Simple Refinement Types

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


Refinement Types = Types + Predicates

Example: Integers equal to 0


{-@ type Zero = {v:Int | v = 0} @-}
{-@ zero :: Zero @-}
zero     :: Int
zero     =  0
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Refinement types via special comments {-@ ... @-}

Example: Natural Numbers


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


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


Contracts: Function Types

Pre-Conditions


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

Unfinished Business


How to describe non empty lists?


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


Next: How to describe properties of structures [continue...]