Stretching First Order Equational Logic: Proofs with Partiality, Subtypes and Retracts

It is widely recognized that equational logic is simple, (relatively) decidable, and (relatively) easily mechanized. But it is also widely thought that equational logic has limited applicability because it cannot handle subtypes or partial functions. We show that a modest stretch of equational logic eeectively handles these features. Space limits preclude a full theoretical treatment, so we often sketch, motivate and exemplify. 1 Introduction First order equational logic (EL) has signiicant conceptual, theoretical and computational advantages, to the extent that I suggest it should be used if it can be used for a given application. But there are many applications where ordinary rst order EL does not seem suuciently expressive. This paper describes an extension that greatly expands its expressiveness and applicability, at little cost to its advantages. EL was untyped at birth 3], but later extended to many sorts in various ways, of which 1] was perhaps rst and 5] notationally simplest; extensions to overloaded function symbols and conditional equations were also important; see 10] for technical and historical details. Section 2 quickly reviews many sorted EL, and Section 3 covers the next important extension, order sorted EL 11], including an inductive proof for a typical partial function. A nal section discusses a further extension to hidden EL.