Specifying and Verifying Higher-order Rust Iterators

. In Rust, programs are often written using iterators , but these pose problems for veriﬁcation: they are non-deterministic , inﬁnite , and often higher-order , eﬀectful and built using adapters . We present a general framework for specifying and reasoning with Rust iterators in ﬁrst-order logic. Our approach is capable of addressing the challenges set out above, which we demonstrate by verifying real Rust iterators, including a higher-order, eﬀectful Map . Using the Creusot veriﬁcation platform, we evaluate our framework on clients of iterators, showing it leads to eﬃcient veriﬁcation of complex functional properties.


Introduction
The Rust language aims to empower systems software programmers by offering them safe and powerful linguistic abstractions to solve their problems.The most notorious of these abstractions, Rust's borrowing mechanism, enables safe usage of pointers without a garbage collector or performance penalty.A close second is perhaps Rust's iterator system, through which Rust provides composable mechanisms to express the traversal and modification of collections.Iterators also underlie Rust's for loop syntax, and are thus the primary manner Rust developers write loops or interact with data structures.It is therefore essential for a verification tool for Rust to provide good support for iterators.
Rust iterators generate sequences of values.Most importantly, they are objects providing a method fn next(&mut self) -> Option<Self::Item>.This method takes a mutable reference (&mut self) to the iterator, allowing it to change its internal state, and optionally returns a value of type Self::Item, the type of the values generated by the iterator.If, instead of returning such a value, the iterator returns None, it means iteration has finished for now, though it may resume again later.Rust's for loops are just syntactic sugar for repeatedly calling next at the beginning of each iteration, until such a call returns None.For example, the following two pieces of code present a Rust loop for iterating over integers between 0 (included) and n (excluded), using a range iterator: for i in 0..n { <body > } let mut iter = 0..n; loop { match iter.next(){ None => break, Some(i) => <body > } } The piece of code on the left-hand side uses an idiomatic for loop, while the other shows its desugared version.
Iterators present unique challenges for verification tools: indeed, because the use of iterators is pervasive in Rust, it is necessary to allow verification of code using iterators with as little interaction as possible.In particular, most common patterns such as iterating over integers in a given range or reading the elements of a vector should not need any annotation other than the loop invariants the user would write if not using iterators.On the other hand, Rust's iterator library is complex, with many features representing as many challenges for verification: iterators can be built from various data structures and modified through iterator adapters, which make it possible to create iterators from simpler ones, by, e.g., skipping the first few elements or applying a given function to each of the elements.
Consider the example below: let mut cnt = 0; let w = vec![1,2,3].iter().map(|x|{cnt+= 1; x + 1}).collect(); assert_eq!(w,vec! [2,3,4]); assert_eq!(cnt,3); On line 2, quite a lot happens at once.First, we produce an iterator over the elements of the vector vec! [1,2,3] with the syntax .iter(),which we transform through a call to map.The method map is an iterator adapter : it returns a new iterator that calls the given closure on each of the elements generated by the underlying iterator, and forwards the value returned by the closure.Interestingly, the closure we pass to map captures mutable state: it modifies the variable cnt.
Finally, the method collect gathers the elements generated into a new vector w.
We aim at requiring only lightweight annotations for verifying this kind of code: the appeal of iterator chains like on line 2 are the ergonomics, they are compact and highly-readable.For verification of iterator-based code to be successful, it must preserve this ergonomics.However, despite its apparent simplicity, this piece of code is challenging to verify: it combines higher-order functions and mutable state, uses potentially overflowing integers, and assertions on line 3 check full functional behavior.
More generally, to support iterators, a verification tool for Rust needs to provide a specification scheme that both provides good ergonomics and overcomes the following technical challenges: -Strong Automation: for verification to be used, it must require little to no user interaction and lead to good verification performance.-Interruptibility: iterators can produce infinite sequences of values and can be interrupted before completion, thus specification and verification must happen as the iterator is used, and not at completion.
-Non-Determinism: iterators can feature both specification or implementation non-determinism, so the sequence of known values might not be known in advance to the verifier.For example, the order of elements generated by an iterator over a hash table may be left unspecified for a client.-Compositionality: iterators can be consumed by adapters, so their specifications need to follow a general pattern which make them composable.For example, the specification of a adapter such as skip(n), which skips the first n elements of a given iterator, should accept the specification of any iterator, and provide a sound and useful specification for the combined iterator.-Higher-Order & Effects: some iterator adapters, such as map, are higher-order, they take a closure as parameter.To verify programs using these adapters, a verification tool should overcome the challenges of higher-order functions, which potentially capture mutable state.

