34 lines
1.4 KiB
Markdown
34 lines
1.4 KiB
Markdown
The Lean Standard Library
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=========================
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The Lean standard library is contained in the following files and directories:
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* [init](init/init.md) : constants and theorems needed for low-level system operations
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* [logic](logic/logic.md) : logical constructs and axioms
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* [data](data/data.md) : concrete datatypes and type constructors
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* [algebra](algebra/algebra.md) : algebraic structures
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* [tools](tools/tools.md) : additional tools
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Modules can be loaded individually, but they are also be loaded by importing the
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following packages:
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* [standard](standard.lean) : constructive logic and datatypes
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* [classical](classical.lean) : classical mathematics
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Lean's default logical framework is a version of the Calculus of Constructions with:
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* an impredicative, proof-irrelevant type `Prop` of propositions
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* universe polymorphism
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* a non-cumulative hierarchy of universes, `Type 1`, `Type 2`, ... above `Prop`
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* inductively defined types
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* quotient types
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The `standard` library does not rely on any axioms beyond this framework, and is
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hence constructive. It includes theories of the natural numbers, integers,
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lists, and so on.
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The `classical` library also imports a choice function axiom, which implies
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the law of the excluded middle and propositional extensionality. See [logic.choice](logic/choice.lean) for details.
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See also the [hott library](../hott/hott.md), a library for homotopy
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type theory based on a predicative foundation.
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