Claus-Peter Wirth

Proof Development with OMEGA

The Ωmega proof development system [2] is the core of several related and well integrated research projects of the Ωmega research group.

Descente Infinie + Deduction

Inductive theorem proving in the form of descente infinie was known to the ancient Greeks and is the standard induction method of a working mathematician since it was reinvented in the middle of the 17th century. We present an integration of descente infinie into state-of-the-art free-variable sequent and tableau calculi. It is well-suited for an efficient interplay of human interaction and automation and combines raising, explicit representation of dependence between variables, the liberalized δ-rule, preservation of solutions, and unrestricted applicability of lemmas and induction hypotheses. The semantical requirements are satisfied for a variety of two-valued logics, such as clausal logic, classical first-order logic, and higher-order modal logic.

A generic modular data structure for proof attempts alternating on ideas and granularity

A practically useful mathematical assistant system requires the sophisticated combination of interaction and automation. Central in such a system is the proof data structure, which has to maintain the current proof state and which has to allow the flexible interplay of various components including the human user. We describe a parameterized proof data structure for the management of proofs, which includes our experience with the development of two proof assistants. It supports and bridges the gap between abstract level proof explanation and low-level proof verification. The proof data structure enables, in particular, the flexible handling of lemmas, the maintenance of different proof alternatives, and the representation of different granularities of proof attempts.

On Notions of Inductive Validity for First-Oder Equational Clauses

We define and discuss various conceivable notions of inductive validity for first-order equational clauses. This is done within the framework of constructor-based positive/negative conditional equational specifications which permits to treat negation and incomplete function definitions in an adequate and natural fashion. Moreover, we show that under some reasonable assumptions all these notions are monotonic w. r. t. consistent extension, in contrast to the case of inductive validity for initial semantics (of unconditional or positive conditional equations). In particular from a practical point of view, this monotonicity property is crucial since it allows for an incremental construction process of complex specifications where consistent extensions of specifications cannot destroy the validity of (already proved) inductive properties. Finally we show how various notions of inductive validity in the literature fit in or are related to our classification.

How to Prove Inductive Theorems? QuodLibet!

QuodLibet is a tactic-based inductive theorem proving system that meets today’s standard requirements for theorem provers such as a command interpreter, a sophisticated graphical user interface, and a carefully programmed inference machine kernel that guarantees soundness. In essence, it is the synergetic combination of the features presented in the following sections that makes QuodLibet a system quite useful in practice; and we hope that it is actuallyas you like it, which is the Latin “quod libet” translated into English. We start by presenting some of the design goals that have guided the development of QuodLibet. Note that the system is not intended to pursue the push bottom technology for inductive theorem proving, but to manage more complicated proofs by an effective interplay between interaction and automation.

Conditional Equational Specifications of Data Types with Partial Operations for Inductive Theorem Proving

We propose a specification language for the formalization of data types with partial or non-terminating operations as part of a rewrite-based framework for inductive theorem proving. The language requires constructors for designating data items and admits positive/negative conditional equations as axioms in specifications. The (total algebra) semantics for such specifications is based on so-called data models. We develop admissibility conditions that guarantee the unique existence of a distinguished data model. Since admissibility of a specification requires confluence of the induced rewrite relation, we provide an effectively testable confluence criterion which does not presuppose termination.

History and Future of Implicit and Inductionless Induction: Beware the Old Jade and the Zombie!

In this survey on implicit induction I recollect some memories on the history of implicit induction as it is relevant for future research on computer-assisted theorem proving, esp. memories that significantly differ from the presentation in a recent handbook article on “inductionless induction”. Moreover, the important references excluded there are provided here. In order to clear the fog a little, there is a short introduction to inductive theorem proving and a discussion of connotations of implicit induction like “descente infinie”, “inductionless induction”, “proof by consistency”, implicit induction orderings (term orderings), and refutational completeness.

Shallow confluence of conditional term rewriting systems

Recursion can be conveniently modeled with left-linear positive/negative-conditional term rewriting systems, provided that non-termination, non-trivial critical overlaps, non-right-stability, non-normality, and extra variables are admitted. For such systems we present novel sufficient criteria for shallow confluence and arrive at the first decidable confluence criterion admitting non-trivial critical overlaps. To this end, we restrict the introduction of extra variables of right-hand sides to binding equations and require the critical pairs to have somehow complementary literals in their conditions. Shallow confluence implies [level] confluence, has applications in functional logic programming, and guarantees the object-level consistency of the underlying data types in the inductive theorem prover QuodLibet.

HERBRAND’s Fundamental Theorem in the Eyes of JEAN VAN HEIJENOORT

Using Heijenoort’s unpublished generalized rules of quantification, we discuss the proof of Herbrand’s Fundamental Theorem in the form of Heijenoort’s correction of Herbrand’s “False Lemma” and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand’s Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it “the central theorem of predicate logic” and considered the form of its expression to be “concise and felicitous”.

lim+, δ + , and Non-Permutability of β-Steps

Using a human-oriented formal example proof of the lim+-theorem (that the sum of limits is the limit of the sum), we exhibit a non-permutability of @b-steps and @d^+-steps (according to Smullyans classification), which is not visible with non-liberalized @d-rules and dissolves into a problem of mere inefficiency with further liberalized @d-rules, such as the @d^+^+-rules. Beside a careful presentation of the human-oriented search for a formal proof of (lim+), our main intention is to show where sequent and tableau calculi are in conflict with human-oriented proof construction.