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import TestGame.Metadata
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import Std.Tactic.RCases
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import Mathlib.Tactic.LeftRight
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set_option tactic.hygienic false
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Game "TestGame"
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World "Proposition"
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Level 13
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Title "Und/Oder"
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Introduction
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"
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Übung macht den Meister: Benutze alle vier Methoden mit UND und ODER
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umzugehen um folgende Aussage zu beweisen.
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| | Und (`∧`) | Oder (`∨`) |
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|---------|:-------------------------|:------------------------|
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| Annahme | `rcases h with ⟨h₁, h₂⟩` | `rcases h with h \\| h` |
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| Goal | `constructor` | `left`/`right` |
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"
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-- Note: The other direction would need arguing by cases.
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Statement or_and_left
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"Angenommen $A \\lor (B \\land C)$ ist wahr, zeige dass
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$(A \\lor B) \\land (A \\lor C)$ wahr ist."
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(A B C : Prop) (h : A ∨ (B ∧ C)) : (A ∨ B) ∧ (A ∨ C) := by
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constructor
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rcases h with h | h
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left
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assumption
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rcases h with ⟨h₁, h₂⟩
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right
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assumption
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rcases h with h | h
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left
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assumption
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rcases h with ⟨h₁, h₂⟩
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right
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assumption
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Message (A : Prop) (B : Prop) (C : Prop) (h : B ∧ C) : (A ∨ B) ∧ (A ∨ C) =>
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"Das `∧` in der Annahme kann mit `rcases h with ⟨h₁, h₂⟩` zerlegt werden."
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Message (A : Prop) (B : Prop) (C : Prop) : (A ∨ B) ∧ (A ∨ C) =>
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"Das `∧` im Goal kann mit `constructor` zerlegt werden."
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Message (A : Prop) (B : Prop) (C : Prop) (h : A ∨ (B ∧ C)) : (A ∨ B) ∧ (A ∨ C) =>
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"Das `∨` in der Annahme kann mit `rcases h with h | h` zerlegt werden."
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Message (A : Prop) (B : Prop) (C : Prop) (h : A ∨ (B ∧ C)) : (A ∨ B) =>
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"Das `∨` in der Annahme kann mit `rcases h with h | h` zerlegt werden."
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Message (A : Prop) (B : Prop) (C : Prop) (h : A ∨ (B ∧ C)) : (A ∨ C) =>
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"Das `∨` in der Annahme kann mit `rcases h with h | h` zerlegt werden."
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Message (A : Prop) (B : Prop) (C : Prop) (h : B ∧ C) : (A ∨ B) =>
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"Das `∧` in der Annahme kann mit `rcases h with ⟨h₁, h₂⟩` zerlegt werden."
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Message (A : Prop) (B : Prop) (C : Prop) (h : B ∧ C) : (A ∨ C) =>
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"Das `∧` in der Annahme kann mit `rcases h with ⟨h₁, h₂⟩` zerlegt werden."
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-- TODO: Message nur Anhand der Annahmen?
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Hint (A : Prop) (B : Prop) : A ∨ B =>
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"`left` oder `right`?"
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Tactics left right assumption constructor rcases
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