Created
March 29, 2014 22:02
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module Rel | |
DecProofType : {a:Type} -> Dec a -> Type | |
DecProofType {a} (Yes _) = a | |
DecProofType {a} (No _) = a -> _|_ | |
decProof : {a:Type} -> (dec : Dec a) -> DecProofType dec | |
decProof (Yes x) = x | |
decProof (No x) = x | |
DecType : {a:Type} -> Dec a -> Type | |
DecType {a} _ = a | |
-- Proofs that something is not an element of a vector | |
using ( x : t, y : t, xs : Vect n t ) | |
data NElem : t -> Vect n t -> Type where | |
Empty : NElem x [] | |
Miss : Not (y = x) -> NElem x xs -> NElem x (y :: xs) | |
strip : NElem x (y :: xs) -> NElem x xs | |
strip (Miss _ x) = x | |
headInEq : NElem x (y :: xs) -> Not (y = x) | |
headInEq (Miss p _) = p | |
total | |
decNElem : DecEq t => (x : t) -> (xs : Vect n t) -> Dec (NElem x xs) | |
decNElem x Nil = Yes Empty | |
decNElem x (y :: ys) with (decEq y x) | |
decNElem x (x :: ys) | (Yes yEqX) = | |
No (\nElemXYYs => headInEq nElemXYYs yEqX) | |
decNElem x (y :: ys) | (No f) with (decNElem x ys) | |
decNElem x (y :: ys) | (No f) | (Yes z) = Yes (Miss f z) | |
decNElem x (y :: ys) | (No f) | (No notNElemXYs) = | |
No (\nElemXYYs => notNElemXYs (strip nElemXYYs)) | |
namespace Uniq | |
mutual | |
-- Vector types where each element is unique | |
data Uniq : Nat -> Type -> Type where | |
Nil : Uniq 0 t | |
Cons : (x : t) -> | |
(xs : Uniq n t) -> | |
{ default (decProof decNElem') | |
isUniq : NElem {n} x (content xs) } -> | |
Uniq (S n) t | |
content : Uniq n t -> Vect n t | |
content Nil = Nil | |
content (Cons a b) = a :: content b | |
decNElem' : DecEq t => {x : t} -> {xs : Uniq n t} -> Dec (NElem x (content xs)) | |
decNElem' {x} {xs} = decNElem x (content xs) |
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