Low* core libraries

Low* is made up of a few primitive libraries that enjoy first-class support in KaRaMeL. These core libraries are typically made up of a model (an .fst file) and an interface (an .fsti file). Verification is performed against the model, but at extraction-time, the model is replaced with primitive C constructs.

The memory model

Beyond the language subset, one defining component of Low* is how it models the C memory.

The F* HyperHeap model

F* is by default equipped with HyperHeap, a hierarchical memory model that divides the heap into a tree of regions. This coarse-grained separation allows the programmer to state modifies clauses at the level of regions, rather than on individual references.

The HyperHeap memory model is described in the 2016 POPL paper, as well as the F* tutorial. We assume that the reader has a passing degree of familiarity with HyperHeap.

The HyperStack model

Low* refines the HyperHeap memory model, adding a distinguished set of regions that model the C call stack. Programs may use stack allocation, heap allocation or both. The HyperStack memory model offers a set of effects that capture the allocation behavior of functions.

The HyperStack memory model comprises the files FStar.Monotonic.HyperStack.fst, FStar.HyperStack.fst and FStar.HyperStack.ST.fst in the ulib directory of F*.


Many verification errors point to definitions in these three files. Being familiar with these modules, their combinators and key concepts helps understand why a given program fails to verify.


We recommend always defining the ST abbreviation at the beginning of your module, in order to shadow the FStar.ST module, which is not Low*.

module ST = FStar.HyperStack.ST
module HS = FStar.HyperStack

The top-level region is the root, and is always a valid region. HS.rid is the type of regions.

let root: HS.rid = HS.roo

Stack frames are modeled as distinguished regions that satisfy the is_stack_region predicate. Allocating in a stack frame, unsurprisingly, results in a stack-allocated variable or array in C. Stack frames may be de-allocated as program execution progresses up the call stack, meaning that the underlying HyperHeap region may disappear.

Regions that are not stack frames may not be de-allocated, and therefore satisfy the is_eternal_region predicate. This includes the root. Allocating in one of these regions amounts to performing a heap allocation in C.

Pushing a new stack frame amount to allocating a new stack region. In the HyperHeap model, creating a new region requires a parent. Thus, when a new stack frame is allocated, its parent is either the top-most stack frame, or the root if no stack frame has been allocated so far.


The root is not a stack region and does not satisfy is_stack_region.

let _ =
  HS.root_is_not_freeable ();
  assert (ST.is_eternal_region root /\ ~ (Monotonic.HyperStack.is_stack_region root))

The most popular effect is the Stack effect, which takes:

  • a precondition over the initial heap, of type HS.mem -> Type, and a
  • post-condition over the initial heap, the result, the final heap, of type HS.mem -> a -> HS.mem -> Type
effect Stack (a:Type) (pre: ST.st_pre) (post: (m0: HS.mem -> Tot (ST.st_post' a (pre m0)))) =
    (fun (p: ST.st_post a) (h: HS.mem) ->
      pre h /\ (forall a h1. (pre h /\ post h a h1 /\ ST.equal_domains h h1) ==> p a h1))

The relevant bit in this otherwise mundane definition is the ST.equal_domains predicate.

let equal_domains (m0 m1: HS.mem) =
  (HS.get_tip m0) == (HS.get_tip m1) /\
  Set.equal (Map.domain (HS.get_hmap m0)) (Map.domain (HS.get_hmap m1)) /\
  ST.same_refs_in_all_regions m0 m1

The equal_domains predicate states that a function in the Stack effect:

  • preserves the tip of the memory, i.e. calling this function leaves the C call stack intact;
  • does not allocate any new region on the heap, i.e. this is a C function that does not heap-allocate;
  • does not allocate in any existing region, i.e. this is a C function that does not grow any existing stack frame on the call stack.

A function that satisfies these conditions is a function that can be safely compiled as a C function. In other words, using the native C call stack is a valid implementation of our model.

let f (x: UInt32.t): Stack UInt32.t (fun _ -> True) (fun _ _ _ -> True) =
  FStar.UInt32.( x *%^ x )

Based on the knowledge above, consider the following failing function.

