heap_lang.md

HeapLang overview

HeapLang is the example language that gets shipped with Iris. It is not the only language you can reason about with Iris, but meant as a reasonable demo language for simple examples.

Language

HeapLang is a lambda-calculus with operations to allocate individual locations, load, store, CmpXchg (compare-and-exchange) and FAA (fetch-and-add). Moreover, it has a fork construct to spawn new threads. In terms of values, we have integers, booleans, unit, heap locations, as well as (binary) sums and products.

Recursive functions are the only binders, so the sum elimination (Case) expects both branches to be of function type and passes them the data component of the sum.

For technical reasons, the only terms that are considered values are those that begin with the Val expression former. This means that, for example, Pair (Val a) (Val b) is not a value -- it reduces to Val (PairV a b), which is. This leads to some administrative redexes, and to a distinction between "value pairs", "value sums", "value closures" and their "expression" counterparts.

However, this also makes values syntactically uniform, which we exploit in the definition of substitution which just skips over Val terms, because values should be closed and hence not affected by substitution. As a consequence, we can entirely avoid even talking about "closed terms", that notion just does not have to come up anywhere. We also exploit this when writing specifications, because we can just write lemmas involving terms of the form Val ?v and Coq can determine ?v by unification (because all values start with the Val constructor).

Notation

Notation for writing HeapLang terms is defined in notation.v. There are two scopes, %E for expressions and %V for values. For example, (a, b)%E is an expression pair and (a, b)%V a value pair. The e of a WP e {{ Q }} is implicitly in %E scope.

We define a whole lot of short-hands, such as non-recursive functions (λ:), let-bindings, sequential composition, and a more conventional match: that has binders in both branches.

The widely used # is a short-hand to turn a basic literal (an integer, a location, a boolean literal or a unit value) into a value. Since values coerce to expressions, # is widely used whenever a Coq value needs to be placed into a HeapLang term.

Tactics

HeapLang comes with a bunch of tactics that facilitate stepping through HeapLang programs as part of proving a weakest precondition. All of these tactics assume that the current goal is of the shape WP e @ E {{ Q }}.

Tactics to take one or more pure program steps:

• wp_pure: Perform one pure reduction step. Pure steps are defined by the PureExec typeclass and include beta reduction, projections, constructors, as well as unary and binary arithmetic operators.
• wp_pures: Perform as many pure reduction steps as possible. This tactic will not reduce lambdas/recs that are hidden behind a definition. If the computation reaches a value, the WP will be entirely removed and the postcondition becomes the new goal.
• wp_rec, wp_lam: Perform a beta reduction. Unlike wp_pure, this will also reduce lambdas that are hidden behind a definition.
• wp_let, wp_seq: Reduce a let-binding or a sequential composition.
• wp_proj: Reduce a projection.
• wp_if_true, wp_if_false, wp_if: Reduce a conditional expression. The discriminant must already be true or false.
• wp_unop, wp_binop, wp_op: Reduce a unary, binary or either kind of arithmetic operator.
• wp_case, wp_match: Reduce Case/match: constructs.
• wp_inj, wp_pair, wp_closure: Reduce constructors that turn expression sums/pairs/closures into their value counterpart.

Tactics for the heap:

• wp_alloc l as "H": Reduce an allocation instruction and call the new location l (in the Coq context) and the points-to assertion H (in the spatial context). You can leave out the as "H" to introduce it as an anonymous assertion, which is equivalent to as "?".
• wp_load: Reduce a load operation. This automatically finds the points-to assertion in the spatial context, and fails if it cannot be found.
• wp_store: Reduce a store operation. This automatically finds the points-to assertion in the spatial context, and fails if it cannot be found.
• wp_cmpxchg_suc, wp_cmpxchg_fail: Reduce a succeeding/failing CmpXchg. This automatically finds the points-to assertion. It also automatically tries to solve the (in)equality to show that the CmpXchg succeeds/fails, and opens a new goal if it cannot prove this goal.
• wp_cmpxchg as H1 | H2: Reduce a CmpXchg, performing a case distinction over whether it succeeds or fails. This automatically finds the points-to assertion. The proof of equality in the first new subgoal will be called H1, and the proof of the inequality in the second new subgoal will be called H2.
• wp_faa: Reduce a FAA. This automatically finds the points-to assertion.

Further tactics:

• wp_bind pat: Apply the bind rule to "focus" the term matching pat. For example, wp_bind (!_)%E focuses a load operation. This is useful in particular when accessing invariants, which is only possible when the WP in the goal is for a single, atomic operation -- wp_bind can be used to bring the goal into the right shape.
• wp_apply pm_trm: Apply a lemma whose conclusion is a WP, automatically applying wp_bind as needed. See the ProofMode docs for an explanation of pm_trm.

There is no tactic for Fork, just do wp_apply wp_fork.

To verify a recursive function, use iLöb. Make sure you do wp_pures before running iLöb; otherwise the induction hypothesis will likely not be applicable when you need it. (This makes sure that all administrative redexes are reduced in your induction hypothesis, just like we state our WP specifications with all of the redexes reduced.)

Notation and lemmas for derived notions involving a thunk

Sometimes, it is useful to define a derived notion in HeapLang that involves thunks. For example, the parallel composition e1 ||| e2 is definable in HeapLang, but that requires thunking e1 and e2 before passing them to par. (This is defined in par.v.) However, this is somewhat subtle because of the distinction between expression lambdas and value lambdas.

The normal e1 ||| e2 notation uses expression lambdas, because clearly we want e1 and e2 to behave normal under substitution (which they would not in a value lambda). However, the specification for parallel composition should use value lambdas, because prior to applying it the term will be reduced as much as possible to achieve a normal form. To facilitate this, we define a copy of the e1 ||| e2 notation in the value scope that uses value lambdas. This is not actually a value, but we still put it in the value scope to differentiate from the other notation that uses expression lambdas. (In the future, we might decide to add a separate scope for this.) Then, we write the canonical specification using the notation in the value scope.

This works very well for non-recursive notions. For while loops, the situation is unfortunately more complex and proving the desired specification will likely be more involved than expected, see this discussion.