Logical formulas

These sections describe some properties of the logical fragment Gappa manipulates. Notice that this fragment is sound, as the generated formal proofs depend on the support libraries, and these libraries are formally proved by relying only on the axioms of basic arithmetic on real numbers.

Undecidability

First, notice that the equality of two expressions is equivalent to checking that their difference is bounded by zero: e - f in [0,0]. Second, the property that a real number is a natural number can be expressed by the equality between its integer part int<dn>(e) and its absolute value |e|.

Thanks to classical logic, a first-order formula can be written in prenex normal form. Moreover, by skolemizing the formula, existential quantifiers can be removed (although Gappa does not allow the user to type arbitrary functional operators in order to prevent mistyping existing operators, the engine can handle them).

As a consequence, a first-order formula with Peano arithmetic (addition, multiplication, and equality, on natural numbers) can be expressed in Gappa's formalism. It implies that Gappa's logical fragment is not decidable.

Expressiveness

Equality between two expressions can be expressed as a bound on their difference: e - f in [0,0]. For inequalities, the difference can be compared to zero: e - f >= 0. The negation of the previous propositions can also be used. Checking the sign of an expression could also be done with bounds; here are two examples: e - |e| in [0,0] and e in [0,1] \/ 1 / e in [0,1]. Logical negations of these formulas can be used to obtain strict inequalities. They can also be defined by discarding only the zero case: not e in [0,0].

Disclaimer: although these properties can be expressed, it does not mean that Gappa is able to handle them efficiently. Yet, if a proposition is proved to be true by Gappa, then it can be considered to be true even if the previous "features" were used in its expression.