[@ expect_failure ]
let g (): Stack unit (fun _ -> True) (fun _ _ _ -> True) =
  let b = B.alloca 0ul 8ul in

F* reports an assertion failure for the is_stack_region predicate. Indeed, the alloca function requires that the tip be a valid stack region, which is false when no stack frame has been pushed on the call stack.

One important insight at this stage is that F* does not “automatically” enrich the verification context with the assumption that upon entering g, we have pushed a new stack frame. This would be the wrong thing to do for a total function; furthermore, there is simply no such support in the language.

Rather, the user is expected to manually indicate which operations need to conceptually happen in a new stack frame. The Low* memory model provides two combinators for this purpose: push_frame and pop_frame. The f function did not need them, because it performed no stateful operation.

We can attempt to fix g by adding a call to push_frame.

[@ expect_failure ]
let g2 (): Stack unit (fun _ -> True) (fun _ _ _ -> True) =
  push_frame ();
  let b = B.alloca 0ul 8ul in

F* now reports an error for the equal_domains predicate above. Indeed, the only way to leave the C call stack intact, and therefore satisfy the requirements of the Stack effect, is to ensure we pop the stack frame we just pushed.

let g3 (): Stack unit (fun _ -> True) (fun _ _ _ -> True) =
  push_frame ();
  let b = B.alloca 0ul 8ul in
  pop_frame ();

g3 now successfully compiles to C:

void g3()
  uint32_t b[8U] = { 0U };

The Stack effect prevents heap allocation, hence ensuring that from the caller’s perspective, any heap (“eternal”) regions remain unchanged.

For code that performs heap allocations, the libraries offer the ST effect. It is similar to the Stack effect, and takes the same form of pre- and post-conditions, but allows heap allocation.

let g4 (): ST unit (fun _ -> True) (fun _ _ _ -> True) =
  push_frame ();
  let b = B.malloc HS.root 0ul 8ul in
  pop_frame ();

The St effect might occasionally be convenient.

effect St (a:Type) = ST a (fun _ -> True) (fun _ _ _ -> True)

One can reflect the memory as an HS.mem at any program point, by using ST.get ().

let test_st_get (): St unit =
  push_frame ();
  let m = ST.get () in
  assert (Monotonic.HyperStack.is_stack_region (HS.get_tip m));
  pop_frame ()

These are the basic building blocks of our memory model. Verifying on top of this memory model involves reflecting the state of the memory at the proof level, using the HS.mem type, and capturing the effect of allocations, updates and de-allocations using suitable pre- and post-conditions. This can be done using a combination of modifies clauses and libraries that reflect low-level constructs, such as buffers and machine integers, at the proof level. All of these are covered in the rest of this chapter.

Advanced: the StackInline effect


Machine integers

Machine integers are modeled as natural numbers that fit within a certain number of bits. This model is dropped by KaRaMeL, in favor of C’s fixed-width types.

Fixed-width integers are found in FStar.UInt{16,32,64,128}.fst and FStar.Int{16,32,64,128}. The FStar.Int.Cast.Full.fst module offers conversion functions between these integer types.


By default, KaRaMeL relies on the non-standard unsigned __int128 C type to implement FStar.UInt128.t. This type is widely supported across GCC and Clang versions, but not by the Microsoft compilers. If you need 128-bit unsigned integers, consider reading krmllib/README.md, which offers both an MSVC-specific alternative, and a portable, albeit slower, implementation.

Machine integers offer the classic set of arithmetic operations. Like in C, unsigned integers have wraparound overflow semantics, exposed via the add_mod function. Signed integers offer no such function. Other undefined behaviors of C are ruled out at the F* level, such as shifting an integer by the bit width.


In addition to classic arithmetic operations, some modules offer constant-time operations such as eq_mask and gte_mask, which allow defining a “secret integer” module on top of these integers, that offers no comparison operator returning a boolean, to avoid timing leaks. See the HACL* libraries for secret integers.

Machine integers modules also define operators, suffixed with ^. For instance, the + operation for UInt32 is +^. Wraparound variants have an extra % character, such as +%^, when available.


