HeapLang.md 6.72 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129
# 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`, `CAS` (compare-and-swap) 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.

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 very 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`](theories/heap_lang/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.

Noteworthy is the fact that functions (`rec:`, `λ:`) in the value scope (`%V`)
are *locked*.  This is to prevent them from being unfolded and reduced too
eagerly.

## Tactics

HeapLang coms with a bunch of tactics that facilitate stepping through HeaLang
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.
- `wp_rec`, `wp_lam`: Perform a beta reduction.  Unlike `wp_pure`, this will
  also reduce locked lambdas.
- `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 assertion that we own it `H` (in the
  spatial context).  You can leave away the `as "H"` to introduce it as an
  anonymous assertion, i.e., that is equivalent to `as "?"`.
- `wp_load`: Reduce a load operation.  This automatically finds the necessary
  ownership in the spatial context, and fails if it cannot be found.
- `wp_store`: Reduce a store operation.  This automatically finds the necessary
  ownership in the spatial context, and fails if it cannot be found.
- `wp_cas_suc`, `wp_cas_fail`: Reduce a succeeding/failing CAS.  This
  automatically finds the necessary ownership.  It also automatically tries to
  solve the (in)equality to show that the CAS succeeds/fails, and opens a new
  goal if it cannot prove this goal.
- `wp_cas as H1 | H2`: Reduce a CAS, performing a case distinction over whether
  it succeeds or fails.  This automatically finds the necessary ownership.  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 necessary ownership.

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](ProofMode.md) for an
  explanation of `pm_trm`.

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

## 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 defineable in
HeapLang, but that requires thunking `e1` and `e2` before passing them to
`par`. (This is defined in [`par.v`](theories/heap_lang/lib/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 but 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].

[discussion]: https://gitlab.mpi-sws.org/iris/iris/merge_requests/210#note_32842