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choose を使わない例を示す #6

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Oct 3, 2023
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choose を使わない例を示す #6

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4052c8a
choose を使わない例を示す
aconite-ac Oct 3, 2023
36890a6
``choose.md``内の「選択公理」を「選択原理」に置換
aconite-ac Oct 3, 2023
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21 changes: 21 additions & 0 deletions Examples/Choose.lean
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Expand Up @@ -14,3 +14,24 @@ example (f : X → Y) (hf : ∀ y, ∃ x, f x = y) : ∃ g : Y → X, ∀ y, f (

exact ⟨g, hg⟩
-- ANCHOR_END: first


-- ANCHOR: no_choose
variable (P : X → Y → Prop)

noncomputable example (h : ∀ x, ∃ y, P x y) : ∃ f : X → Y, ∀ x, P x (f x) := by
-- `f` を作る
let f' : (x : X) → {y // P x y} := fun x ↦
have hne_st : Nonempty {y // P x y} :=
let ⟨y, py⟩ := h x; ⟨⟨y, py⟩⟩
Classical.choice hne_st

let f : X → Y := fun x ↦ (f' x).val

-- 上記で作った関数が条件を満たすことを示す
have h₁ : ∀ x, P x (f x) := by
intro x
exact (f' x).property

exists f
-- ANCHOR_END: no_choose
8 changes: 8 additions & 0 deletions src/choose.md
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Expand Up @@ -6,4 +6,12 @@ needs: `import Mathlib.Tactic.Choose`

```lean
{{#include ../Examples/Choose.lean:first}}
```

## 補足

`choose` が自動で示してくれることは選択原理 `Classical.choice` を使って手動で示すことができます．たとえば次のようになります．

```lean
{{#include ../Examples/Choose.lean:no_choose}}
```