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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 -}
- let
- fill : (i : I) (A : I -> Type)
- (phi : I) (u : (i : I) -> Partial phi (A i))
- -> Sub (A i0) phi (u i0) -> A i
- = \i A phi u a0 ->
- comp (\j -> A (i && j)) (phi || ~i) (\j -> [ phi -> u (i && j), ~i -> a0 ]) a0
- in let
- trans : (A : Type) (a b c : A)
- -> Path (\i -> A) a b
- -> Path (\i -> A) b c
- -> Path (\i -> A) a c
- = \A a b c p q i ->
- comp (\j -> A) (i || ~i)
- (\j -> [ ~i -> a, i -> q j ])
- (p i)
- in let
- elimI : (P : I -> Type)
- (a : P i0)
- (b : P i1)
- -> Path P a b
- -> (i : I) -> P i
- = \P a b p i -> p i
- in let
- contrI : (i : I) -> Path (\i -> I) i0 i
- = \i -> elimI (\x -> Path (\i -> I) i0 x) (\i -> i0) (\i -> i) (\i j -> i && j) i
- in let
- IisContr : (i : I) * ((j : I) -> Path (\i -> I) i j)
- = (i0, contrI)
- in let
- compPath : (A : I -> Type)
- (phi : I) (u : (i : I) -> Partial phi (A i))
- -> (a0 : Sub (A i0) phi (u i0)) -> Path A a0 (comp A phi u a0)
- = \A phi u a0 j -> fill j A phi u a0
- in compPath -}
-
- let
- fill : (i : I) (A : I -> Type)
- (phi : I) (u : (i : I) -> Partial phi (A i))
- -> Sub (A i0) phi (u i0) -> A i
- = \i A phi u a0 ->
- comp (\j -> A (i && j)) (phi || ~i) (\j -> [ phi -> u (i && j), ~i -> a0 ]) a0
- in
- let
- pres : (A : I -> Type)
- (T : I -> Type)
- (f : (i : I) -> T i -> A i)
- (phi : I)
- (t : (i : I) -> Partial phi (T i))
- (t0 : Sub (T i0) phi (t i0))
- -> (let c1 : A i1 = comp A phi (\j -> [phi -> f j (t j)]) (f i0 t0) in
- let c2 : A i1 = f i1 (comp T phi (\j -> [phi -> t j]) t0) in
- Sub (Path (\i -> A i1) c1 c2) phi (\j -> f i1 (t i1)))
- = \A T f phi t t0 j ->
- let v : (i : I) -> T i = \i -> fill i T phi (\j -> [phi -> t j]) t0
- in comp A (phi || j) (\u -> [phi || j -> f u (v u)]) (f i0 t0)
- in pres
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