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strictly positive 性についての補足 #1439

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Feb 1, 2025
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strictly positive 性についての補足 #1439

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25 changes: 12 additions & 13 deletions LeanByExample/Declarative/Inductive.lean
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Expand Up @@ -282,23 +282,22 @@ error: (kernel) arg #1 of 'Foo.mk' has a non positive occurrence of the datatype

strictly positive 性に違反するような帰納型を仮に定義できたとすると、矛盾が導かれてしまいます。[`unsafe`](#{root}/Modifier/Unsafe.md) 修飾子で実際に試してみることができます。 -/

-- 任意に型 A が与えられたとして固定する
opaque A : Type

/-- strictly positive 制約を破っている帰納型 -/
unsafe inductive Bad where
| mk (f : Bad → Bad)

namespace Bad
-- Bad の項が構築できる
unsafe def default : Bad := Bad.mk id
| mk (f : Bad → A)

-- Bad の項が構築できれば矛盾が導ける
unsafe def contradiction (b : Bad) : False := by
induction b
case mk f ih =>
apply ih default
unsafe def selfApply (b : Bad) : A :=
match b with
| Bad.mk f => f b

-- Bad の項が構築できるので矛盾する
unsafe example : False := contradiction default
-- ω を計算すると ω 自身が出てくる
-- つまり無限ループしていることに注意
unsafe def ω : A := selfApply (Bad.mk selfApply)

end Bad
-- A は任意の型なのに、Inhabited インスタンスが構築できてしまった！
unsafe instance allTypeInhabited : Inhabited A := ⟨ω⟩

/- [^hitchhiker]: [The Hitchhiker’s Guide to Logical Verification](https://github.com/blanchette/interactive_theorem_proving_2024) を参考にいたしました。-/