32 lines
959 B
Text
32 lines
959 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,
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L2 : (f y x) = (g A f x y) := Refl (g A f x y)
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in Trans 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 : Π 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)) := Trans L4 L1
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in Trans L5 L3
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10
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