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- let
- sym : (A : Type) (x y : A) -> Path (\x -> A) x y -> Path (\x -> A) y x
- = λ A x y p i -> p (~ i)
- in let
- funext : (A : Type) (B : A -> Type) (f g : (x : A) -> B x) -> ((x : A) -> Path (\i -> B x) (f x) (g x)) -> Path (\i -> (x : A) -> B x) f g
- = λ A B f g h i x -> h x i
- in let
- i0IsI1 : Path (\x -> I) i0 i1
- = λ i -> i
- in let
- singContr : (A : Type) (a b : A) (p : Path (\j -> A) a b) -> Path (\i -> (x : A) * (Path (\j -> A) a x)) (a, \i -> a) (b, p)
- = λ A a b p i -> (p i, λ j -> p (i && j))
- in let
- transport : (A : I -> Type) (a : A i0) -> A i1
- = \A a -> comp A i0 (\i -> []) a
- in let
- Jay : (A : Type) (x : A)
- (P : (y : A) -> Path (\i -> A) x y -> Type)
- (d : P x (\i -> x))
- (y : A) (p : Path (\i -> A) x y)
- -> P y p
- = \A x P d y p -> transport (\i -> P (p i) (\j -> p (i && j))) d
- in Jay
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