Contributions
In order to reach this goal, we propose a new specification scheme for iterators in Rust.Our contributions can be summarized as follows: -In Section 3, we provide a general specification scheme for Rust iterators in first-order logic.It supports possibly non-deterministic, infinite and interruptible iterators.It is inspired by Filliâtre and Pereira's specification of iterators in Why3 [5], but it is adapted to our style of specification using a prophetic mutable value semantics [13] for Rust.This style of specification is particularly well suited to handle mutable values (of which iterators are an instance), by leveraging the non-aliasing guarantees provided by Rust's type system.-In Section 4.1, we show that this scheme can be trivially instantiated for basic iterators such as a range of integers.-In Section 4.2, we show how this scheme can be instantiated to give full functional specification to mutating iterators.These iterators allow to mutate the content of a data structure by iterating over mutable references pointing to the content of the data structure.-In Section 4.3, we show that our specification scheme is composable, so that it can be used to specify iterator adapters transforming arbitrary iterators into more complex ones.We give two examples: take, which truncates an iterator to at most a given number of elements, and skip, which skips a given number of elements at the beginning of iteration.-To support higher-order iterator adapters, we provide a specification mechanism for closures in Section 5.This mechanism distinguishes the three kinds of closures of Rust (Fn, FnMut and FnOnce), and allows specifying the side effects a closure may have on its environment by making explicit the effect of a call on the state of the closure.It allows reducing the verification conditions for closures to first-order logic, enabling usage of off-the-shelf automation.-In Section 6, we explain how we can combine the techniques presented in previous sections to specify higher-order iterator adapters, by taking map as an example.This provides a way to verify the functional correctness of programs using higher-order iterators, while requiring lightweight annotations.-We provide a freely available1 implementation of our proposal in Creusot [4].This tool is a state-of-the-art verification platform for safe Rust code, allowing users to verify programs by adding contracts to their functions.This implementation extends Creusot's handling of for loops to benefit from structural invariants provided by the specification of iterators.We evaluate it in Section 7 on several benchmarks.

Specifications in Rust Programs
Before explaining the specification of iterators, we introduce the style of specification we use in this paper.One important aspect of specifications of imperative programs is their memory model, that is the way they handle pointers and mutations performed through them.Following previous work [7,8,4], we choose to leverage the non-aliasing guarantees of Rust's type system.Because of the non-aliasing guarantees, a given memory location can be mutated through at most one reference at a given point in time, excluding all "spooky actions at a distance" that are customary with pointer aliasing.Therefore, it is possible to give a mutable value semantics [13] to Rust programs, meaning that, even though Rust programs can perform mutation of memory, they can be reasoned about in a purely applicative manner.As a result, the Rust type Box<T> of heap-allocated pointers, and the Rust type &T of read-only references are simply modeled by wrappers over values of type T in our specifications.As shown in previous work [4,7,8], this interpretation of Rust programs is key to verifying complex Rust programs, because it avoids the use of any kind of separation logic or dynamic frames, which are challenging to automate.The handling of mutable references &mut T requires caution.Such references represent the temporary borrow of ownership of a memory location, so that mutations through such a reference will be observed by the initial owner once the borrow ends.To correctly model the propagation of mutations from the mutable reference to the borrowed variable, this style of specification models a mutable reference r: &mut T as a pair of a current value *r of type T (representing the current value pointed to by the reference) and of a prophecy ^r, representing the value the reference will point to when the borrow ends.
This prophetic interpretation makes it possible to give precise specifications to functions that manipulate mutable references.For example, the function push adding a new element at the end of a vector in place can be specified as follows: #[ensures(@^self == (@*self).concat(Seq::singleton(v)))]fn push(&mut self, v: T); Here, we use the operator @ to refer to the model of a vector, i.e., the mathematical sequence of its elements.The postcondition thus ensures that the content of the final vector pointed to by self, denoted by ^self, is modeled by the sequence of elements of the initial vector *self, concatenated with the new element v.
We sometimes use purely mathematical functions and predicates, annotated with the #[logic] and #[predicate] attributes.
We use Rust traits to give composable specifications to iterators.They are analogous to Haskell's typeclasses, enabling ad-hoc polymorphism.For example, an order relation can be specified as a trait containing both a mathematical order relation with its laws (reflexivity, antisymmetry and transitivity), and a program function specified as returning the value prescribed by the logical predicate.
To aid in specification and verification of code, we use ghost code, code which exists only during verification and has no influence on runtime behavior.

