Conservative Definitions for Higher-order Logic with Ad-hoc Overloading

Sammanfattning: With an ever growing dependency on computer systems, the need to guarantee their correct behaviour increases. Mathematically rigorous techniques like formal verification offer a way to derive a system's mathematical properties for example with the help of a theorem prover. A theorem prover is a type of software that assists a user in deriving theorems expressed in a formal language. With a theorem prover one should never be able to prove something that is contradictory, as otherwise the proof effort is worthless. This property is called consistency and is essential for formal developments.Theorem provers enjoy high confidence, since they often rely on a trusted logical kernel that is enriched with new symbols in a controlled way. Instead of asserting the existence of mathematical objects with their desired properties, new symbols are introduced through definitions. These definitions are checked by this kernel, and expected to act as abbreviations. Any theorem that is expressible without a definition should not need that definition in its proof. A definition satisfying this property is called conservative. Conservative definitions are especially important as they preserve consistency, so that a proof of a contradiction is not possible after adding these definitions.Isabelle/HOL is a prominent theorem prover with the unique feature of permitting different definitions of one constant at non-overlapping types. In addition to these so-called overloading definitions, a symbol may be used before it is defined. These features mean that consistency or conservativity arguments for simpler logics do not immediately apply to Isabelle/HOL. In the past, design flaws in the definitional mechanism made theories in the Isabelle/HOL theorem prover inconsistent, e.g. it was possible to derive a contradiction from cyclic definitions.With this thesis we show that (overloading) definitions in higher-order logic (HOL) are conservative, hence not source of inconsistency. We define and prove a notion of conservativity that applies to theories of overloading definitions in higher-order logic and formalise aspects of our results in a theorem prover. As a practical implication, when searching for a proof of a formula, our syntactic conservativity criterion allows to exclude irrelevant symbols. In particular, our results confirm that Isabelle/HOL theories are consistent, thus users cannot introduce contradictions through definitions. As a specialisation of our work, our notion of conservativity holds for variants of HOL without overloading. Overall, our work contributes to the understanding of higher-order logic with overloading definitions.