cubical-0.1.0: examples/exists.cub

module exists where

import prelude

-- existence: a new modality

exists : (A : U) (B : A -> U) -> U
exists A B = inh (Sigma A B)

exElim : (A : U) (B : A -> U) (C : U) -> prop C -> (Sigma A B -> C) ->
         exists A B -> C
exElim A B C p f = inhrec (Sigma A B) C p f

atmostOne : (A : U) (B : A -> U) -> U
atmostOne A B = (a b : A) -> B a -> B b -> Id A a b

exactOne : (A : U) (B : A -> U) -> U
exactOne A B = and (exists A B) (atmostOne A B)

lemInh : (A : U) -> prop A -> inh A -> A
lemInh A h = inhrec A A h (\x -> x)