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2ae330785b
* [contrib] Add misc libraries to contrib Expose some `private` function in libs/base that I needed, and seem like their visibility was forgotten I'd appreciate a code review, especially to tell me I'm re-implementing something that's already elsewhere in the library Mostly extending existing functionality: * `Data/Void.idr`: add some utility functions for manipulating absurdity. * `Decidable/Decidable/Extra.idr`: add support for double negation elimination in decidable relations * `Data/Fun/Extra.idr`: + add `application` (total and partil) for n-ary functions + add (slightly) dependent versions of these operations * `Decidable/Order/Strict.idr`: a strict preorder is what you get when you remove the diagonal from a pre-order. For example, `<` is the associated preorder for `<=` over `Nat`. Analogous to `Decidable.Order`. The proof search mechanism struggled a bit, so I had to hack it --- sorry. Eventually we should move `Data.Fun.Extra.Pointwise` to `Data.Vect.Quantifiers` in base but we don't have any interesting uses for it at the moment so it's not urgent. Co-authored by @gallais
43 lines
1.3 KiB
Idris
43 lines
1.3 KiB
Idris
module Data.Rel
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import Data.Fun
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||| Build an n-ary relation type from a Vect of Types
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public export
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Rel : Vect n Type -> Type
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Rel ts = Fun ts Type
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||| Universal quantification of a n-ary Relation over its
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||| arguments to build a (function) type from a `Rel` type
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||| ```
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||| λ> All [Nat,Nat] LTE
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||| (x : Nat) -> (x : Nat) -> LTE x x
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||| ```
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public export
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All : (ts : Vect n Type) -> (p : Rel ts) -> Type
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All [] p = p
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All (t :: ts) p = (x : t) -> All ts (p x)
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||| Existential quantification of a n-ary relation over its
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||| arguments to build a dependent pair (eg. Sigma type).
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||| Given a (type of) relation `p : [t_1, t_2 ... t_n] x r` where `t_i` and `r` are
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||| types, `Ex` builds the type `Σ (x_1 : t_1). Σ (x_2 : t_2) ... . r`
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||| For example:
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||| ```
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||| λ> Ex [Nat,Nat] LTE
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||| (x : Nat ** (x : Nat ** LTE x x))
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||| ```
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||| Which is the type of a pair of natural numbers along with a proof that the first
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||| is smaller or equal than the second.
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public export
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Ex : (ts : Vect n Type) -> (p : Rel ts) -> Type
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Ex [] p = p
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Ex (t :: ts) p = (x : t ** Ex ts (p x))
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||| Map a type-level function over the co-domain of a n-ary Relation
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public export
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liftRel : (ts : Vect n Type) -> (p : Rel ts) -> (Type -> Type) -> Type
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liftRel ts p f = All ts $ map @{Nary} f p
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