2014-01-05 20:05:08 +00:00
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import Int.
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variable P : Int -> Int -> Bool
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2013-12-16 23:43:09 +00:00
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2014-01-06 05:45:31 +00:00
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set::opaque exists false.
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2014-01-01 19:00:32 +00:00
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2014-01-05 20:05:08 +00:00
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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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2014-01-06 03:10:21 +00:00
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forall::intro (fun a,
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forall::intro (fun b,
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forall::elim (not::not::elim (forall::elim (not::not::elim R1) a)) b))
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2013-12-16 23:43:09 +00:00
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2014-01-05 20:05:08 +00:00
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axiom Ax : forall x, exists y, P x y
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2013-12-16 23:43:09 +00:00
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2014-01-05 20:05:08 +00:00
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theorem T2 : exists x y, P x y :=
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2014-01-06 03:10:21 +00:00
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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) := forall::intro (fun a,
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forall::intro (fun b,
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forall::elim (not::not::elim (forall::elim (not::not::elim R) a)) b)),
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L2 : exists y, P 0 y := forall::elim Ax 0
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in exists::elim L2 (fun (w : Int) (H : P 0 w),
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absurd H (forall::elim (forall::elim L1 0) w))).
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2013-12-16 23:43:09 +00:00
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2014-01-05 20:05:08 +00:00
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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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2014-01-06 03:10:21 +00:00
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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) := forall::intro (fun a,
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forall::intro (fun b,
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forall::elim (not::not::elim (forall::elim (not::not::elim R) a)) b)),
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L2 : exists y, P a y := forall::elim H1 a
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in exists::elim L2 (fun (w : A) (H : P a w),
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absurd H (forall::elim (forall::elim L1 a) w))).
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