diff --git a/formalizations/guarded-cubical/ErrorDomains.agda b/formalizations/guarded-cubical/ErrorDomains.agda index c278d824b8b61adc10a9aab4bd045b9785af769f..a1b7ad42ebbd929cbbe1a76f134557679a773dc5 100644 --- a/formalizations/guarded-cubical/ErrorDomains.agda +++ b/formalizations/guarded-cubical/ErrorDomains.agda @@ -128,27 +128,21 @@ trivialize2 {X} _R_ hSym hTrans hCong hθ = fix trivialize2' -- (θ (next (fix θ))) R -- (fix θ) --- alternatively, we can drop symmetry if we assume that the relation --- is reflexive, or at least that fix θ is related to itself. +-- don't need symmetry trivialize3 : {X : Type} (_R_ : L℧ X -> L℧ X -> Type) -> transitive _R_ -> congruence _R_ -> - fix θ R fix θ -> ((x : L℧ X) -> x R (θ (next x))) -> ((x : L℧ X) -> x R (fix θ)) -trivialize3 {X} _R_ hTrans hCong fix-ok hθR = fix trivialize3' +trivialize3 {X} _R_ hTrans hCong hθR = fix trivialize3' where - lem : θ (next (fix θ)) R fix θ - lem = subst (λ x → x R fix θ) (fix-eq θ) fix-ok - trivialize3' : ▹ ((x : L℧ X) -> x R (fix θ)) → (x : L℧ X) -> x R (fix θ) trivialize3' IH lx = - hTrans - (hθR lx) + subst (λ y → lx R y) (sym (fix-eq θ)) (hTrans - (hCong (λ t → IH t lx)) - lem) + (hθR lx) + (hCong (λ t → IH t lx))) --------------------------------------------------------------------------