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?













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 [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' = 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 = x + 1 -- x = 3 |- {v = x + 1} <: Nat where x = 3













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


{-@ 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

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













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 Predicate Subtyping













Unfinished Business


How to describe non empty lists?


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


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