Reasoning on Iteration
In this section, we present the general mechanism we use to specify iterators (Section 3.1), and how this kind of specification is used in a for loop (Section 3.2).

Specifying Iterators
In Rust, the mechanism of iterators is captured by a trait named Iterator, whose simplified definition can be given as: As can be seen in Figure 1, we extend2 the iterator trait with the purely logical predicates produces and completed.We require that any implementation of this trait satisfies the laws produces_refl and produces_trans: such laws are lemmas stated as specifications of purely logical functions (i.e., the preconditions should imply the postconditions).The next method is then specified thanks to the two predicates.Any implementation of the Iterator trait needs to give a logical definition of produces and completed predicates, prove the laws, give a program definition for next and finally prove that it satisfies its specification.
Iterators are specified as state machines: a value of an iterator type is seen as a state; fn produces_refl(a: Self); specify mutations that happen when an iterator returns None3 .This added expressivity in the specification allows us to express properties of unfused iterators which may intermittently produce None during iteration.The produces transition relation is annotated with sequences of generated values rather than with unique values so that a user can reason about interesting properties of sequences as a whole rather than directly reasoning about the notion of transitive closure, which automated solvers do not handle well.The price to pay is the laws of reflexivity and transitivity which the implementers have to prove.

Structural Invariant of for Loops
Part of the appeal of for loops is the structure they provide over the looping process.When a programmer sees a for, they can conclude that the body will be executed once for each element in the iterator.Unlike with while loops, it is not possible to decrement the loop index or otherwise perform unpredictable looping patterns.This informal reasoning can be formalized as a loop invariant, provided structurally by the for loop itself.The iterator at the i-th iteration is the result of calling next exactly i times on some initial state.In our formalism, given an initial iterator state initial and a current iterator state iter, we can state this invariant as ∃ p, initial p iter.This invariant holds for any for loop over any iterator: it can be derived from the laws produces_refl and produces_trans.
When using our extension to Creusot, every for loop benefits from this structural invariant: we change the way these loops are desugared into the more primitive loop construct, by adding ghost variables init_iter and produced and the new invariant init_iter.produces(produced,iter).More precisely, a simple for loop for x in iter {<body >} is desugared into: Interestingly, the ghost variable produced can be referred to in a user invariant to relate the state of the loop with the iteration state.In the piece of code in Figure 2, we use a variable count to count the number of elements generated by an iterator, and use such an invariant to verify its intended meaning.

Examples of Specifications of Simple Iterators
In Section 3, we have presented a general framework to specify iterators and use them in for loops.In this section, we present several simple examples of iterators defined in this framework.

The Range Iterator
We start with a simple Range iterator, whose purpose is to iterate over the integers in a given range.The notation a..b used idiomatically in Rust is a syntactic sugar for this kind of iterators.The original definition from the Rust standard library is generic over the type of integers used, but, for the sake of simplicity, we use a monomorphic version here: struct Range { start: usize, end: usize } If self.start≥ self.end, the next method returns None.Otherwise, it increments self.start and returns the initial value of Some(self.start).Note that the upper bound of the range, end, is excluded in the iteration.
In order to instantiate our iterator specification scheme with Range, we use the produces and completed predicates defined by: Transitivity and reflexivity are easily verified.Rust's standard library also contains ranges whose upper bound is included rather than excluded, and ranges without an upper bound.They can all be specified using similar techniques.
Note that with these definitions, the structural invariant of for loops directly implies that the loop index (the last produced value) is in the range.In addition, if the range is non-empty, one can deduce that the last iterated value is end − 1.These two properties usually require an additional invariant if the loop is encoded using the while construct.For an illustration consider Figure 2.

