2014-01-05 20:05:08 +00:00
|
|
|
import Int.
|
|
|
|
|
variable P : Int -> Int -> Bool
|
2013-12-16 23:43:09 +00:00
|
|
|
|
2014-01-05 20:05:08 +00:00
|
|
|
setopaque exists false.
|
2014-01-01 19:00:32 +00:00
|
|
|
|
2014-01-05 20:05:08 +00:00
|
|
|
theorem T1 (R1 : not (exists x y, P x y)) : forall x y, not (P x y) :=
|
2013-12-16 23:43:09 +00:00
|
|
|
ForallIntro (fun a,
|
|
|
|
|
ForallIntro (fun b,
|
|
|
|
|
ForallElim (DoubleNegElim (ForallElim (DoubleNegElim R1) a)) b))
|
|
|
|
|
|
2014-01-05 20:05:08 +00:00
|
|
|
axiom Ax : forall x, exists y, P x y
|
2013-12-16 23:43:09 +00:00
|
|
|
|
2014-01-05 20:05:08 +00:00
|
|
|
theorem T2 : exists x y, P x y :=
|
2013-12-16 23:43:09 +00:00
|
|
|
Refute (fun R : not (exists x y, P x y),
|
|
|
|
|
let L1 : forall x y, not (P x y) := ForallIntro (fun a,
|
|
|
|
|
ForallIntro (fun b,
|
|
|
|
|
ForallElim (DoubleNegElim (ForallElim (DoubleNegElim R) a)) b)),
|
|
|
|
|
L2 : exists y, P 0 y := ForallElim Ax 0
|
|
|
|
|
in ExistsElim L2 (fun (w : Int) (H : P 0 w),
|
|
|
|
|
Absurd H (ForallElim (ForallElim L1 0) w))).
|
|
|
|
|
|
2014-01-05 20:05:08 +00:00
|
|
|
theorem T3 (A : (Type U)) (P : A -> A -> Bool) (a : A) (H1 : forall x, exists y, P x y) : exists x y, P x y :=
|
2013-12-16 23:43:09 +00:00
|
|
|
Refute (fun R : not (exists x y, P x y),
|
|
|
|
|
let L1 : forall x y, not (P x y) := ForallIntro (fun a,
|
|
|
|
|
ForallIntro (fun b,
|
|
|
|
|
ForallElim (DoubleNegElim (ForallElim (DoubleNegElim R) a)) b)),
|
|
|
|
|
L2 : exists y, P a y := ForallElim H1 a
|
|
|
|
|
in ExistsElim L2 (fun (w : A) (H : P a w),
|
|
|
|
|
Absurd H (ForallElim (ForallElim L1 a) w))).
|
