A.1.1 Type universes
We postulate a hierarchy of universes
denoted by primitive constants
The first two rules for universes say that they form a cumulative hierarchy of types:
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for ,
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if and , then ,
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:
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if , and is a new variable,11By “new” we mean that it does not appear in or .then .
In the body of the book, an equality judgment between types and is usually abbreviated to . 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:
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if and then .