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79 lines
2.7 KiB
79 lines
2.7 KiB
{- {- let
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sym : (A : Type) (x y : A) -> Path (\x -> A) x y -> Path (\x -> A) y x
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= λ A x y p i -> p (~ i)
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in let
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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
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= λ A B f g h i x -> h x i
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in let
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i0IsI1 : Path (\x -> I) i0 i1
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= λ i -> i
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in let
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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)
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= λ A a b p i -> (p i, λ j -> p (i && j))
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in let
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transport : (A : I -> Type) (a : A i0) -> A i1
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= \A a -> comp A i0 (\i -> []) a
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in let
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Jay : (A : Type) (x : A)
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(P : (y : A) -> Path (\i -> A) x y -> Type)
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(d : P x (\i -> x))
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(y : A) (p : Path (\i -> A) x y)
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-> P y p
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= \A x P d y p -> transport (\i -> P (p i) (\j -> p (i && j))) d
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in -}
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let
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fill : (i : I) (A : I -> Type)
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(phi : I) (u : (i : I) -> Partial phi (A i))
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-> Sub (A i0) phi (u i0) -> A i
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= \i A phi u a0 ->
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comp (\j -> A (i && j)) (phi || ~i) (\j -> [ phi -> u (i && j), ~i -> a0 ]) a0
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in let
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trans : (A : Type) (a b c : A)
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-> Path (\i -> A) a b
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-> Path (\i -> A) b c
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-> Path (\i -> A) a c
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= \A a b c p q i ->
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comp (\j -> A) (i || ~i)
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(\j -> [ ~i -> a, i -> q j ])
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(p i)
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in let
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elimI : (P : I -> Type)
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(a : P i0)
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(b : P i1)
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-> Path P a b
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-> (i : I) -> P i
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= \P a b p i -> p i
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in let
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contrI : (i : I) -> Path (\i -> I) i0 i
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= \i -> elimI (\x -> Path (\i -> I) i0 x) (\i -> i0) (\i -> i) (\i j -> i && j) i
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in let
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IisContr : (i : I) * ((j : I) -> Path (\i -> I) i j)
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= (i0, contrI)
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in let
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compPath : (A : I -> Type)
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(phi : I) (u : (i : I) -> Partial phi (A i))
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-> (a0 : Sub (A i0) phi (u i0)) -> Path A a0 (comp A phi u a0)
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= \A phi u a0 j -> fill j A phi u a0
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in compPath -}
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let
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fill : (i : I) (A : I -> Type)
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(phi : I) (u : (i : I) -> Partial phi (A i))
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-> Sub (A i0) phi (u i0) -> A i
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= \i A phi u a0 ->
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comp (\j -> A (i && j)) (phi || ~i) (\j -> [ phi -> u (i && j), ~i -> a0 ]) a0
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in
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let
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pres : (A : I -> Type)
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(T : I -> Type)
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(f : (i : I) -> T i -> A i)
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(phi : I)
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(t : (i : I) -> Partial phi (T i))
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(t0 : Sub (T i0) phi (t i0))
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-> (let c1 : A i1 = comp A phi (\j -> [phi -> f j (t j)]) (f i0 t0) in
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let c2 : A i1 = f i1 (comp T phi (\j -> [phi -> t j]) t0) in
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Sub (Path (\i -> A i1) c1 c2) phi (\j -> f i1 (t i1)))
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= \A T f phi t t0 j ->
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let v : (i : I) -> T i = \i -> fill i T phi (\j -> [phi -> t j]) t0
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in comp A (phi || j) (\u -> [phi || j -> f u (v u)]) (f i0 t0)
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in pres
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