. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Expressions 11 3.1 Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5 Other Commands 37 5.1 Universes and Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Reference Manual, Release 3.3.0 10 Chapter 2. Lexical Structure CHAPTER THREE EXPRESSIONS 3.1 Universes Every type in Lean is, by definition, an expression of type Sort u for some universe level u. A

type theory known as the Calculus of Constructions, with a countable hierarchy of non-cumulative universes and inductive types. By the end of this chapter, you will understand much of what this means. 2 discuss Prop in the next chapter. We want some operations, however, to be polymorphic over type universes. For example, list α should make sense for any type α, no matter which type universe α lives in also just syntactic sugar for Sort (u+1). Prop has some special features, but like the other type universes, it is closed under the arrow constructor: if we have p q : Prop, then p → q : Prop. There are

1 3.9 Cumulativity 3.9.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is 3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing universes. For example, List Set is not a subtype of List Set1. Agda also does Language Reference Agda User Manual, Release 2.6.4.1 3.40 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

modules. 3.9 Cumulativity 3.9.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is 3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing universes. For example, List Set is not a subtype of List Set1. Agda also does Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.40 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

modules. 3.9 Cumulativity 3.9.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is 3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing universes. For example, List Set is not a subtype of List Set1. Agda also does Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.40 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

modules. 3.9 Cumulativity 3.9.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is 3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing universes. For example, List Set is not a subtype of List Set1. Agda also does Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.40 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

modules. 3.9 Cumulativity 3.9.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is 3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing universes. For example, List Set is not a subtype of List Set1. Agda also does Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.39 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

p 3.8 Cumulativity 3.8.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is P = ∀ x → ∀? n (P x) 3.8.3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing uni- verses. For example, List Set is not a subtype Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.37 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

p 3.8 Cumulativity 3.8.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. 64 Chapter 3. Language Reference Agda User Manual, Release 2.6 Manual, Release 2.6.2.2 3.8.3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing uni- verses. For example, List Set is not a subtype Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.38 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and

p 3.8 Cumulativity 3.8.1 Basics Since version 2.6.1, Agda supports optional cumulativity of universes under the --cumulativity flag. {-# OPTIONS --cumulativity #-} When the --cumulativity flag is P = ∀ x → ∀? n (P x) 3.8.3 Limitations Currently cumulativity only enables subtyping between universes, but not between any other types containing uni- verses. For example, List Set is not a subtype Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.37 Sort System Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and