[Merged by Bors] - feat(data/equiv/encodable): ulower lowers countable types to Type 0
#2574
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I plan to use this type to refactor the many-one degrees so that there is only one |
gebner
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ulower lowers countable types to Type 0
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May 7, 2020
…0` (#2574) Given a type `α : Type u`, we can lift it into a higher universe using `ulift α : Type (max u v)`. This PR introduces an analogous construction going in the other direction. Given an encodable (= countable) type `α : Type u`, we can lower it to the very bottom using `ulower α : Type 0`. The equivalence is primitive recursive if the type is primcodable. The definition of the equivalence was already present as `encodable.equiv_range_encode`. However it is very inconvenient to use since it requires decidability instances (`encodable.decidable_range_encode`) that are not enabled by default because they would introduce overlapping instances that are not definitionally equal.
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ulower lowers countable types to Type 0ulower lowers countable types to Type 0
anrddh
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May 15, 2020
…0` (leanprover-community#2574) Given a type `α : Type u`, we can lift it into a higher universe using `ulift α : Type (max u v)`. This PR introduces an analogous construction going in the other direction. Given an encodable (= countable) type `α : Type u`, we can lower it to the very bottom using `ulower α : Type 0`. The equivalence is primitive recursive if the type is primcodable. The definition of the equivalence was already present as `encodable.equiv_range_encode`. However it is very inconvenient to use since it requires decidability instances (`encodable.decidable_range_encode`) that are not enabled by default because they would introduce overlapping instances that are not definitionally equal.
anrddh
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May 16, 2020
…0` (leanprover-community#2574) Given a type `α : Type u`, we can lift it into a higher universe using `ulift α : Type (max u v)`. This PR introduces an analogous construction going in the other direction. Given an encodable (= countable) type `α : Type u`, we can lower it to the very bottom using `ulower α : Type 0`. The equivalence is primitive recursive if the type is primcodable. The definition of the equivalence was already present as `encodable.equiv_range_encode`. However it is very inconvenient to use since it requires decidability instances (`encodable.decidable_range_encode`) that are not enabled by default because they would introduce overlapping instances that are not definitionally equal.
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Jan 13, 2021
…eter (#2630) The file `reduce.lean` defines many-one degrees for computable reductions. At the moment every primcodable type `α` has a separate type of degrees `many_one_degree α`. This is completely antithetical to the notion of degrees, which are introduced to classify problems up to many-one equivalence. This PR defines a single `many_one_degree` type that lives in `Type 0`. We use the `ulower` infrastructure from #2574 which shows that every type is computably equivalent to a subset of natural numbers. The function `many_one_degree.of` which assigns to every set of a primcodable type a degree is still universe polymorphic. In particular, we show that `of p = of q ↔ many_one_equiv p q`, etc. in maximal generality, where `p` and `q` are subsets of different types in different universes. See previous discussion at #1203.
b-mehta
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Jan 16, 2021
…eter (#2630) The file `reduce.lean` defines many-one degrees for computable reductions. At the moment every primcodable type `α` has a separate type of degrees `many_one_degree α`. This is completely antithetical to the notion of degrees, which are introduced to classify problems up to many-one equivalence. This PR defines a single `many_one_degree` type that lives in `Type 0`. We use the `ulower` infrastructure from #2574 which shows that every type is computably equivalent to a subset of natural numbers. The function `many_one_degree.of` which assigns to every set of a primcodable type a degree is still universe polymorphic. In particular, we show that `of p = of q ↔ many_one_equiv p q`, etc. in maximal generality, where `p` and `q` are subsets of different types in different universes. See previous discussion at #1203.
cipher1024
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Mar 15, 2022
…0` (leanprover-community#2574) Given a type `α : Type u`, we can lift it into a higher universe using `ulift α : Type (max u v)`. This PR introduces an analogous construction going in the other direction. Given an encodable (= countable) type `α : Type u`, we can lower it to the very bottom using `ulower α : Type 0`. The equivalence is primitive recursive if the type is primcodable. The definition of the equivalence was already present as `encodable.equiv_range_encode`. However it is very inconvenient to use since it requires decidability instances (`encodable.decidable_range_encode`) that are not enabled by default because they would introduce overlapping instances that are not definitionally equal.
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Given a type
α : Type u, we can lift it into a higher universe usingulift α : Type (max u v). This PR introduces an analogous construction going in the other direction. Given an encodable (= countable) typeα : Type u, we can lower it to the very bottom usingulower α : Type 0. The equivalence is primitive recursive if the type is primcodable.The definition of the equivalence was already present as
encodable.equiv_range_encode. However it is very inconvenient to use since it requires decidability instances (encodable.decidable_range_encode) that are not enabled by default because they would introduce overlapping instances that are not definitionally equal.