Extensional Constructs in Intensional Type Theory [Paperback]

Extensional Constructs in Intensional Type Theory presents a novel approach to the treatment of equality in Martin-Loef type theory (a basis for important work in mechanised mathematics and program verification). Martin Hofmann attempts to reconcile the two different ways that type theories deal with identity types. The book will be of interest particularly to researchers with mainly theoretical interests and implementors of type theory based proof assistants, and also fourth year undergraduates who will find it useful as part of an advanced course on type theory.Extensional Constructs in Intensional Type Theory presents a novel approach to the treatment of equality in Martin-Loef type theory (a basis for important work in mechanised mathematics and program verification). Martin Hofmann attempts to reconcile the two different ways that type theories deal with identity types. The book will be of interest particularly to researchers with mainly theoretical interests and implementors of type theory based proof assistants, and also fourth year undergraduates who will find it useful as part of an advanced course on type theory.1. Introduction.- 1.1 Definitional and propositional equality.- 1.2 Extensional constructs.- 1.3 Method.- 1.3.1 The use of categorical models.- 1.3.2 Syntactic models.- 1.4 Applications.- 1.4.1 Application to machine-assisted theorem proving.- 1.5 Overview.- 2. Syntax and semantics of dependent types.- 2.1 Syntax for a core calculus.- 2.1.1 Raw syntax.- 2.1.2 Judgements.- 2.1.3 Notation.- 2.1.4 Derived rules and meta-theoretic properties.- 2.2 High-level syntax.- 2.2.1 Telescopes.- 2.2.2 Elements of telescopes and context morphisms.- 2.2.3 Definitions and substitution.- 2.3 Further type formers.- 2.3.1 Unit type.- 2.3.2 ?-types.- 2.3.3 Function and cartesian product types.- 2.3.4 The Calculus of Constructions.- 2.3.5 Universes.- 2.3.6 Quotient types.- 2.4 Abstract semantics of type theory.- 2.4.1 Syntactic catels$