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46 lines
992 B
46 lines
992 B
{-# PRIMITIVE Type #-}
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{-# PRIMITIVE Pretype #-}
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I : Pretype
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{-# PRIMITIVE Interval I #-}
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i0 : I
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i1 : I
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{-# PRIMITIVE i0 #-}
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{-# PRIMITIVE i1 #-}
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iand : I -> I -> I
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{-# PRIMITIVE iand #-}
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ior : I -> I -> I
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{-# PRIMITIVE ior #-}
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inot : I -> I
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{-# PRIMITIVE inot #-}
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PathP : (A : I -> Pretype) -> A i0 -> A i1 -> Type
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{-# PRIMITIVE PathP #-}
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Path : {A : Pretype} -> A -> A -> Type
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Path {A} = PathP (\i -> A)
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refl : {A : Pretype} {x : A} -> Path x x
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refl {A} {x} i = x
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sym : {A : I -> Pretype} {x : A i0} {y : A i1} -> PathP A x y -> PathP (\i -> A (inot i)) y x
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sym p i = p (inot i)
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the : (A : Pretype) -> A -> A
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the A x = x
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iElim : {A : I -> Pretype} {x : A i0} {y : A i1} -> PathP A x y -> (i : I) -> A i
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iElim p i = p i
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Singl : (A : Type) -> A -> Type
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Singl A x = (y : A) * Path x y
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isContr : Type -> Type
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isContr A = (x : A) * ((y : A) -> Path x y)
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singContr : {A : Type} {a : A} -> isContr (Singl A a)
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singContr {A} {a} = ((a, \i -> a), \y i -> (y.2 i, \j -> y.2 (iand i j)))
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