75 lines
2.3 KiB
TeX
75 lines
2.3 KiB
TeX
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\chapter{Abstract functional extensionality}
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\label{app:abstract-funext}
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In two places in my formalization was the computational behaviours of
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functional extensionality used. The reduction behaviour can be
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disabled by marking functional extensionality as abstract. Below the
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fully normalized goal and context with functional extensionality
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marked abstract has been shown. The excerpts are from the module
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%
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\begin{center}
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\sourcelink{Cat.Category.Monad.Voevodsky}
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\end{center}
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%
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where this is also written as a comment next to the proofs. When
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functional extensionality is not abstract the goal and current value
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are the same. It is of course necessary to show the fully normalized
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goal and context otherwise the reduction behaviours is not forced.
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\subsubsection*{First goal}
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Goal:
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\begin{verbatim}
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PathP (λ _ → §2-3.§2 omap (λ {z} → pure))
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(§2-fromMonad
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(.Cat.Category.Monad.toKleisli ℂ
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(.Cat.Category.Monad.toMonoidal ℂ (§2-3.§2.toMonad m))))
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(§2-fromMonad (§2-3.§2.toMonad m))
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\end{verbatim}
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Have:
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\begin{verbatim}
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PathP
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(λ i →
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§2-3.§2 K.IsMonad.omap
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(K.RawMonad.pure
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(K.Monad.raw
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(funExt (λ m₁ → K.Monad≡ (.Cat.Category.Monad.toKleisliRawEq ℂ m₁))
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i (§2-3.§2.toMonad m)))))
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(§2-fromMonad
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(.Cat.Category.Monad.toKleisli ℂ
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(.Cat.Category.Monad.toMonoidal ℂ (§2-3.§2.toMonad m))))
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(§2-fromMonad (§2-3.§2.toMonad m))
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\end{verbatim}
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\subsubsection*{Second goal}
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Goal:
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\begin{verbatim}
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PathP (λ _ → §2-3.§1 omap (λ {X} → pure))
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(§1-fromMonad
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(.Cat.Category.Monad.toMonoidal ℂ
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(.Cat.Category.Monad.toKleisli ℂ (§2-3.§1.toMonad m))))
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(§1-fromMonad (§2-3.§1.toMonad m))
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\end{verbatim}
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Have:
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\begin{verbatim}
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PathP
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(λ i →
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§2-3.§1
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(RawFunctor.omap
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(Functor.raw
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(M.RawMonad.R
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(M.Monad.raw
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(funExt
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(λ m₁ → M.Monad≡ (.Cat.Category.Monad.toMonoidalRawEq ℂ m₁)) i
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(§2-3.§1.toMonad m))))))
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(λ {X} →
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fst
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(M.RawMonad.pureNT
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(M.Monad.raw
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(funExt
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(λ m₁ → M.Monad≡ (.Cat.Category.Monad.toMonoidalRawEq ℂ m₁)) i
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(§2-3.§1.toMonad m))))
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X))
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(§1-fromMonad
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(.Cat.Category.Monad.toMonoidal ℂ
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(.Cat.Category.Monad.toKleisli ℂ (§2-3.§1.toMonad m))))
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(§1-fromMonad (§2-3.§1.toMonad m))
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\end{verbatim}
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