lean2/tests/lean/exists3.lean

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import Int.
variable P : Int -> Int -> Bool
set::opaque exists false.
theorem T1 (R1 : not (exists x y, P x y)) : forall x y, not (P x y) :=
forall::intro (fun a,
forall::intro (fun b,
forall::elim (not::not::elim (forall::elim (not::not::elim 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) := forall::intro (fun a,
forall::intro (fun b,
forall::elim (not::not::elim (forall::elim (not::not::elim R) a)) b)),
L2 : exists y, P 0 y := forall::elim Ax 0
in exists::elim L2 (fun (w : Int) (H : P 0 w),
absurd H (forall::elim (forall::elim 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) := forall::intro (fun a,
forall::intro (fun b,
forall::elim (not::not::elim (forall::elim (not::not::elim R) a)) b)),
L2 : exists y, P a y := forall::elim H1 a
in exists::elim L2 (fun (w : A) (H : P a w),
absurd H (forall::elim (forall::elim L1 a) w))).