# Isabelle¶

Isabelle is an interactive theorem prover. It comes with an IDE where users can type mathematical specifications and proofs which will be checked continuously. Specifications consist of a sequence of type and value definitions. All high-level tools inside Isabelle must justify their proofs and constructions against a small inference kernel. This greatly diminishes the risk of unsound proofs.

Additionally, Isabelle features powerful proof automation, for example:

- classical and equational reasoning
- decision procedures (e.g. for linear arithmetic)
- interfaces to automated provers (e.g. Z3 and CVC4)

However, most non-trivial proofs will require the user to give proof hints. In general, interactive provers like Isabelle trade better guarantees for weaker automation.

Normally, users write Isabelle specifications manually. Leon comes with a bridge to Isabelle which allows it to be used as a solver for program verification.

## Installation¶

You don’t have to obtain a copy of Isabelle. Leon is able to download and
install Isabelle itself. The installation happens in the appropriate folder for
your operating system, e.g. `%APPDATA%` under Windows or `$HOME/.local`
under Linux.

## Basic usage¶

For most purposes, Isabelle behaves like any other solver. When running
*Verification*, you can pass `isabelle` as a solver (see
*Command Line Options* for details). Isabelle is a bit slower to start up than
the other solvers, so please be patient.

## Advanced usage¶

Isabelle has some peculiarities. To accomodate for those, there are a set of
specific annotations in the object `leon.annotation.isabelle`.

### Mapping Leon entities to Isabelle entities¶

Isabelle – or its standard logic *HOL*, to be precise – ships with a rather
large existing library. By default, the Isabelle backend would just translate
all Leon definitions into equivalent Isabelle definitions, which means that
existing proofs about them are not applicable. For example, Isabelle/HOL already
has a concatenation function for lists and an associativity proof for it. The
annotations `@typ`, `@constructor` and `@function` allow the user to
declare a mapping to existing Isabelle entities. For lists and their append
function, this looks like this:

```
@isabelle.typ(name = "List.list")
sealed abstract class List[T] {
@isabelle.function(term = "List.append")
def ++(that: List[T]): List[T] = (this match {
case Nil() => that
case Cons(x, xs) => Cons(x, xs ++ that)
})
}
@isabelle.constructor(name = "List.list.Cons")
case class Cons[T](h: T, t: List[T]) extends List[T]
@isabelle.constructor(name = "List.list.Nil")
case class Nil[T]() extends List[T]
```

The above is a simplified excerpt from Leon’s standard library. However, it is no problem to map functions with postconditions to their Isabelle equivalents.

By default, mapping for types is loosely checked, whereas mapping for functions is checked:

- For types, it is checked that there are exactly as many constructors in Leon as in Isabelle and that all constructors are annotated. It is also checked that the given constructor names actually exist and that they have the same number of arguments, respectively. There is no check whether the field types are equivalent. However, in most cases, such a situation would be caught later because of type errors.
- For functions, a full equivalence proof is being performed. First, the function will be translated to Isabelle as usual. Second, the system will try to prove that both definitions are equivalent. If that proof fails, the system will emit a warning and ignore the mapping.

The checking behaviour can be influenced with the `strict` option (see
*Command Line Options*).

### Scripting source files¶

The `script` annotation allows to embed arbitrary Isabelle text into Leon
source files. In the following example, this is used together with mapping:

```
@isabelle.script(
name = "Safe_Head",
source = """
fun safe_head where
"safe_head [] = None" |
"safe_head (x # _) = Some x"
lemma safe_head_eq_head[simp]:
assumes "~ List.null xs"
shows "safe_head xs = Some (hd xs)"
using assms
by (cases xs) auto
"""
)
@isabelle.function(term = "Safe_Head.safe_head")
def safeHead[A](xs: List[A]): Option[A] = xs match {
case Nil() => None()
case Cons(x, _) => Some(x)
}
```

`script` annotations are processed only for functions which are directly or
indirectly referenced from the source file which is under verification by Leon.
The effect of a script is equivalent to defining an Isabelle theory with the
name and content as specified in the annotation, importing the `Leon` theory.
Theories created via script annotations must be independent of each other, but
are processed before everything else. As a consequence, any entities defined
in scripts are available for all declarations.

Note

Invalid proofs will raise an error, but skipped proofs (with `sorry`) are
not reported.

### Proof hints¶

The system uses a combination of tactics to attempt to prove postconditions of
functions. Should these fail, a custom proof method can be specified with the
`proof` annotation.

```
@isabelle.proof(method = """(induct "<var xs>", auto)""")
def flatMapAssoc[A, B, C](xs: List[A], f: A => List[B], g: B => List[C]) =
(xs.flatMap(f).flatMap(g) == xs.flatMap(x => f(x).flatMap(g))).holds
```

The method string is interpreted as in Isabelle:

```
lemma flatMapAssoc: ...
by (induct "<var xs>", auto)
```

Note

In annotations, the function parameters are not in scope. That means that
referring to the actual Scala variable `xs` is impossible. Additionally,
in Isabelle, `xs` will not be called `xs`, but rather `xs'76` (with
the number being globally unique). To be able to refer to `xs`, the system
provides the special input syntax `<var _>`, which turns an identifier
of a variable into its corresponding variable in Isabelle. Think of it as a
quotation for Scala in Isabelle. There is also a counterpart for constants:
`<const _>`.

The `proof` annotations admits a second argument, `kind`, which specifies a
comma-seperated list of verification conditions it should apply to. The empty
string (default) means all verification conditions.

### Influencing the translation of functions¶

By default, the system will only translate the body of a function, that is,
without pre- and postconditions, to Isabelle. Sometimes, the precondition is
required for termination of the function. Since Isabelle doesn’t accept
function definitions for which it can’t prove termination, the presence of the
precondition is sometimes necessary. This can be achieved by annotating the
function with `@isabelle.fullBody`. If, for other reasons, termination can’t
be proven, the annotation `@isabelle.noBody` leaves the function unspecified:
It can still be called from other functions, but no proofs depending on the
outcome of the functions will succeed.

## Advanced example¶

The following example illustrates the definition of a tail-recursive function.
The challenge when proving correctness for these kinds of functions is that
“simple” induction on the recursive argument is often not sufficient, because
the other arguments change in the recursive calls. Hence, it is prudent to
specify a proof hint. In this example, an induction over the definition of the
`lenTailrec` function proves the goal:

```
def lenTailrec[T](xs: List[T], n: BigInt): BigInt = xs match {
case Nil() => n
case Cons(_, xs) => lenTailrec(xs, 1 + n)
}
@isabelle.proof(method = """(induct "<var xs>" "<var n>" rule: [[leon_induct lenTailrec]], auto)""")
def lemma[A](xs: List[A], n: BigInt) =
(lenTailrec(xs, n) >= n).holds
```

The attribute `[[leon_induct _]]` summons the induction rule for the
specified function.

## Limitations¶

- Mutually-recursive datatypes must be “homogeneous”, that is, they all must have exactly the same type parameters; otherwise, they cannot be translated.
- Recursive functions must have at least one declared parameter.
- Polymorphic recursion is unsupported.
- The
constandleon_inductsyntax take a mangled identifier name, according to the name mangling rules of Scala (and some additional ones). The mangling doesn’t change the name if it only contains alphanumeric characters.- The
constandleon_inductsyntax does not work for a given functionfif there is another functionfdefined anywhere else in the program.