levels
Jon Eugster
2 years ago
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f636a8d833
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68606aab99
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import Adam.Metadata
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import Mathlib
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Game "Adam"
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World "Function"
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Level 10
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Title "Inverse"
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Introduction
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"
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Eigentlich hast du nur beiläufig Robo gefragt, ob bijektiv nicht auch bedeute, dass
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eine Inverse Funktion bestehe. Jetzt steht ihr aber schon seit einer halben Stunde rum
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und der Gelehrte möchte wissen, wie das den genau ginge.
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Offensichtlich kennt er diese Aussage als `Function.bijective_iff_has_inverse` aus seinen Büchern,
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aber er möchte, dass du ihm das hier und jetzt nochmals von Grund auf zeigst.
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"
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open Function
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set_option pp.rawOnError true
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-- bijective_iff_has_inverse
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Statement {A B : Type} (f : A → B) :
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Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f := by
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Branch
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exfalso
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Hint "Das war eine blöde Idee
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dd
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ddds
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"
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Hint (hidden := true) "constructor"
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constructor
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Hint "intro h" -- does not show
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intro h
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Hint "rcases h with ⟨h₁, h₂⟩" -- shows too late: 1, 2 after
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rcases h with ⟨h₁, h₂⟩
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Hint (strict := true) "let g := fun x => (h₂ x).choose" -- shows correct + 1 after
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let g := fun x => (h₂ x).choose
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Hint "use g"
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use g
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Hint "constructor" -- does not show
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constructor
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Hint "intro x" -- does not show
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intro x
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Hint "simp" -- Error updating: Error fetching goals: Rpc error: InternalError: unknown universe metavariable '?_uniq.286465'. Try again.
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simp
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Hint "apply h₁"
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apply h₁
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Hint "apply Exists.choose_spec (h₂ (f x))"
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apply Exists.choose_spec (h₂ (f x))
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Hint "intro y" -- Error updating: Error fetching goals: Rpc error: InternalError: unknown universe metavariable '?_uniq.286465'. Try again.
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intro y
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Hint "simp"
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simp
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Hint "apply Exists.choose_spec (h₂ y)"
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apply Exists.choose_spec (h₂ y)
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Hint "intro ⟨g, h₁, h₂⟩"
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intro ⟨g, h₁, h₂⟩
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Hint "constructor"
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constructor
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Hint "intro a b hab"
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intro a b hab
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Hint (strict := true) "have h : g (f a) = g (f b)"
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have h : g (f a) = g (f b)
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Hint "apply congrArg"
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apply congrArg
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Hint "assumption"
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assumption
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Hint "rw [h₁, h₁] at h"
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rw [h₁, h₁] at h
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Hint "assumption"
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assumption
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Hint "intro x"
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intro x
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Hint "use g x"
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use g x
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Hint "rw [h₂]"
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rw [h₂]
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-- NewDefinition LeftInverse RightInverse
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-- NewLemma Exists.choose Exists.choose_spec congrArg congrFun
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-- DisabledLemma Function.bijective_iff_has_inverse
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Hint (A B: Type) (f : A → B) (h : Bijective f) :
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(∃ g, LeftInverse g f ∧ RightInverse g f) → Bijective f =>
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"Test"
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Conclusion
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"Endlich entkommt ihr dem Tempel.
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**Robo**: Da würden mich keine zehn Pferde nochmals hineinbringen!
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**Du**: Von wegen Pferden, wie viele PS hat eigentlich unser Raumschiff?"
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