96 lines
2.3 KiB
Markdown
96 lines
2.3 KiB
Markdown
Presentation
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====
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Find one clear goal.
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Remember crowd-control.
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Leave out:
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lemPropF
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Outline
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-------
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Introduction
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A formalization of Category Theory in cubical Agda.
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Cubical Agda: A constructive interpretation of functional
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extensionality and univalence
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Talk about structure of library:
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===
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What can I say about reusability?
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Meeting with Andrea May 18th
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============================
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App. 2 in HoTT gives typing rule for pathJ including a computational
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rule for it.
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If you have this computational rule definitionally, then you wouldn't
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need to use `pathJprop`.
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In discussion-section I mention HITs. I should remove this or come up
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with a more elaborate example of something you could do, e.g.
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something with pushouts in the category of sets.
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The type Prop is a type where terms are *judgmentally* equal not just
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propositionally so.
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Maybe mention that Andreas Källberg is working on proving the
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initiality conjecture.
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Intensional Type Theory (ITT): Judgmental equality is decidable
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Extensional Type Theory (ETT): Reflection is enough to make judgmental
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equality undecidable.
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Reflection : a ≡ b → a = b
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ITT does not have reflections.
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HTT ~ ITT + axiomatized univalence
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Agda ~ ITT + K-rule
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Coq ~ ITT (no K-rule)
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Cubical Agda ~ ITT + Path + Glue
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Prop is impredicative in Coq (whatever that means)
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Prop ≠ hProp
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Comments about abstract
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-----
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Pattern matching for paths (?)
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Intro
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-----
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Main feature of judgmental equality is the conversion rule.
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Conor explained: K + eliminators ≡ pat. matching
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Explain jugmental equality independently of type-checking
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Soundness for equality means that if `x = y` then `x` and `y` must be
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equal according to the theory/model.
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Decidability of `=` is a necessary condition for typechecking to be
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decidable.
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Canonicity is a nice-to-have though without canonicity terms can get
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stuck. If we postulate results about judgmental equality. E.g. funext,
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then we can construct a term of type natural number that is not a
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numeral. Therefore stating canonicity with natural numbers:
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∀ t . ⊢ t : N , ∃ n : N . ⊢ t = sⁿ 0 : N
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is a sufficient condition to get a well-behaved equality.
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Eta-equality for RawFunctor means that the associative law for
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functors hold definitionally.
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Computational property for funExt is only relevant in two places in my
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whole formulation. Univalence and gradLemma does not influence any
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proofs.
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