The unary minus is broken for machine integers. This does not parse: let x = UInt32.(-^ 0ul)

Operators follow the standard precedence rules of F*, which are outlined on its wiki. Operators are resolved in the current scope; we recommend the use of module abbreviations and the let-open notation M.( ... ).

module U32 = FStar.UInt32

let z = U32.(16ul -^ 8ul )


By default, operations require that the caller prove that the result fits in the given integer width. For instance, U32.add has (requires (size (v a + v b) n)) as a precondition. Look at U32.add_modulo for no precondition.

Machine integers can be reflected as natural numbers of type nat using the v function. It is generally more convenient to perform proofs on natural numbers.

let test_v (): unit =
  let x = 0ul in
  assert (U32.v x = 0)

The buffer library

LowStar.Buffer is the workhorse of Low*, and allows modeling C arrays on the stack and in the heap. LowStar.Buffer models C arrays as follows:

let lseq (a: Type) (l: nat) : Type =
  (s: Seq.seq a { Seq.length s == l } )

type buffer (a:Type) =
  | MkBuffer: max_length:UInt32.t
    -> content:reference (lseq a (U32.v max_length))
    -> idx:UInt32.t
    -> length:UInt32.t{U32.(v idx + v length <= v max_length)}
    -> buffer a

In other words, buffers are modeled as a reference to a sequence, along with a starting index idx, and a length, which captures how much of an allocation slice one is currently pointing to.

This is a model: at compilation-time, KaRaMeL implements buffers using C arrays.

The length is available in ghost (proof) code only: just like in C, one cannot compute the length of a buffer at run-time. Therefore, a typical pattern is to use refinements to tie together a buffer and its length, as we saw with the initial memcpy example.

let do_something (x: B.buffer UInt64.t) (l: U32.t { U32.v l = B.length x }): St unit =

Allocating a buffer on the stack is done using the alloca function, which takes an initial value and a length. alloca requires that the top of the stack be a valid stack frame.

let test_alloc_stack (): Stack unit (fun _ -> True) (fun _ _ _ -> True) =
  push_frame ();
  let b = B.alloca 0UL 8ul in
  pop_frame ();

Allocating a buffer on the heap is done using the malloc function, which takes a region, an initial value and a length. The region is purely for proof and separation purposes, and has no effect on the generated code. A buffer created with malloc can be freed with free.

let test_alloc (): St unit =
  let b = B.malloc HS.root 0UL 8ul in
  B.free b

Pointer arithmetic is performed by the means of the sub function. Under the hood, the sub function returns a buffer that points to the same underlying reference, but has different idx and length fields.

let test_sub (): St unit =
  let b = B.malloc HS.root 0UL 8ul in
  let b_l = B.sub b 0ul 4ul in // idx = 0; length = 4
  let b_r = B.sub b 4ul 4ul in // idx = 4; length = 4
  B.free b

Just like in C, one can only free the base pointer, i.e. this is an error:

[@ expect_failure ]
let test_sub_error (): St unit =
  let b = B.malloc HS.root 0UL 8ul in
  let b_l = B.sub b 0ul 4ul in // idx = 0; length = 4
  B.free b_l

Reading and modifying a buffer is performed by means of the index and upd functions. These are exposed as the .() and .()<- operators respectively, defined in LowStar.BufferOps. This latter module module only contains those operators, and is meant to be used with open to bring operators into scope without further polluting the context with any definition from LowStar.Buffer.

let test_index (): St unit =
  let b = B.malloc HS.root 0UL 8ul in
  b.(0ul) <- UInt64.add_mod b.(0ul) b.(0ul);
  B.free b

Buffers are reflected at the proof level using sequences, via the as_seq function, which returns the contents of a given buffer in a given heap, i.e. a sequence slice ranging over the interval [idx; idx + length).

let test_as_seq (): St unit =
  let b = B.malloc HS.root 0UL 1ul in
  let h = ST.get () in
  assert (Seq.equal (B.as_seq h b) (Seq.cons 0UL Seq.createEmpty));
  B.free b

B.get is an often-convenient shorthand to index the value of a given buffer in a given heap.

let test_get (): St unit =
  let b = B.malloc HS.root 0UL 1ul in
  let h = ST.get () in
  assert (B.get h b 0 = 0UL);
  B.free b

C NULL pointers

LowStar.Buffer also exposes a model of the C NULL pointer, null – this is what you should use if you need zero-length buffers. The NULL pointer is always live, and always has length 0. The pointer and pointer_or_null functions define convenient aliases, while the (!*) operator (defined in LowStar.BufferOps) guarantees that the dereference will be pretty-printed with a * C dereference, as opposed to an access at array index 0. Pointers can always be tested for nullity via the is_null p function, which is guaranteed to be pretty-printed as p != NULL.

