module elimEquiv where

import univalence

-- a corollary of equivalence

allTransp : (A B : U) -> hasSection (Id U A B) (Equiv A B) (IdToEquiv A B)
allTransp A B = equivSec (Id U A B) (Equiv A B)  (IdToEquiv A B) (univAx A B)

-- an induction principle for isEquiv

transpRef : (A : U) -> Id (A->A) (id A) (transport A A (refl U A))
transpRef A = funExt A (\ _ -> A) (id A) (transport A A (refl U A)) (transportRef A)

elimIsEquiv : (A:U) -> (P : (B:U) -> (A->B) -> U) -> P A (id A) -> 
              (B :U) -> (f : A -> B) -> isEquiv A B f -> P B f
elimIsEquiv A P d B f if = rem2 B (f,if)
 where 
  rem1 : P A (transport A A (refl U A))
  rem1 = subst (A->A) (P A) (id A) (transport A A (refl U A)) (transpRef A) d

  rem : (B:U) -> (p:Id U A B) -> P B (transport A B p)
  rem = J U A (\ B p ->  P B (transport A B p)) rem1

  rem2 : (B:U) -> (p:Equiv A B) -> P B p.1
  rem2 B = allSection (Id U A B) (Equiv A B) (IdToEquiv A B) (allTransp A B) (\ p -> P B p.1) (rem B)