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

SMTs to Automate Type Checking


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


Refinements Drawn from Decidable logics.


For automatic, decidable (and thus predictable) SMT 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 [NUPRL, PVS]


\[\boxed{\Gamma \vdash t_1 \preceq t_2}\]


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

Example: Natural Numbers


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


And so, we can type:

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

Example: Natural Numbers


\[ \begin{array}{rcrccll} \mathbf{VC\ is\ Valid:} & x = 3 & \Rightarrow & v = x + 1 & \Rightarrow & 0 \leq v & \mbox{(by SMT)} \\ & & & & & \\ \mathbf{So:} & x = 3 & \vdash & \{v:Int\ |\ v = x + 1\} & \preceq & \Nat & \\ \end{array} \]

And so, we can type:

{-@ four :: Nat @-} four :: Int four = x + 1 -- x = 3 |- {v = x + 1} <: Nat where x = 3

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


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

Unfinished Business


How to describe non empty lists?


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


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