The modifies clauses library

The current heap model of F* is based on a select-update theory: the heap is reflected as a map, allocation adds a key in the map, assignment updates the map, and reading selects from the map.

Proving properties of programs therefore requires the programmer to reason about the heap model. However, stating precise post-conditions that refer to a particular heap after a particular update does not scale up to large programs: we want to reason abstractly about modifications, and use a library of composable predicates that allow one to generically reason about a given modification to the heap.

This is where the LowStar.Modifies library comes in handy. The modifies clauses library allows one to reason about allocation, de-allocation, modifications using a single unified modifies clause. An abstract notion of a memory location allows composing predicates, and deriving properties such as: “if I modify a location l1 disjoint from l2, then the contents of the memory at address l2 remain unchanged”.

Abstract memory locations

The LowStar.Modifies library abstracts over memory locations. Memory locations have type loc. Locations form a monoid, where loc_none is the empty location and loc_union combines two location to form the union of the two.

Several injections exist to create locations; for now, we will mostly use loc_buffer, which injects a LowStar.Buffer.t into an abstract location.

Inclusion and disjointness

The LowStar.Modifies module provides an inclusion relation, via loc_includes. This allows the programmer to state, for instance, that the location of a stack-allocated buffer is included in its stack frame.

Perhaps more useful is the loc_disjoint predicates, which allows the programmer to state that two memory locations do not overlap.

The modifies clause

The modifies clause is of the form modifies l h0 h1 where l is an abstract memory location, h0 is the initial heap and h1 is the resulting heap. Here is an example:

module M = LowStar.Modifies

let example_modifies_callee (b1 b2: B.buffer UInt32.t) : Stack unit
  (requires (fun h -> B.live h b1 /\ B.live h b2 /\ B.length b1 == 1 /\ B.length b2 == 1 /\ B.disjoint b1 b2))
  (ensures (fun h _ h' ->
    M.modifies (M.loc_union (M.loc_buffer b1) (M.loc_buffer b2)) h h' /\
    B.live h' b1 /\ B.live h' b2 /\
    B.get h' b1 0 == 18ul /\ B.get h' b2 0 == 42ul
= b2.(0ul) <- 42ul;
  b1.(0ul) <- 18ul

The pre- and post-conditions of the example_modifies_callee function state that, if b1 and b2 are two disjoint live buffers of length 1, then example_modifies changes their contents to 18ul and 42ul, respectively. In itself, the modifies clause tells nothing, but it starts becoming useful when the example_modifies_callee function is called by another function:

let example_modifies_caller (b0: B.buffer UInt32.t) : Stack unit
  (requires (fun h -> B.live h b0 /\ B.length b0 == 3))
  (ensures (fun h _ h' ->
    M.modifies (M.loc_buffer b0) h h' /\
    B.live h' b0 /\
    B.get h' b0 0 == B.get h b0 0
= let b1 = B.sub b0 1ul 1ul in
  let b2 = B.sub b0 2ul 1ul in
  example_modifies_callee b1 b2;
  assert (forall h . B.get h b0 0 == B.get h (B.gsub b0 0ul 1ul) 0)

This function takes a buffer b0 of length 3, and from it, extracts two disjoint buffers, b1 and b2, as the sub-buffers of b0 of length 1 at offsets 1 and 2, respectively. Since they are both live and disjoint, they can then be passed to example_modifies_callee. Then, the post-condition of example_modifies_caller about the contents of the cell of b0 at offset 0 is due to the fact that that cell of b0 is disjoint from both b1 and b2 (because it is the cell of the sub-buffer of b0 at offset 0, as suggested by the assert), and so, by virtue of the modifies clause of example_modifies_callee, its value is preserved.