Halakhah for Yoma 156:4
אלא אמר רבא לעולם דכ"ע מנעל הוא ובשבת בהא פליגי מר סבר גזרינן דילמא משתמיט ואתי לאתויי ד' אמות ומר סבר לא גזרינן
Said Abaye: There, the reference is where it [the wooden leg] has pads, and [the prohibition to go out with them on the Day of Atonement is] due to the comfort [they bring]. Said Raba to him: But if it be no object of wear [without them], would the pads make it one? And, furthermore, is any comfort not coming from shoes forbidden on the Day of Atonement?
Sefer HaChinukh
And concerning the shoe (sandal), our teachers - God should protect them - explained that a shoe is always of leather. And that is what is forbidden on Yom Kippur, but not of a different type. And the principle of the matter according to some commentators is that anything that is fitting for the matter of release (chalitsah) - meaning to say, that it is of leather - is forbidden on Yom Kippur, and permitted to go out into a public domain with them (wearing them) on Shabbat. But anything that is not fitting for release - such as cork and reeds and palms (see Yoma 78b, that it is a shoe of grasses) and other types of grasses (plants) - are permitted on Yom Kippur, so long as they do not go out with them [to a public domain] in a place without an eruv, as we consider them like a burden (and not a piece of clothing, which is permitted). But from the commentators, there are [also] many honored ones that permit one to go out into the public domain with all of them.
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