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fix incorrectly named isIsoToIso, add IsoToIsIso #1173

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Nov 28, 2024
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fix incorrectly named isIsoToIso, add IsoToIsIso #1173

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17 changes: 11 additions & 6 deletions Cubical/Foundations/Isomorphism.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -104,14 +104,19 @@ isoToEquiv : Iso A B → A ≃ B
isoToEquiv i .fst = i .Iso.fun
isoToEquiv i .snd = isoToIsEquiv i

isoToIsIso : {f : A → B} → isIso f → Iso A B
isoToIsIso {f = f} fIsIso .Iso.fun = f
isoToIsIso fIsIso .Iso.inv = fIsIso .fst
isoToIsIso fIsIso .Iso.rightInv = fIsIso .snd .fst
isoToIsIso fIsIso .Iso.leftInv = fIsIso .snd .snd
IsoToIsIso : (f : Iso A B) → isIso (f .Iso.fun)
IsoToIsIso f .fst = f .Iso.inv
IsoToIsIso f .snd .fst = f .Iso.rightInv
IsoToIsIso f .snd .snd = f .Iso.leftInv

isIsoToIso : {f : A → B} → isIso f → Iso A B
isIsoToIso {f = f} fIsIso .Iso.fun = f
isIsoToIso fIsIso .Iso.inv = fIsIso .fst
isIsoToIso fIsIso .Iso.rightInv = fIsIso .snd .fst
isIsoToIso fIsIso .Iso.leftInv = fIsIso .snd .snd

isIsoToIsEquiv : {f : A → B} → isIso f → isEquiv f
isIsoToIsEquiv fIsIso = isoToIsEquiv (isoToIsIso fIsIso)
isIsoToIsEquiv fIsIso = isoToIsEquiv (isIsoToIso fIsIso)

isoToPath : Iso A B → A ≡ B
isoToPath {A = A} {B = B} f i =
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