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import Adam.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 "Adam"
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World "Implication"
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Level 9
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Title "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 h₁ \\| h₂` |
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| Goal | `constructor` | `left`/`right` |
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"
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Statement and_or_imp
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"Angenommen $(A \\land B) \\lor (A \\Rightarrow C)$ und $A$ sind wahr, zeige dass
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$B \\lor (C \\land A)$ wahr ist."
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(A B C : Prop) (h : (A ∧ B) ∨ (A → C)) (hA : A) : (B ∨ (C ∧ A)) := by
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rcases h with h₁ | h₂
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left
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rcases h₁ with ⟨x, y⟩
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assumption
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right
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constructor
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apply h₂
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assumption
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assumption
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HiddenHint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B ∨ (A → C)) (hA : A) : B ∨ (C ∧ A) =>
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"Ein ODER in den Annahmen teilt man mit `rcases h with h₁ | h₂`."
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-- If starting with `left`.
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B ∨ (A → C)) : B =>
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"Da kommst du nicht mehr weiter..."
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-- If starting with `right`.
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B ∨ (A → C)) : (C ∧ A) =>
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"Da kommst du nicht mehr weiter..."
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HiddenHint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B) (hA : A) : B ∨ (C ∧ A) =>
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"`left` oder `right`?"
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HiddenHint (A : Prop) (B : Prop) (C : Prop) (h : B) (hA : A) : B ∨ (C ∧ A) =>
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"`left` oder `right`?"
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B) (hA : A) : B ∨ (C ∧ A) =>
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"Ein UND in den Annahmen kann man mit `rcases h with ⟨h₁, h₂⟩` aufteilen."
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B) (hA : A) : B =>
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"Ein UND in den Annahmen kann man mit `rcases h with ⟨h₁, h₂⟩` aufteilen."
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B) : C =>
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"Sackgasse."
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A ∧ B) : C ∧ A =>
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"Hmmm..."
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Hint (A : Prop) (B : Prop) (C : Prop) (h : A → C) : C ∧ A =>
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"Ein UND im Goal kann mit `constructor` aufgeteilt werden."
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NewTactic left right assumption constructor rcases
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