# Simple Refinement Types

## Simple Refinement Types

Refinement Types = Types + Predicates

## Types

b := Int         -- base types
| Bool
| ...
| a, b, c     -- type variables

t := {x:b | p}   -- refined base
| x:t -> t    -- refined function

p := ...         -- predicate in decidable logic


## Predicates

p := e           -- atom
| e1 == e2    -- equality
| e1 <  e2    -- ordering
| (p && p)    -- and
| (p || p)    -- or
| (not p)     -- negation


## Expressions

e := x, y, z,...    -- variable
| 0, 1, 2,...    -- constant
| (e + e)        -- addition
| (e - e)        -- subtraction
| (c * e)        -- linear multiplication
| (f e1 ... en)  -- uninterpreted function


Refinement Logic: QF-UFLIA

Quantifier-Free Logic of Uninterpreted Functions and Linear Arithmetic

## Example: Integers equal to 0

{-@ type Zero = {v:Int | v = 0} @-} {-@ zero :: Zero @-} zero = 0

Refinement types via special comments {-@ ... @-}

## Example: List of Natural Numbers

{-@ type Nat = {v:Int | 0 <= v} @-} {-@ nats :: [Nat] @-} nats = [0, 1, 2, 3]

## Exercise: Positive Integers

{-@ type Pos = {v:Int | 0 <= v} @-} {-@ poss :: [Pos] @-} poss = [0, 1, 2, 3]

Q: First, can you fix Pos so poss is rejected?

Q: Next, can you modify poss so it is accepted?

# Refinement Type Checking

## A Term Can Have Many Types

What is the type of 0 ?

{-@ zero  :: Zero @-}
zero      = 0

{-@ zero' :: Nat  @-}
zero'     = zero


## Predicate Subtyping

(NUPRL, PVS)

In environment $$\Gamma$$ the type $$t_1$$ is a subtype of the type $$t_2$$

$\boxed{\Gamma \vdash t_1 \preceq t_2}$

Environment $$\Gamma$$ is a sequence of binders

$\Gamma \doteq \overline{\bindx{x_i}{P_i}}$

## Predicate Subtyping

$\boxed{\Gamma \vdash t_1 \preceq t_2}$

$\begin{array}{rll} {\mathbf{If\ VC\ is\ Valid}} & \bigwedge_i P_i \Rightarrow Q \Rightarrow R & (\mbox{By SMT}) \\ & & \\ {\mathbf{Then}} & \overline{\bindx{x_i}{P_i}} \vdash \reft{v}{b}{Q} \preceq \reft{v}{b}{R} & \\ \end{array}$

## Example: Natural Numbers

        type Nat = {v:Int | 0 <= v}


$\begin{array}{rcrccll} \mathbf{VC\ is\ Valid:} & \True & \Rightarrow & v = 0 & \Rightarrow & 0 \leq v & \mbox{(by SMT)} \\ % & & & & & \\ \mathbf{So:} & \emptyset & \vdash & \Zero & \preceq & \Nat & \\ \end{array}$

Hence, we can type:

{-@ zero' :: Nat @-} zero' = zero -- zero :: Zero <: Nat

## SMT Automates Subtyping

Eliminates boring proofs ... makes verification practical.

# Contracts: Function Types

## Pre-Conditions

{-@ impossible :: {v:_ | false} -> a @-} impossible msg = error msg

No value satisfies false $$\Rightarrow$$ no valid inputs for impossible

Program type-checks $$\Rightarrow$$ impossible never called at run-time

## Exercise: Pre-Conditions

Let's write a safe division function

{-@ safeDiv :: Int -> Int -> Int @-} safeDiv _ 0 = impossible "divide-by-zero" safeDiv x n = x div n

Q: Yikes! Can you fix the type of safeDiv to banish the error?

## Precondition Checked at Call-Site

avg2 x y = safeDiv (x + y) 2

Precondition

{-@ safeDiv :: n:Int -> d:NonZero -> Int @-}


Verifies As

$\inferrule{}{(v = 2) \Rightarrow (v \not = 0)} {\bindx{x}{\Int}, \bindx{y}{\Int} \vdash \reftx{v}{v = 2} \preceq \reftx{v}{v \not = 0}}$

## Exercise: Check That Data

calc :: IO () calc = do putStrLn "Enter numerator" n <- readLn putStrLn "Enter denominator" d <- readLn putStrLn \$ "Result = " ++ show (safeDiv n d) calc

Q: Can you fix calc so it typechecks?

## Precondition Checked at Call-Site

avg :: [Int] -> Int avg xs = safeDiv total n where total = sum xs n = length xs -- returns a Nat

Rejected as n can be any Nat

$0 \leq n \Rightarrow (v = n) \not \Rightarrow (v \not = 0)$

## size returns positive values

Specify post-condition as output type

{-@ size :: [a] -> Pos @-} size [_] = 1 size (_:xs) = 1 + size xs -- size _ = impossible "size"

## Postconditions Checked at Return

{-@ size    :: [a] -> Pos @-}
size []     = 1                        -- (1)
size (_:xs) = 1 + n  where n = size xs -- (2)


Verified As

$\begin{array}{rll} \True & \Rightarrow (v = 1) & \Rightarrow (0 < v) & \qquad \mbox{at (1)} \\ (0 < n) & \Rightarrow (v = 1 + n) & \Rightarrow (0 < v) & \qquad \mbox{at (2)} \\ \end{array}$

## Verifying avg

avg' xs = safeDiv total n where total = sum xs n = size xs

Verifies As

$(0 < n) \Rightarrow (v = n) \Rightarrow (v \not = 0)$

## Recap

Refinement Types

Types + Predicates

Specify Properties

Via Refined Input- and Output- Types

Verify Properties

Via SMT based Predicate Subtyping

How to prevent calling size with empty lists?