A transient data structure is a package of an ephemeral data structure, a persistent data structure, and fast conversions between them. We describe the specification and proof of a transient stack and its iterators. This data structure is a scaleddown version of the general-purpose transient sequence data structure implemented in the OCaml library Sek. Internally, it relies on fixed-capacity arrays, or chunks, which can be shared between several ephemeral and persistent stacks. Dynamic tests are used to determine whether a chunk can be updated in place or must be copied: a chunk can be updated if it is uniquely owned or if the update is monotonic. Using CFML, which implements Separation Logic with Time Credits inside Coq, we verify the functional correctness and the amortized time complexity of this data structure. Our verification effort covers iterators, which involve direct pointers to internal chunks. The specification of iterators describes what the operations on iterators do, how much they cost, and under what circumstances an iterator is invalidated.

We present a Separation Logic with space credits for reasoning about heap space in a sequential call-by-value lambda-calculus equipped with garbage collection and mutable state. A key challenge is to design sound, modular, lightweight mechanisms for establishing the unreachability of a block. Prior work in this area uses pointed-by assertions to keep track of the predecessors of every block, but is carried out in the setting of an assembly-like programming language. We take up the challenge in the setting of a high-level language, where a key problem is to identify and reason about the memory locations that the garbage collector considers as roots. For this purpose, we propose novel "stackable" assertions, which keep track of the existence of stack-to-heap pointers without explicitly recording their origin. Furthermore, we explain how to reason about closures -- concrete heap-allocated data structures that implement the abstract concept of a first-class function. We demonstrate the expressiveness and tractability of our program logic via a range of examples, including recursive functions on linked lists, objects implemented using closures and mutable internal state, recursive functions in continuation-passing style, and three stack implementations that exhibit different space bounds. These last three examples illustrate reasoning about the reachability of the items stored in a container as well as amortized reasoning about space. All of our results are proved in Coq on top of Iris.