solve: HoTT 3.2

[kiwiyou]

[gemini-code-assist]

Summary of Changes

Hello @kiwiyou, I'm Gemini Code Assist¹! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request introduces a new formal proof for a fundamental theorem in Homotopy Type Theory (HoTT), specifically demonstrating that the sum type of two sets is also a set. The proof meticulously constructs an auxiliary type family to manage equality within sum types, then establishes its propositional nature and equivalence to the standard equality. Finally, it leverages advanced HoTT concepts like univalence and transport to formally derive the desired set property for the sum type.

Highlights

• New Formal Proof Added: A new file, HoTT-3.2.typ, has been added containing a formal proof for Homotopy Type Theory (HoTT) 3.2. This proof demonstrates that if A and B are sets, then their sum type A + B is also a set.
• Encode-Decode Method for Sum Type Equality: The proof introduces and utilizes an 'encode-decode' method with a code(x, y) type family to handle equality within sum types, specifically addressing degenerate cases like inl = inr.
• Proof of Propositional Nature and Equivalence: The proof formally establishes that code(x, y) is a proposition (isProp(code(x, y))) and demonstrates its equivalence to the standard equality (x eq(A + B) y).
• Application of Univalence and Transport: The final step of the proof leverages the principles of univalence and transport to formally construct and verify the isSet(A + B) property, concluding that the sum of two sets is indeed a set.

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[gemini-code-assist]

Code Review

This pull request adds a solution for HoTT exercise 3.2, which is well-structured and uses typst features effectively. I've found a few issues in the mathematical text that could affect the correctness and clarity of the proof's description. Specifically, there's a likely typo in the definition of decode and some inaccuracies in the explanatory text for the equivalence proof. I've provided suggestions to correct these points.

[kiwiyou]

/gemini review

[gemini-code-assist]

Code Review

This pull request adds a solution to exercise 3.2 from the HoTT book. The proof is well-structured and follows the encode-decode method correctly to show that the sum of two sets is a set. The overall logic is sound. I've added a couple of suggestions to improve the notational clarity in the derivation steps of the proof, making them more formally rigorous.

[gemini-code-assist]

The first step in this chain of equivalences is an instantiation based on the path induction mentioned in the preceding text, rather than a formal equivalence. This can be confusing. For better clarity and formal rigor, it's better to start the calculation directly from the instantiated expression.

    encode(inl(a), inl(a), decode(inl(a), inl(a), refl_a)) \
    &equiv encode(inl(a), inl(a), ap_inl (refl_a)) \
    &equiv encode(inl(a), inl(a), refl_inl(a)) \
    &equiv refl_a \
    &equiv c

[gemini-code-assist]

Similar to the previous proof, this chain of equivalences starts with an instantiation rather than a formal equivalence step. To improve clarity and rigor, the calculation should begin directly with the instantiated expression.

    decode(inl(a), inl(a), encode(inl(a), inl(a), refl_inl(a))) \
    &equiv decode(inl(a), inl(a), refl_a) \
    &equiv ap_inl (refl_a) \
    &equiv refl_inl(a) \
    &equiv p