CCHM/test.itt

{- 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 IisContr