lean2/tests/lean/scope.lean
2014-01-08 17:25:14 -08:00

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import Int.
scope
variable A : Type
variable B : Type
variable f : A -> A -> A
definition g (x y : A) : A := f y x
variable h : A -> B
variable hinv : B -> A
axiom Inv (x : A) : hinv (h x) = x
axiom H1 (x y : A) : f x y = f y x
theorem f_eq_g : f = g := funext (fun x, (funext (fun y,
let L1 : f x y = f y x := H1 x y,
L2 : f y x = g x y := refl (g x y)
in trans L1 L2)))
theorem Inj (x y : A) (H : h x = h y) : x = y :=
let L1 : hinv (h x) = hinv (h y) := congr2 hinv H,
L2 : hinv (h x) = x := Inv x,
L3 : hinv (h y) = y := Inv y,
L4 : x = hinv (h x) := symm L2,
L5 : x = hinv (h y) := trans L4 L1
in trans L5 L3.
end
print environment 3.
eval g Int Int::sub 10 20