in the program success  program is correct difficult enterprise. Testing and proving should both be part of the development process of reliable software.

Properties of programs Definedness and termination Evaluating an expression can have one of two outcomes: the evaluation can halt, or terminate, to give a result, or the evaluation can go on forever. The proofs we consider state a property that holds for all defined values (partial correctness). Finiteness In a lazy language we have two kinds of special elements: infinite objects, e.g., infinite lists, partially defined objects. Programs as formulas A definition square :: Int -> Int square x = x*x leads to the following formula  x::Int (square x = x*x)

Verification Principle of extensionality: Two functions f and g are equal if they have the same value at every argument. Principle of induction for natural numbers: In order to prove that a logical property P(n) holds for all natural numbers n we have to do two things: Base case: Prove P(0). Induction step: Prove P(n+1) on the assumption that P(n) holds. Principle of structural induction for lists: In order to prove that a logical property P(xs) holds for all finite lists xs we have to do two things: Base case: Prove P([]). Induction step: Prove P(x:xs) on the assumption that P(xs) holds.

Reasoning about algebraic types Verification for algebraic types follows the example of lists. Principle of structural induction for algebraic types: In order to prove that a logical property P(x) holds for all finite elements of an algebraic type T: Base case: Prove P(C) for all non-recursive constructors C of T. Induction step: Prove P(Cr y1 … yn) for all recursive constructors Cr of T on the assumption that P(y1) and … and P(yn) holds. Example: data Tree a = Empty | Node a (Tree a) (Tree a) Base case: Prove P(Empty) Induction step: Prove P(Node x t1 t2) for all x of type a on the assumption that P(t1) and … and P(t2) holds.

List induction revisited To prove a property for all finite or partial lists (fp-lists) we can use the following principle: Principle of structural induction for fp-lists: In order to prove that a logical property P(xs) holds for all fp-lists xs we have to do three things: Base case: Prove P([]) and P(undef). Induction step: Prove P(x:xs) on the assumption that P(xs) holds. fp-lists as an approximation of infinite lists: [a1,a2,a3,…] is approximated by the collection undef, a1:undef, a1:a2:undef, a1:a2:a3:undef, … For some properties (admissible or continuous predicates) it is enough to show the property for all approximations to know that it will be valid for all infinite lists as well. In particular, this is true for all equations.

Case study Consider again the following data type for expressions: data Expr = Lit Int | IVar Var | Let Var Expr Expr | Expr :+: Expr | Expr :-: Expr | Expr :*: Expr | Expr :\: Expr deriving Show The meaning (value) of such an expression is evaluated using a Store. Store is an abstract data type providing several functions.

Case study (cont’d) USER IMPLEMENTOR initial :: Store value :: Store -> Var -> Int update :: Store -> Var -> Int -> Store USER IMPLEMENTOR eval :: Expr -> Store -> Int eval (Lit n) store = n eval (IVar v) store = value store v eval (Let v e1 e2) store = eval e2 (update store v (eval e1 store)) eval (e1 :+: e2) store = eval e1 store + eval e2 store eval (e1 :-: e2) store = eval e1 store - eval e2 store eval (e1 :*: e2) store = eval e1 store * eval e2 store eval (e1 :\: e2) store = eval e1 store `div` eval e2 store

Case study (cont’d) Several questions arise: What are the natural properties which should be fulfilled by Expr and the eval function? What are natural properties of the functions provided by the ADT Store? Are those properties sufficient to show the properties of eval?

Case study (cont’d) Consider the following function: subst :: Expr -> Expr -> Var -> Expr subst (Lit n) _ _ = Lit n subst (IVar v) e w = if v==w then e else (IVar v) subst (Let v e1 e2) e w = Let v (subst e1 e w) (if v==w then e2 else subst e2 e w) subst (e1 :+: e2) e w = subst e1 e w :+: subst e2 e w subst (e1 :-: e2) e w = subst e1 e w :-: subst e2 e w subst (e1 :*: e2) e w = subst e1 e w :*: subst e2 e w subst (e1 :\: e2) e w = subst e1 e w :\: subst e2 e w A natural property would be eval (subst e1 e2 v) store = eval e1 (update store v (eval e2 store))

Case study (cont’d) Consider the following properties: value initial v = 0 value (update store v n) v = n v /= w  value (update store v n) w = value store w Notice, every element of Store can be generated using initial and update. These functions are “abstract” constructor functions for the ADT Store. For all other functions (the function value) axioms in terms of the constructor functions are provided. This will give a sufficient set of axioms. Yes!!! (see derivation in class).

Proving in Coq