## 15.3 Technical data

### 15.3.1 Overview of the tactic

- The goal is negated twice and the first negation is introduced as an hypothesis.
- Hypothesis are decomposed in simple equations or inequations. Multiple goals may result from this phase.
- Equations and inequations over `nat` are translated over `Z`, multiple goals may result from the translation of substraction.
- Equations and inequations are normalized.
- Goals are solved by the OMEGA decision procedure.
- The script of the solution is replayed.

### 15.3.2 Overview of the OMEGA decision procedure

The OMEGA decision procedure involved in the `Omega` tactic uses a small subset of the decision procedure presented in

> "The Omega Test: a fast and practical integer programming algorithm for dependence analysis", William Pugh, Communication of the ACM , 1992, p 102-114.

Here is an overview. The reader is refered to the original paper for more information.

- Equations and inequations are normalized by division by the GCD of their coefficients.
- Equations are eliminated, using the Banerjee test to get a coefficient equal to one.
- Note that each inequation defines a half space in the space of real value of the variables.
- Inequations are solved by projecting on the hyperspace defined by cancelling one of the variable. They are partitioned according to the sign of the coefficient of the eliminated variable. Pairs of inequations from different classes define a new edge in the projection.
- Redundant inequations are eliminated or merged in new equations that can be eliminated by the Banerjee test.
- The last two steps are iterated until a contradiction is reached (success) or there is no more variable to eliminate (failure).

It may happen that there is a real solution and no integer one. The last steps of the Omega procedure (dark shadow) are not implemented, so the decision procedure is only partial.