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    • CommentRowNumber1.
    • CommentAuthorAlizter
    • CommentTimeJan 19th 2019

    cleaned up page and added a small ideas section too

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 19th 2019

    I didn’t understand ’apd’. Is it a typo?

    • CommentRowNumber3.
    • CommentAuthorAlizter
    • CommentTimeJan 19th 2019
    • (edited Jan 19th 2019)

    apd is dependent ap

    apd f(p)apd_f(p) is the application of f: (a:A)B(a)f : \prod_{(a:A)} B(a) to p:x= Ayp : x =_A y

    So instead of applying a function to both sides, we apply a dependent function (the result isn’t a path type anymore but a dependent path). See page ~92 in the HoTT book.

    In this case this was written before I modified the page.

    Edit:

    For most things you will only ever use circle recursion which tells you how to build a function out of the circle. Circle induction tells you how to build a dependent function out the circle.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 19th 2019

    Thanks!

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJan 19th 2019

    For most things you will only ever use circle recursion which tells you how to build a function out of the circle.

    I’m not sure what you mean by that. I can’t think offhand of much that you can prove about the circle using recursion but not induction.

    • CommentRowNumber6.
    • CommentAuthorAlizter
    • CommentTimeJan 20th 2019

    There isn’t much you can prove but when I was first learning about it in HoTT I found it a lot easier and intuitive to use circle recursion. I wouldn’t have gotten very far with induction without some more work. So perhaps I should of said: it’s easier to use circle recursion than induction. I’ll take back what I said about most things.

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