27 lines
No EOL
912 B
Text
27 lines
No EOL
912 B
Text
import Int.
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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 := funext (fun x, (funext (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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end
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print environment 3.
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eval g Int Int::sub 10 20 |