cubical-0.2.0: examples/description.cub

module description where

import exists
import set

exAtOne : (A : U) (B : A -> U) -> exactOne A B -> atmostOne A B
exAtOne A B z = z.2

propSig : (A : U) (B : A -> U) -> propFam A B -> atmostOne A B ->
          prop (Sigma A B)
propSig A B h h' au bv = eqPropFam A B h au bv (h' au.1 bv.1 au.2 bv.2)

descrAx : (A : U) (B : A -> U) -> propFam A B -> exactOne A B -> Sigma A B
descrAx A B h z = lemInh (Sigma A B) (propSig A B h z.2) z.1

iota : (A : U) (B : A -> U) (h : propFam A B) (h' : exactOne A B) -> A
iota A B h h' = (descrAx A B h h').1

iotaSound : (A : U) (B : A -> U) (h : propFam A B) (h' : exactOne A B) -> B (iota A B h h')
iotaSound A B h h' = (descrAx A B h h').2

iotaLem : (A : U) (B : A -> U) (h : propFam A B) (h' : exactOne A B) ->
          (a : A) -> B a -> Id A a (iota A B h h')
iotaLem A B h h' a p = exAtOne A B h' a (iota A B h h') p (iotaSound A B h h')