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The Type Theory of Lean [Thesis]

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Author(s): Mario Carneiro
Year: 2019

Language: English

Introduction
Type theory in programming languages
Set theoretic models of type theory
The axioms
Typing
Definitional equality
Reduction
let binders ( reduction)
Definitions ( reduction)
Inductive types
Inductive specifications
Large elimination
The recursor
The computation rule ( reduction)
Non-primitive axioms
Quotient types
Propositional extensionality
Axiom of choice
Differences from Coq
Properties of the type system
Undecidability of definitional equality
Algorithmic equality is not transitive
Failure of subject reduction
Regularity
Unique typing
The reduction
The Church-Rosser theorem
Reduction of inductive types to W-types
The menagerie
Translating type families
Translating subsingleton eliminators
The remainder
Soundness
Proof splitting
Modeling Lean in ZFC
Definition of W-types in ZFC
Definition of acc in ZFC
Soundness
Type injectivity