IterMut: Mutating Iteration Over a Vector
Our approach to iterators can be used to iterate over elements of a vector.But instead of presenting the simple case of a read-only vector iterator, we study a more general iterator, IterMut, permitting to both read and write vector elements while iterating; the simpler case of the read-only iterator uses the same ideas.
This iterator produces mutable references for each element of a vector in turn.The state of this iterator is a mutable reference to the slice (i.e., a fragment of a vector) of elements that remain to be iterated: To define the production relation of IterMut, we use a helper function tr, which transposes a mutable reference to a slice into a sequence of mutable references to its elements.Its defining property is: With the help of tr, the produces and completed relations of IterMut are simple to express: It means that the iterator it produces a sequence of mutable references, which must be the initial segment of tr(it.inner),into a final state it such that tr(it.inner) is the sequence of mutable references that are left to be generated.Such an iterator is completed when the inner slice is empty.This compact specification is enough to reason about mutating through the returned pointers as in the following example: That is, we are able to prove with a simple loop invariant that this loop sets to 0 all the elements of the vector.
The reasoning that occurs to prove this program is as follows.First, at the end of a loop iteration, we know that the final value of the borrow x is equal to 0 since we have just written 0 and this value will not change since x goes out of scope.Together with the invariant of the preceding iteration, this is enough to prove that the invariant is maintained.Second, after the loop has executed, the final iterator state is empty, so we know produced contains the complete sequence of borrows to elements of v. But, thanks to the loop invariant, the prophetic value of each of these borrows is 0. So we can deduce that the final content of v is a sequence of zeros.

Iterator Adapters
Because all iterators implement the same trait Iterator which gives them a specification, we can easily build adapters which wrap and transform the behavior of an iterator.
It is important to note that, following Rust's standard library, these adapters are generic over the type of the underlying iterator; individual values of a type cannot have different predicates.While the verification tool cannot know the concrete definitions of produces or completed for the wrapped iterator, it knows it must satisfy the Iterator trait interface.
The simplest example is Take<I> (where I is another iterator), which truncates an iterator to produce at most n elements.The state of Take<I> is a record with two fields: a counter n for the remaining elements to take and an iterator iter to take from.The specification predicates of Take<I> are defined as follows: The subtle definition here is that of completed(it): if the counter is 0, then next does nothing.But, following Rust's implementation, if the counter is not 0, then it is first decremented even if the call to the underlying iterator returns None.
Again, when instantiated to a specific underlying iterator type, we can substitute the definitions of ( ) and completed(−) for the underlying iterator, to get a concrete definition of these predicates for Take<I>, which are easier to handle by automated solvers.
Another adapter is Skip<I>, whose goal is to skip the first n elements of an iterator.Similarly to Take<I>, the state is a record with two fields: a number n of elements to skip and an underlying iterator iter.
The relation of Skip<I> is defined as follows: The first disjunct is needed to ensure reflexivity of ( ).The second disjunct describes what happens after a non-empty sequence of calls.If we produced some sequence of elements v, then we must have been able to skip n elements first, which we existentially quantify over.
If the Skip<I> iterator is completed, the underlying iterator has also completed, but potentially after having generated some skipped elements that we existentially quantify over: Using Skip<I> and Take<I> we are able to prove an algebraic property of iterators: if we take n elements and then skip n elements from that iterator, we must necessarily get the empty iterator.

