## Providing hints

### Rewriting rules

Internally, Gappa tries to compute the range of mathematical terms. For example, if the tool has to bound the product of two factors, it will check if it knows the ranges of both factors. If it does, it will apply the theorem about real multiplication in order to compute the range of the product.

Unfortunately, there may be some expressions that Gappa cannot bound tightly. This usually happens because it has no result on a sub-term or because the expression is badly correlated. In this case, the user can provide an intermediate expression with the following hint.

primary -> secondary;

If Gappa finds a property about the secondary expression, it will use it as if it was about the primary expression. This transformation is valid as long as both expressions are equal. So, when generating a theorem proof, Gappa adds this equality as a hypothesis. It is then up to the user to prove it in order to be able to apply the theorem.

To detect mistyping early, Gappa checks if both expressions can be normalized to the same one according to field rules (and are therefore equal) and warn if they are not. (It will not generate a proof of their equality though.) Note that Gappa does not check if the divisors that appear in the expressions are always nonzero; it just warns about them.

If an equality requires some conditions to be verified, they can be added to the rule:

sqrt(x * x) -> -x { x <= 0 };

### Approximated expressions

As mentioned before, Gappa has an internal database of rewriting rules. Some of these rules need to know about accurate and approximate terms. Without this information, they do not match any expression. For example, Gappa will replace the term B by b + -(b - B) only it knows a term b that is an approximation of the term B.

Gappa has two heuristics to detect terms that are approximations of other terms. First, rounded values are approximations of their non-rounded counterparts. Second, any absolute or relative error that appears as a hypothesis of a logical sub-formula will define a pair of accurate and approximate terms.

Since these pairs create lots of already proved rewriting rules, it is helpful for the user to define its own pairs. This can be done with the following syntax.

approximate ~ accurate;

In the following example, the comments show two hints that could be added if they had not already been guessed by Gappa. In particular, the second one enables a rewriting rule that completes the proof.

@floor = int<dn>;
{ x - y in [-0.1,0.1] -> floor(x) - y in ? }
# floor(x) ~ x;
# x ~ y;

### Dichotomy search

The last kind of hint can be used when Gappa is unable to prove a formula but would be able to prove it if the hypothesis ranges were not so wide. Such a failure is usually caused by a bad correlation between the sub-terms of the expressions. This can be solved by rewriting the expressions. But the failure can also happen when the proof of the formula is not the same everywhere on the domain, as in the following example. In both cases, the user can ask Gappa to split the ranges into tighter ranges and see if it helps proving the formula.

@rnd = float< ieee_32, ne >;
x = rnd(x_);
y = x - 1;
z = x * (rnd(y) - y);
{ x in [0,3] -> |z| in ? }

With this script, Gappa will answer that $|z| \le 3 \cdot 2^{-24}$. This is not the best answer because Gappa does not notice that rnd(y) - y is always zero when $\frac{1}{2} \le x \le 3$. The user needs to ask for a bisection with respect to the expression rnd(x_).

There are three types of bisection. The first one splits an interval in as many equally wide sub-intervals as asked by the user. The second one splits an interval along the points provided by the user.

$x in 6; # split the range of "x" in six sub-intervals$ x in (0.5,1.9999999); # split the range of "x" in three sub-intervals, the middle one being [0.5,~2]
$x; # equivalent to:$ x in 4;

The third kind of bisection tries to find by dichotomy the best range split such that a goal of the logical formula holds true. This requires the range of this goal to be specified, and the enclosed expression has to be indicated on the left of the $ symbol. { x in [0,3] -> |z| <= 1b-26 } |z|$ x;

Contrarily to the first two kinds of bisection, this third one keeps the range improvements only if the goal was finally reached. If there was a failure in doing so, all the improvements are discarded. Gappa will display the sub-range on which the goal was not proved. There is a failure when Gappa cannot prove the goal on a range and this range cannot be split into two half ranges, either because its internal precision is not enough anymore or because the maximum depth of dichotomy has been reached. This depth can be set with the -Edichotomy=999 option.

More than one bisection hint can be used. And hints of the third kind can try to satisfy more than one goal at once.

a, b, c $u; a, d$ v;

These two hints will be used one after the other. In particular, none will be used when Gappa is doing a bisection with the other one. By adding terms on the right of the $ symbol, more complex hints can be built. Beware of combinatorial explosions though. The following hint is an example of a complex bisection: it asks Gappa to find a set of sub-ranges on u such that the goal on b is satisfied when the range on v is split into three sub-intervals. b$ u, v in 3

Rather than mentioning simple terms on the left-hand side of hints, one can also write a logical formula. As a consequence, Gappa no longer infers the termination condition from the goal but instead performs a bisection until the formula is satisfied. This is especially useful if no enclosures of the terms appear in the goal, or if the termination criteria is not even an expression enclosure.

rnd(a) = a \$ u