lean2/tests/lean/exists3.lean

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Import Int.
Variable P : Int -> Int -> Bool
SetOpaque exists false.
Theorem T1 (R1 : not (exists x y, P x y)) : forall x y, not (P x y) :=
ForallIntro (fun a,
ForallIntro (fun b,
ForallElim (DoubleNegElim (ForallElim (DoubleNegElim R1) a)) b))
Axiom Ax : forall x, exists y, P x y
Theorem T2 : exists x y, P x y :=
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))).
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 :=
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))).