Closures in Rust
Unlike traditional functional languages, Rust has no function type for closures.Two closures, even with identical bodies, are not of the same type: closures are each given a unique, anonymous type representing the captured environment.This design is motivated by the need to fully resolve closures during compilation: the compiler is always able to identify exactly which piece of code is used at every call site.To abstract over closures and write higher-order functions, Rust provides three traits that the closure type may implement: FnOnce, FnMut, and Fn.They describe the different ways a closure's environment can be passed during a call: by ownership, by mutable reference or by immutable reference.The compiler automatically provides the relevant instances when a user writes a closure.
Traditionally, verifying higher-order code with mutable state has needed seperation logic or dynamic frames, but because of Rust's mutable value semantics we can avoid these tools.Instead, we provide a specification for higher-order functions in first-order logic, which generates simple verification conditions (see code of Section 7).Specifically, we extend FnOnce, FnMut, and Fn with logical predicates that capture the pre-and post-conditions of closures.We begin by considering the simplest case, The predicates precondition and postcondition_once refer to the specification added to the call_once method used to call the closure.
A call to a FnOnce closure consumes it.On the other hand, FnMut allows a mutable closure to be called multiple times.Here is our extended FnMut trait: Because every FnMut closure is also an FnOnce closure, we can reuse the precondition predicate to specify call_mut.However, we need a new predicate for the richer postconditions that become possible: since the closure is called using a mutable borrow, the postcondition specify changes made to captured variables.
Rust compiles closures via closure conversion, the state of each closure becomes a struct holding references to all captured variables.However, this struct can only be modified in a restricted fashion: we can only mutate the values pointed by the captures, and not the captures themselves.In particular, this means the prophecies of captures remain constant.We capture this property in an unnesting predicate F::unnest(a, b).It expresses that the prophecies in the state of type F have not changed from a to b.This property is both reflexive and transitive which we capture via laws.The unnesting predicate is essential to link the states of a closure throughout repeated calls.Without it we would lose track of the contained prophecies.
In addition to these predicates, our FnMut trait contains laws we elided: unnest is implied by postcondition_mut, and postcondition_mut is linked to the postcondition predicate of the FnOnce trait.
Finally, Fn imposes that the closure is immutable.Each call upholds the postcondition and leaves the state intact.Again, in the following, we elided laws relating postcondition, postcondition_mut and postcondition_once: To verify it, we need f.precondition(a) to be true at each iteration, so we need an invariant which implies it.This exposes the key property that must be true of our closure: the postcondition at iteration n must be able to establish the precondition for iteration n + 1.In the vocabulary of iterators: This expresses that if we eventually produce an element e 1 which satisfies the precondition of the initial closure *f , then combined with the postcondition of f , we must be able to establish the precondition for the final closure ˆf with the following element e 2 .Quantifying over a prefix s in the iteration from a known initial state i ensures this property holds for all possible subsequent iterations.
To encode this property in Map, we use a type invariant, which allows specifying a property that values of a type must uphold.Values of type Map are records with two fields: field func contains the closure state, and field iter contains the underlying iterator.The invariant states that (1) the precondition for the next call will be verified; (2) the preservation property above holds for the current state it; (3) these two invariants are reestablished if the underlying iterator returns None (this is usually trivial since the underlying iterator often is fused: it cannot generate new elements once it returns None); and (4) the type invariant of the underlying iterator holds.
These invariants are initially required as a precondition of the map method used to create the Map iterator.In order to be tackled by automated solvers, this verification condition need to be unfolded: it is therefore crucial that closures and their pre-and post-conditions are statically resolved thanks to the unique anonymous closure types in Rust.
The specification predicates for Map can now be stated: In , we quantify existentially over two pieces of information: the sequence of values v produced by the underlying iterator and the sequence of mutable references of states fs that the closure traverses.We require that fs forms a chain, the final state of each element being the same as the current value of the following one.Finally, we require the closure pre-and post-conditions for every iteration, and that the first and last state are related by the unnesting relation.The definition of completed(−), on the other hand, straightforwardly states that the underlying iterator is completed.Interestingly, the user of this specification can use the precondition of the closure to encode closure invariants that she wishes to maintain along the iteration (as with loop invariants).This specification for Map allows us to specify many use cases, so long as the supplied closure is "history-free": its specification does not depend on the sequence of previously generated values, like in x.map(|a : u32| a + 5).While this is certainly the most common usage of map, we sometimes need a more powerful specification.
Extending Map With Ghost Information.If we attempt to use the previous specification of Map to verify the counter example of Section 1, we will rapidly encounter an issue: to establish that cnt properly counts the number of iterations would require a (manual) induction on the iterated sequence.While the prior specification allows the closure to specify the impact of an immediate call, it has no way of reasoning on the position in the iteration.In our prior thought experiment using a generator, we have no way of writing an invariant which depends on produced, as we allowed for usual for loops.
To make the verification of this kind of code simpler, we extend the signature of Map to provide to the closure the sequence of elements generated by the underlying iterator since the creation of the mapping iterator object.This information does not change the behavior of the program: we make it ghost, so it can only be used in specifications.
The extended version, MapExt, is thus given an additional ghost field, produced, containing this sequence.The relation ( ) is extended to account for this ghost information, by adding a conjunct stating that it .produced= it.produced•vand passing the additional ghost parameter it.produced•v[0..i−1] to the pre-and post-conditions.The completed() relation is extended by adding the conjunct (ˆit).produced= ε (the produced field is reset when the iterator returns None).The type invariants are adapted accordingly.This extra information avoids the need for an explicit induction after the fact to establish that we have properly counted the number of iterations: the postcondition of the last call to next is enough.This mechanism is useful in a wide variety of situations, beyond reasoning on the length of the sequence.

