cubical-0.1.0: examples/UnotSet.cub

module UnotSet where

import BoolEqBool

-- proves that U is not a set

negUIP : neg (set U)
negUIP uipU = tnotf lem5
  where
  eqreflnot : Id (Id U Bool Bool) (refl U Bool) eqBoolBool
  eqreflnot = uipU Bool Bool (refl U Bool) eqBoolBool

  frefl : Bool -> Bool
  frefl = transport Bool Bool (refl U Bool)

  fnot : Bool -> Bool
  fnot = transport Bool Bool eqBoolBool

  lem1 : Id (Bool -> Bool) frefl fnot
  lem1 = cong (Id U Bool Bool) (Bool -> Bool) (transport Bool Bool) 
              (refl U Bool) eqBoolBool eqreflnot

  lem2 : Id Bool true (frefl true)
  lem2 = transportRef Bool true

  lem3 : Id Bool false (fnot true)
  lem3 = transpEquivEq Bool Bool not sNot tNot true

  lem4 : Id Bool (frefl true) (fnot true)
  lem4 = cong (Bool -> Bool) Bool (\f -> f true) frefl fnot lem1

  lem5 : Id Bool true false
  lem5 = compDown Bool true (frefl true) false (fnot true) lem2 lem3 lem4