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import TestGame.Metadata
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import Std.Tactic.RCases
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import Mathlib.Tactic.Contrapose
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import Mathlib.Tactic.Use
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import Mathlib.Tactic.Ring
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import TestGame.ToBePorted
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Game "TestGame"
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World "Predicate"
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Level 8
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Title "Für alle"
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Introduction
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"
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Zum `∃` gehört auch das \"für alle\" `∀` (`\\forall`).
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Ein `∀` im Goal kann man mit `intro` angehen, genau wie bei einer Implikation `→`.
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"
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Statement
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" Für alle natürlichen Zahlen $x$ gilt, falls $x$ gerade ist, dann ist $x + 1$
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ungerade." : ∀ (x : ℕ), (even x) → odd (1 + x) := by
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intro x h
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unfold even at h
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unfold odd
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rcases h with ⟨y, hy⟩
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use y
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rw [hy]
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ring
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Message (n : ℕ) (hn : odd n) (h : ∀ (x : ℕ), (odd x) → even (x + 1)) : even (n + 1) =>
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"`∀ (x : ℕ), (odd x) → even (x + 1)` ist eigentlich das gleiche wie
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`(x : ℕ) → `"
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Conclusion "Für-alle-Statements, Implikationen und Lemmas/Theoreme verhalten sich alle
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praktisch gleich. Mehr dazu später."
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Tactics ring intro unfold
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