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单词 A11TypeUniverses
释义

A.1.1 Type universes


We postulateMathworldPlanetmath a hierarchy of universesPlanetmathPlanetmath denoted by primitive constants

𝒰0,𝒰1,𝒰2,

The first two rules for universes say that they form a cumulative hierarchy of types:

  • 𝒰m:𝒰n for m<n,

  • if A:𝒰m and mn, then A:𝒰n,

and the third expresses the idea that an object of a universe can serve as a type and stand to theright of a colon in judgments:

  • if ΓA:𝒰n, and x is a new variable,11By “new” we mean that it does not appear in Γ or A.then (Γ,x:A)𝖼𝗍𝗑.

In the body of the book, an equality judgment AB:𝒰n between typesA and B is usually abbreviated to AB. This is an instance oftypical ambiguity, as we can always switch to a larger universe, which however does not affect the validity of the judgment.

The following conversion rule allows us to replace a type by one equal to it in a typing judgment:

  • if a:A and AB then a:B.

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