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42 lines
1.2 KiB
Idris
42 lines
1.2 KiB
Idris
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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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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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