2013-12-30 21:35:37 +00:00
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Import int.
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2013-12-19 01:40:21 +00:00
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Scope
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Variable A : Type
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Variable B : Type
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Variable f : A -> A -> A
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Definition g (x y : A) : A := f y x
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Variable h : A -> B
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Variable hinv : B -> A
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Axiom Inv (x : A) : hinv (h x) = x
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Axiom H1 (x y : A) : f x y = f y x
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Theorem f_eq_g : f = g := Abst (fun x, (Abst (fun y,
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let L1 : f x y = f y x := H1 x y,
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L2 : f y x = g x y := Refl (g x y)
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in Trans L1 L2)))
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Theorem Inj (x y : A) (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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EndScope
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Show Environment 3.
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Eval g Int Int::sub 10 20
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