2014-01-01 21:52:25 +00:00
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Import Int.
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2014-01-01 19:35:21 +00:00
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Variable P : Int -> Int -> Bool
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Theorem T1 (R1 : not (exists x y, P x y)) : forall x y, not (P x y) :=
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ForallIntro (fun a,
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ForallIntro (fun b,
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ForallElim (NotExistsElim (ForallElim (NotExistsElim R1) a)) b))
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Axiom Ax : forall x, exists y, P x y
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Theorem T2 : exists x y, P x y :=
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Refute (fun R : not (exists x y, P x y),
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let L1 : forall x y, not (P x y) := ForallIntro (fun a,
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ForallIntro (fun b,
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ForallElim (NotExistsElim (ForallElim (NotExistsElim R) a)) b)),
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L2 : exists y, P 0 y := ForallElim Ax 0
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in ExistsElim L2 (fun (w : Int) (H : P 0 w),
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Absurd H (ForallElim (ForallElim L1 0) w))).
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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 :=
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Refute (fun R : not (exists x y, P x y),
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let L1 : forall x y, not (P x y) := ForallIntro (fun a,
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ForallIntro (fun b,
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ForallElim (NotExistsElim (ForallElim (NotExistsElim R) a)) b)),
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L2 : exists y, P a y := ForallElim H1 a
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in ExistsElim L2 (fun (w : A) (H : P a w),
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Absurd H (ForallElim (ForallElim L1 a) w))).
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