Contents 1 Introduction to type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 1.1 Early versions of type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 1.2 Type theory with -notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 1.3 The Axiom of Choice and Skolemization . . . . . . . . . . . . . . . . . . . . . . 973 1.4 The expressiveness of type theory . . . . . . . . . . . . . . . . . . . . . . . . . 975 1.5 Set theory as an alternative to type theory . . . . . . . . . . . . . . . . . . . . 976 2 Metatheoretical foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 2.1 The Unifying Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 2.2 Expansion proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 2.3 Proof translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982 2.4 Higher-order uni cation . . . . . . . . . . . .

Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higher-order logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and those resulting from instantiations. An expansion tree is called an expansion tree proof (ET-proof) if it encodes a tautology, and, in the form not using skolem functions, an “imbedding ” relation among the critical variables be acyclic. The relative completeness result for expansion tree proofs not using skolem functions, i.e. if A is provable in higher-order logic then A has such an expansion tree proof, is based on Andrews ’ formulation of Takahashi’s proof of the cut-elimination theorem for higher-order logic. If the occurrences of skolem functions in instantiation terms are restricted appropriately, the use of skolem functions in place of critical variables is equivalent to the requirement that the imbedding relation is acyclic. This fact not only resolves the open question of what

Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level , where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domain-specific inference rules as well. Furthermore, these rules can be represented compactly in a tre structure. Finally, we describe a system called PROVERB , which substantially sh...

Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not easily separable. In this contribution we advocate an integration of computer algebra into mechanised reasoning systems at the proof plan level. This approach allows to view the computer algebra algorithms as methods, that is, declarative representations of the problem solving knowledge speci c to a certain mathematical domain. Automation can be achieved in many cases bysearching for a hierarchic proof plan at the methodlevel using suitable domain-speci c control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebra system produces high-level protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expanded to proofs at di erent levels of abstraction, so the approach iswell-suited for producing a high-level verbalised explication as well as for a low-level machine checkable calculus-level proof. We present an implementation of our ideas and exemplify them using an automatically solved example. Changes in the criterion of `rigour of the proof ' engender major revolutions in mathematics.

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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, non-analytic proofs are represented by natural deductions. A non-deterministic translation algorithm between expansion proofs and H-deductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cut-elimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higher-order system, but is proven for the first-order fragment. We extend the translations to a non-analytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a non-extensional equality is definable in our system of type theory. Next we extend analytic and non-analytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a

We report on an experiment in exploring properties of residue classes over the integers with the combined effort of a multi-strategy proof planner and two computer algebra systems. An exploration module classifies a given set and a given operation in terms of the algebraic structure they form. It then calls the proof planner to prove or refute simple properties of the operation. Moreover, we use different proof planning strategies to implement various proving techniques: from naive testing of all possible cases to elaborate techniques of equational reasoning and reduction to known cases.

This paper concerns a knowledge structure called method , within a computational model for human oriented deduction. With human oriented theorem proving cast as an interleaving process of planning and verification, the body of all methods reflects the reasoning repertoire of a reasoning system. While we adopt the general structure of methods introduced by Alan Bundy, we make an essential advancement in that we strictly separate the declarative knowledge from the procedural knowledge. This is achieved by postulating some standard types of knowledge we have identified, such as inference rules, assertions, and proof schemata, together with corresponding knowledge interpreters. Our approach in effect changes the way deductive knowledge is encoded: A new compound declarative knowledge structure, the proof schema, takes the place of complicated procedures for modeling specific proof strategies. This change of paradigm not only leads to representations easier to understand, it also enables us...

In this article we formally describe a declarative approach for encoding plan operators in proof planning, the so-called methods. The notion of method evolves from the much studied concept tactic and was first used by Bundy. While significant deductive power has been achieved with the planning approach towards automated deduction, the procedural character of the tactic part of methods, however, hinders mechanical modification. Although the strength of a proof planning system largely depends on powerful general procedures which solve a large class of problems, mechanical or even automated modification of methods is nevertheless necessary for at least two reasons. Firstly methods designed for a specific type of problem will never be general enough. For instance, it is very difficult to encode a general method which solves all problems a human mathematician might intuitively consider as a case of homomorphy. Secondly the cognitive ability of adapting existing methods to suit novel situa...

Abstract. We present a uniform algorithm for transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fitting’s prefixed tableaus we first develop a procedure for extracting a prefixed sequent proof from a given matrix proof. By considering the additional restrictions on the order of rule applications we then extend this procedure into an algorithm which generates a conventional sequent proof. Our algorithm is based on unified representations of matrix characterizations for various logics as well as of prefixed and usual sequent calculi. The peculiarities of a logic are encoded by certain parameters which are summarized in tables to be consulted by the algorithm. 1