cubical-0.2.0: examples/axChoice.cub

module axChoice where

import contr

-- an interesting isomorphism/equality

idTelProp : (A:U) (B:A -> U) (C:(x:A) -> B x -> U) -> 
              Id U ((x:A) -> Sigma (B x) (C x)) (Sigma ((x:A) -> B x) (\ f -> (x:A) -> C x (f x)))
idTelProp A B C = isoId T0 T1 f g sfg rfg 
 where
  T0 : U 
  T0 = (x:A) -> Sigma (B x) (C x) 

  T1 : U 
  T1 = Sigma ((x:A) -> B x) (\ f -> (x:A) -> C x (f x))

  f : T0 -> T1
  f s = (\ x -> (s x).1, \ x -> (s x).2)

  g : T1 -> T0
  g z = \ x -> (z.1 x, z.2 x)

  sfg : (y:T1) -> Id T1 (f (g y)) y
  sfg z = refl T1 z -- rem2 u v 

  rfg : (s:T0) -> Id T0 (g (f s)) s
  rfg s = refl T0 s

-- we deduce from this equality that isEquiv f is a proposition

propIsEquiv : (A B : U) -> (f : A -> B) -> prop (isEquiv A B f)
propIsEquiv A B f = subst U prop ((y:B) -> contr' (fiber A B f y)) (isEquiv A B f) rem rem1
 where 
   rem : Id U ((y:B) -> contr' (fiber A B f y)) (isEquiv A B f) 
   rem = idTelProp B (fiber A B f) (\ y -> \ s -> (v :fiber A B f y) -> Id (fiber A B f y) s v)

   rem1 : prop ((y:B) -> contr' (fiber A B f y))
   rem1 = isPropProd B (\ y -> contr' (fiber A B f y)) (\ y -> contr'IsProp (fiber A B f y))