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doc: Classical.choice #3871

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Apr 22, 2024
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doc: Classical.choice #3871

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doc: Classical.choice
Kha Apr 10, 2024
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Kha Apr 10, 2024
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Update src/Init/Classical.lean
Kha Apr 17, 2024
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5 changes: 5 additions & 0 deletions src/Init/Classical.lean
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Expand Up @@ -15,6 +15,11 @@ namespace Classical
noncomputable def indefiniteDescription {α : Sort u} (p : α → Prop) (h : ∃ x, p x) : {x // p x} :=
choice <| let ⟨x, px⟩ := h; ⟨⟨x, px⟩⟩

/--
Given that there exists an element satisfying `p`, returns one such element.
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I stumbled a bit over "returns" here, which sounds very much like computation to me! I don't have a suggestion for a better word off the top of my head, though.

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denotes? selects?

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chooses? :)

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represents?

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is logically equivalent to an arbitrary such element

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I like the last two suggestions a lot here

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What about:

Suggested change
Given that there exists an element satisfying `p`, returns one such element.
Given that there exists an element of `α` that satisfies `p`, the result of `choose` is logically equivalent to an
arbitrary such element.

This makes the non-computational nature of it very apparent. We're also out in ultra-nitpick land here :)

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I'm not sure "the result of choose is logically equivalent to an arbitrary such element" is a well formed thing to say. I think I know the theorem it's alluding to but this is a really misleading way to say it. I think you might want to get some mathematician eyes on this

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Better to have the slight odor of computation than to write something that's incorrect :) I think that the discussion on this has now exceeded the very minor downsides of the word "returns", and am supportive of a merge as-is

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Using "logically equivalent" that way is sketchy and doesn't quite make sense for the technical meaning of logical equivalence. I like: "Given that there exists an element of α satisfying p, choose p denotes an otherwise arbitrary such element."


This is a straightforward consequence of, and equivalent to, `Classical.choice`.
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-/
noncomputable def choose {α : Sort u} {p : α → Prop} (h : ∃ x, p x) : α :=
(indefiniteDescription p h).val

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