# Simple Refinement Types

## Simple Refinement Types

Refinement Types = Types + Predicates

## Example: Integers equal to 0

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

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

## Example: 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?

## Type Checking

{-@ type Pos = {v:Int | 0 < v} @-}

{-@ poss :: [Pos]               @-}
poss     =  [1, 2, 3]


Type Checking Via Implication Checking.

v = 1 => 0 < v
v = 2 => 0 < v
v = 3 => 0 < v


## SMT Automates Implication Checking

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

{-@ type NonZero = {v:Int | v /= 0} @-} {-@ 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

${(v = 2) \Rightarrow (v \not = 0)}$

## 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

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

How to talk about list length in logic?

## Recap

Refinement Types Types + Predicates

Specify Properties

Via Refined Input- and Output- Types

Verify Properties

Via SMT based Implication Checking

{-@ avg :: {v:[a]| 0 < length v } -> Pos @-}