lean2/tests/lean/exists8.lean

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