{-# PRIMITIVE Type #-}
{-# PRIMITIVE Pretype #-}

I : Pretype
{-# PRIMITIVE Interval I #-}

i0 : I
i1 : I
{-# PRIMITIVE i0 #-}
{-# PRIMITIVE i1 #-}

iand : I -> I -> I
{-# PRIMITIVE iand #-}

ior : I -> I -> I
{-# PRIMITIVE ior #-}

inot : I -> I
{-# PRIMITIVE inot #-}

PathP : (A : I -> Pretype) -> A i0 -> A i1 -> Type
{-# PRIMITIVE PathP #-}

Path : {A : Pretype} -> A -> A -> Type
Path {A} = PathP (\i -> A)

refl : {A : Pretype} {x : A} -> Path x x
refl {A} {x} i = x

sym : {A : I -> Pretype} {x : A i0} {y : A i1} -> PathP A x y -> PathP (\i -> A (inot i)) y x
sym p i = p (inot i)

the : (A : Pretype) -> A -> A
the A x = x

iElim : {A : I -> Pretype} {x : A i0} {y : A i1} -> PathP A x y -> (i : I) -> A i
iElim p i = p i

Singl : (A : Type) -> A -> Type
Singl A x = (y : A) * Path x y

isContr : Type -> Type
isContr A = (x : A) * ((y : A) -> Path x y)

singContr : {A : Type} {a : A} -> isContr (Singl A a)
singContr {A} {a} = ((a, \i -> a), \y i -> (y.2 i, \j -> y.2 (iand i j)))