# Simple Refinement Types

{-@ LIQUID "--no-termination" @-} module SimpleRefinements where import Prelude hiding (abs, max)

# Simple Refinement Types

Refinement Types = Types + Predicates

# Example: Integers equal to 0

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

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

# Example: Natural Numbers

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

# Exercise: Positive Integers

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


# A Term Can Have Many Types

What is the type of 0 ?

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

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


# SMT Automates Implication Checking

Eliminates boring proofs ... makes verification practical.

# Pre-Conditions

{-@ impossible :: {v:_ | false} -> a @-} impossible :: String -> 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 :: Int -> Int -> Int safeDiv _ 0 = undefined -- 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 :: Int -> Int -> Int 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 @-}