31 lines
906 B
Text
31 lines
906 B
Text
Set: pp::colors
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Set: pp::unicode
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Imported 'Int'
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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 : ∀ x y : A, f x y = f y x) : f = g A f :=
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funext (λ x : A,
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funext (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 L1 ⋈ L2))
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theorem Inj (A B : Type)
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(h : A → B)
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(hinv : B → A)
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(Inv : ∀ x : A, hinv (h x) = x)
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(x y : A)
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(H : h x = h y) : x = 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) := L4 ⋈ L1
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in L5 ⋈ L3
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10
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