17.1 Description of Omega
Omega solves a goal in Presburger arithmetic, ie a universally quantified formula made of equations and inequations. Equations may be specified either on the typenat of natural numbers or on
the type Z of binary-encoded integer numbers. Formulas on
nat are automatically injected into Z. The procedure
may use any hypothesis of the current proof session to solve the goal.Multiplication is handled by Omega but only goals where at least one of the two multiplicands of products is a constant are solvable. This is the restriction meaned by ``Presburger arithmetic''.
If the tactic cannot solve the goal, it fails with an error message. In any case, the computation eventually stops.
17.1.1 Arithmetical goals recognized by Omega
Omega applied only to quantifier-free formulas built from the connectors
/\, \/, ~, ->
on atomic formulas. Atomic formulas are built from the predicates
=, le, lt, gt, ge
on
nat or from the predicates
=, <, <=, >, >=
on
Z. In expressions of type nat, Omega recognizes
plus, minus, mult, pred, S, O
and in expressions of type
Z, Omega recognizes
+,_,*, Zs, and constants.
All expressions of type
nat or Z not built on these
operators are considered abstractly as if they
were arbitrary variables of type nat or Z.17.1.2 Messages from Omega
When Omega does not solve the goal, one of the following errors is generated:Error messages:
-
Omega can't solve this system
This may happen if your goal is not quantifier-free (if it is universally quantified, try Intros first; if it contains existentials quantifiers too, Omega is not strong enough to solve your goal). This may happen also if your goal contains arithmetical operators unknown from Omega. Finally, your goal may be really wrong !
- Omega: Not a quantifier-free goal
If your goal is universally quantified, you should first apply Intro as many time as needed.
- Omega: Unrecognized predicate or connective:ident
- Omega: Unrecognized atomic proposition:prop
- Omega: Can't solve a goal with proposition variables
- Omega: Unrecognized proposition
- Omega: Can't solve a goal with non-linear products
- Omega: Can't solve a goal with equality on type
17.1.3 Control over the output
There are some flags that can be set to get more information on the procedure-
Timeto get the time used by the procedure Systemto visualize the normalized systems.Actionto visualize the actions performed by the OMEGA procedure (see 17.3).
Use Set Omega flag to set the flag flag. Use Unset Omega flag to unset it and Switch Omega flag to toggle it.