mirror of https://github.com/nmvdw/HITs-Examples
99 lines
2.8 KiB
Plaintext
99 lines
2.8 KiB
Plaintext
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module CL where
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private
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data CL' : Set where
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K' : CL'
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S' : CL'
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app' : CL' -> CL' -> CL'
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CL : Set
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CL = CL'
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K : CL
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K = K'
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Sc : CL
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Sc = S'
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app : CL -> CL -> CL
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app = app'
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postulate
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KConv : {x y : CL} -> app (app K x) y == x
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SConv : {x y z : CL} -> app (app (app Sc x) y) z == app (app x z) (app y z)
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CLind : (Y : CL -> Set)
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(KY : Y K)
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(SY : Y Sc)
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(appY : (x y : CL) -> Y x -> Y y -> Y (app x y))
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(KConvY : (x y : CL) (a : Y x) (b : Y y) -> PathOver Y KConv (appY (app K x) y (appY K x KY a) b) a)
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(SConvY : (x y z : CL) (a : Y x) (b : Y y) (c : Y z) ->
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PathOver Y SConv
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(appY
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(app (app Sc x) y)
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z
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(appY
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(app Sc x)
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y
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(appY Sc x SY a)
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b
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)
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c
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)
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(appY (app x z) (app y z) (appY x z a c) (appY y z b c))
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)
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(x : CL) -> Y x
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CLind Y KY SY appY _ _ K' = KY
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CLind Y KY SY appY _ _ S' = SY
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CLind Y KY SY appY KConvY SConvY (app' x x₁) = appY x x₁ (CLind Y KY SY appY KConvY SConvY x) (CLind Y KY SY appY KConvY SConvY x₁)
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postulate
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CLind_βKConv : (Y : CL -> Set)
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(KY : Y K)
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(SY : Y Sc)
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(appY : (x y : CL) -> Y x -> Y y -> Y (app x y))
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(KConvY : (x y : CL) (a : Y x) (b : Y y) -> PathOver Y KConv (appY (app K x) y (appY K x KY a) b) a)
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(SConvY : (x y z : CL) (a : Y x) (b : Y y) (c : Y z) ->
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PathOver Y SConv
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(appY
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(app (app Sc x) y)
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z
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(appY
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(app Sc x)
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y
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(appY Sc x SY a)
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b
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)
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c
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)
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(appY (app x z) (app y z) (appY x z a c) (appY y z b c))
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)
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(x y : CL) ->
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apd (CLind Y KY SY appY KConvY SConvY) KConv == KConvY x y (CLind Y KY SY appY KConvY SConvY x) (CLind Y KY SY appY KConvY SConvY y)
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CLind_βSConv : (Y : CL -> Set)
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(KY : Y K)
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(SY : Y Sc)
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(appY : (x y : CL) -> Y x -> Y y -> Y (app x y))
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(KConvY : (x y : CL) (a : Y x) (b : Y y) -> PathOver Y KConv (appY (app K x) y (appY K x KY a) b) a)
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(SConvY : (x y z : CL) (a : Y x) (b : Y y) (c : Y z) ->
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PathOver Y SConv
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(appY
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(app (app Sc x) y)
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z
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(appY
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(app Sc x)
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y
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(appY Sc x SY a)
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b
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)
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c
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)
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(appY (app x z) (app y z) (appY x z a c) (appY y z b c))
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)
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(x y z : CL) ->
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apd (CLind Y KY SY appY KConvY SConvY) SConv == SConvY x y z (CLind Y KY SY appY KConvY SConvY x) (CLind Y KY SY appY KConvY SConvY y) (CLind Y KY SY appY KConvY SConvY z)
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