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import TestGame.Metadata
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import TestGame.Options.BigOperators
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import Mathlib.Algebra.BigOperators.Fin
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import Mathlib.Tactic.Ring
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Game "TestGame"
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World "Sum"
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Level 4
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Title "Summe aller ungeraden Zahlen"
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Introduction
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"
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Hier nochmals eine Übung zur Induktion.
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"
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set_option tactic.hygienic false
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open BigOperators
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Statement odd_arithmetic_sum
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"Zeige folgende Gleichung zur Summe aller ungeraden Zahlen:
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$\\sum_{i = 0}^n (2n + 1) = n ^ 2$."
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(n : ℕ) : (∑ i : Fin n, (2 * (i : ℕ) + 1)) = n ^ 2 := by
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induction' n with n hn
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simp
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rw [Fin.sum_univ_castSucc]
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simp
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rw [hn]
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rw [Nat.succ_eq_add_one]
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ring
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HiddenHint (n : ℕ) : (∑ i : Fin n, (2 * (i : ℕ) + 1)) = n ^ 2 =>
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"
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Fange wieder mit `induction {n}` an.
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"
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HiddenHint : ∑ i : Fin Nat.zero, ((2 : ℕ) * i + 1) = Nat.zero ^ 2 =>
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"
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Den Induktionsanfang kannst du wieder mit `simp` beweisen.
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"
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HiddenHint (n : ℕ) : ∑ i : Fin (Nat.succ n), ((2 : ℕ) * i + 1) = Nat.succ n ^ 2 =>
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"
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Den Induktionsschritt startest du mit `rw [Fin.sum_univ_castSucc]`.
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"
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HiddenHint (n : ℕ) (hn : ∑ i : Fin n, (2 * (i : ℕ) + 1) = n ^ 2) :
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∑ x : Fin n, (2 * (x : ℕ) + 1) + (2 * n + 1) = Nat.succ n ^ 2 =>
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"
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Hier kommt die Induktionshypothese {hn} ins Spiel.
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"
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HiddenHint (n : ℕ) : n ^ 2 + (2 * n + 1) = Nat.succ n ^ 2 =>
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"
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Mit `rw [Nat.succ_eq_add_one]` und `ring` kannst du hier abschliessen.
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"
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