36 lines
No EOL
1.5 KiB
Markdown
36 lines
No EOL
1.5 KiB
Markdown
The Lean Library
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The Lean library is contained in the following modules and directories:
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* [general_notation](general_notation.lean) : commonly shared notation
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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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* [hott](hott/hott.md) : homotopy type theory
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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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* [hott](hott/default.lean) : homotopy type theory
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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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* univerve 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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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 imports the law of the excluded middle, choice functions,
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and propositional extensionality. See `logic/axioms` for details.
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The `hott` library is compatible with the standard library, but favors "proof
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relevant" versions of the logical connectives, based on `Type` rather than
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`Prop`. See `hott` for details. |