Evaluation
In this section we measure the performance of both the proofs of iterators and their clients, using the Creusot [4] tool for verification of Rust programs.It allows for verification of Rust programs, and requires some annotations to verify the functional correctness of Rust programs.Verification is performed by translating annotated Rust code into a pure, first-order functional program.Then, Creusot uses Why3 [15] to generate verification conditions, which are discharged using automated solvers such as CVC5, Z3 or Alt-Ergo.
The results in Figure 3, were gathered using a Macbook Pro with an M1 Pro CPU and 32 GB of RAM, running macOS 12.2.Why3 was limited to using four provers simultaneously among Z3 4.11.2,CVC5 1.0.2, and Alt-Ergo 2.4.1.
Why3 supports proof transformations: manual tactics which can be used in combination with automated solvers.Because we wish to obtain ergonomic specifications which work well with automation, we minimize their use.Nevertheless, certain complex proofs required minor manual work, which we clearly indicate.The left table in Figure 3 contains a selection of the iterators and adapters we have verified.The Range, IterMut, Skip and Take iterators are implementations of the iterators described in Sections 4.1 to 4.3.The Fuse adapter is responsible for transforming any iterator into a fused one, which will always return None after the first, never resuming iteration.Two versions of Map are provided, the first is the standard library Map, which is restricted to closures whose preconditions are 'history-free', the version in MapExt is provided with ghost information about previous calls as explained in Section 6.
Some manual proof steps were required to prove several iterators.For Skip<I> and Fuse, the manual tactics consist only of telling Why3 to access lemmas about sequences.For Map and MapExt, tactics were used to instantiate quantifiers within the production relation.We think that the use of ghost variables and of the SMT theory of sequences could lift the use of manual tactics.
We also verified several clients of iterators, sometimes featuring combinations of several iterators.The example decuple_range maps a Range, multiplying elements by 10, collecting the results into a vector and verifying functional correctness; counter is an annotated version of the example in the introduction, verifying that we can use mutable state to count the elements of an iterator; concat_vec uses extend to append an iterator to the end of a vector; all_zero uses IterMut to zero every cell of a vector; take_skip checks that if we truncate an iterator to the first n elements and then skip them, the resulting iterator must be empty.We have larger scale examples where iterators are used in the context of a larger verified development: hillel is a port of a prior Creusot solution to Hillel Wayne's verification challenges [16]; knights_tour is the same for the Knight's Tour problem.In both of these cases, updating the code to use for-loops and iterators actually reduced the number of lines of specification.
Because our lines of specification include the assertions which test functional properties, we believe the resulting overhead is reasonable, especially in our client examples.Additionally, our specifications for iterators seem to have low impact on verification times.We compared hillel and knights_tour with alternative versions that only differ by using traditional while loops instead of iterators, verification times are 0.91 and 1.14 respectively.This provides evidence that integrating our iterators does not cause prohibitive increases in verification time.

