26 lines
867 B
Text
26 lines
867 B
Text
Set: pp::colors
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Set: pp::unicode
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Assumed: A
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Assumed: B
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Assumed: f
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Defined: g
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Assumed: h
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Assumed: hinv
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Assumed: Inv
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Assumed: H1
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Proved: f_eq_g
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Proved: Inj
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Definition g (A : Type) (f : A → A → A) (x y : A) : A := f y x
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Theorem f_eq_g (A : Type) (f : A → A → A) (H1 : Π x y : A, f x y = f y x) : f = g A f :=
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Abst (λ x : A,
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Abst (λ y : A,
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let L1 : f x y = f y x := H1 x y, L2 : f y x = g A f x y := Refl (g A f x y) in Trans L1 L2))
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Theorem Inj (A B : Type) (h : A → B) (hinv : B → A) (Inv : Π x : A, hinv (h x) = x) (x y : A) (H : h x = h y) : x =
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y :=
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let L1 : hinv (h x) = hinv (h y) := Congr2 hinv H,
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L2 : hinv (h x) = x := Inv x,
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L3 : hinv (h y) = y := Inv y,
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L4 : x = hinv (h x) := Symm L2,
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L5 : x = hinv (h y) := Trans L4 L1
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in Trans L5 L3
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10
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