33 lines
959 B
Text
33 lines
959 B
Text
|
Set: pp::colors
|
||
|
|
Set: pp::unicode
|
||
|
|
Assumed: A
|
||
|
|
Assumed: B
|
||
|
|
Assumed: f
|
||
|
|
Defined: g
|
||
|
|
Assumed: h
|
||
|
|
Assumed: hinv
|
||
|
|
Assumed: Inv
|
||
|
|
Assumed: H1
|
||
|
|
Proved: f_eq_g
|
||
|
|
Proved: Inj
|
||
|
|
Definition g (A : Type) (f : A → A → A) (x y : A) : A := f y x
|
||
|
|
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) :=
|
||
|
|
Abst (λ x : A,
|
||
|
|
Abst (λ y : A,
|
||
|
|
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 Trans L1 L2))
|
||
|
|
Theorem Inj (A B : Type)
|
||
|
|
(h : A → B)
|
||
|
|
(hinv : B → A)
|
||
|
|
(Inv : Π x : A, (hinv (h x)) = x)
|
||
|
|
(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
|
||
|
|
10
|