Related and Future Work
RustHorn [7] and RustHornBelt [8] show how the non-aliasing guarantees of Rust can be used for reducing the verification of Rust programs into the proof of first-order logic formulas.These works serve as theoretical foundations for Creusot [4], which we use to evaluate our specification scheme for iterators.
Prusti [1] is a semi-automatic verifier for Rust built on the Viper [10] separation logic verification platform.Prusti models mutable borrowing and ownership using separation logic permissions, unlike our choice of using a prophetic mutable value semantics.This leads to differences in the specification languages: whereas Creusot uses the ^operator to reason about borrows, Prusti uses a notion called pledges.Pledges are assertions which must be true at the end of a specific lifetime.At the time of writing, pledges are not fully first-class in Prusti's specification logic: they are used through a kind of postcondition.In particular a ghost predicate like produces cannot contain a pledge.The ^operator can be used anywhere in specifications, which allows us to give a natural specification to mutating iterators like IterMut (Section 4.2).
The verification of higher-order programs has been studied by Régis-Gianas and Pottier [14], who verify them using higher-order logic.Prusti supports closures by modeling them in Viper's separation logic [17].Like our approach, Prusti transforms specifications of higher-order programs into first-order ver-ification conditions, but in separation logic.They introduce several constructs to specify closures: history invariants, specification entailment, and call descriptions.We instead enable users to refer to pre-and post-conditions of closures via a trait.While we not have the constructs Prusti provides primitively for closures, we believe these constructs can be encoded using our primitives, at the cost of lower ergonomics.Our approach is more expressive: unlike Prusti's call descriptions, we can distinguish the order of calls (see Section 6).Also, Prusti's approach for borrows makes it difficult to handle iterators such as IterMut.
Like us, Aeneas [6] verifies Rust programs by translation to a functional language, and targets traditional proof assistants such as Coq, or F * .They use a technique called backward functions to interpret mutable borrows.To our knowledge, Aeneas supports neither closures nor iterators.
The formalization of iterators is a well-studied subject with implementations in a variety of imperative and functional languages: WhyML [5], Eiffel [11], Java [9], and OCaml [12].Of particular relevance is the approach developed by Filliâtre and Pereira [5], which specifies iterators in WhyML using a ghost field visited : seq 'a and two predicates permitted : cursor 'a -> bool and completed : cursor 'a -> bool where cursor 'a is an iterator for values of type 'a.This work leverages Why3's regions system to distinguish individual cursors over time.In contrast, in our context, we lose object identity: there is no way to identify that two iterator values are two successive states of the same iterator.We thus generalize this approach to our setting by explicitly providing pre-and post-states in produces.Our work is also more expressive: we specify and verify higher-order iterators using potentially mutable closures, which are ruled out by Why3's region system.The framework of iteration described by Polikarpova, Tschannen, and Furia [11] is limited to finite, deterministic iteration: the user must provide up front the sequence of abstract values the iterator will produce.Pottier [12] presents an implementation of iterators for a hash map written in OCaml.They do this by working in the separation logic CFML [2], utilizing Coq's powerful but manual reasoning mechanisms for theorem proving.While Pottier does not provide a general specification of iterators (cascades) with mutable state, CFML should permit it, though usage may require a challenging proof.
Future Work.While we have specified and proved key iterators, many more remain.The filter adapter is interesting as each call to next may make an unbounded number of steps with the underlying iterator using the provided mutable closure.Rust provides a hierarchy of traits that further refine iterators like DoubleEndedIterator, and ExactSizeIterator.The recent integration of generic associated types enables new, more flexible forms of iteration like lending iterators.We believe these would naturally integrate into our framework, but remain to be done.Finally, while we believe we have developed a correct, and simple approach to specify closures, the ergonomics leave much room for improvement.Improving this will help make our specifications more concise and user-friendly.In particular, we would like to explore automatic inference of pre-and postconditions of simple closures.

trait
Iterator { type Item; fn next(&mut self) -> Option<Self::Item>; } This trait describes the interface an iterator should implement: an iterator should give a type Item of generated elements, and should implement a method next which optionally returns the next generated element, and possibly mutates in place the internal state of the iterator through the mutable reference &mut self.