Created
December 6, 2025 10:58
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| variable {α : Type} | |
| /-- リストの長さが偶数であることを表す述語 -/ | |
| @[grind] | |
| inductive EvenLength : List α → Prop | |
| | nil : EvenLength [] | |
| | cons_cons {a b : α} {l : List α} (h : EvenLength l) : | |
| EvenLength (a :: b :: l) | |
| example (xs : List α) : EvenLength xs ↔ xs.length % 2 = 0 := by | |
| revert xs | |
| suffices | |
| (∀ (xs : List α)(a : α), ((EvenLength xs ↔ xs.length % 2 = 0) ∧ (EvenLength (a :: xs) ↔ (a :: xs).length % 2 = 0))) by { | |
| intro xs | |
| rcases xs with _ | ⟨a, xs⟩<;> grind | |
| -- simp; constructor; simp | |
| -- exact (this xs a).2 | |
| } | |
| intro xs | |
| induction xs with | |
| | nil => | |
| grind | |
| -- intro x | |
| -- simp; constructor; constructor | |
| -- intro F; rcases F | |
| | cons y l IH => | |
| intro x; simp | |
| rcases IH y with ⟨IHl, IHy⟩ | |
| grind | |
| -- constructor; assumption | |
| -- rw [Nat.add_assoc]; simp | |
| -- rw [<- IHl] | |
| -- constructor<;> intro H | |
| -- · rcases H; assumption | |
| -- · constructor; assumption |
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