LP, the Larch Prover — Sample proof: useful kinds of axioms


The axioms in \dflink{set1.lp} fall into several categories:

Induction rules
The first axiom, sort Set generated by {}, insert, asserts that all elements of sort S can be obtained by finitely many applications of insert to {}. It provides the basis for defintions and proofs by induction.

Explicit definitions
The second axiom, {e} = insert(e, {}), is a single \llink{formula} that defines the operator {__} (as a constructor for a singleton set).

Inductive definitions
The next two pairs of axioms provide induction definitions of the membership operator \in and the subset operator \subseteq. Inductive definitions generally consist of one formula per generator.

Implicit definitions
The formula involving the union operator (\union), together with the other axioms, completely constrains the interpretation of that operator.

Constraining properties
The final axiom (that of the extensionality principle) expresses the fact that if two sets have the same elements, then they must be the same set.