TSTP Solution File: SEV028^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV028^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n105.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:36 EDT 2014
% Result : Timeout 300.03s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV028^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:36:31 CDT 2014
% % CPUTime : 300.03
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1de1dd0>, <kernel.Type object at 0x1cac0e0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1de17a0>, <kernel.DependentProduct object at 0x1cac638>) of role type named cQ
% Using role type
% Declaring cQ:(a->(a->Prop))
% FOF formula (((and (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), ((cQ Xx) Xx))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))) of role conjecture named cTHM558_pme
% Conjecture to prove = (((and (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), ((cQ Xx) Xx))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((and (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), ((cQ Xx) Xx))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))))']
% Parameter a:Type.
% Parameter cQ:(a->(a->Prop)).
% Trying to prove (((and (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), ((cQ Xx) Xx))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))):(((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))):(((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))):(((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found x1:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x1:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found (fun (x2:((cQ Xy) Xz))=> x1) as proof of (P b)
% Found (fun (x1:((cQ Xx) Xy)) (x2:((cQ Xy) Xz))=> x1) as proof of (((cQ Xy) Xz)->(P b))
% Found (fun (x1:((cQ Xx) Xy)) (x2:((cQ Xy) Xz))=> x1) as proof of (((cQ Xx) Xy)->(((cQ Xy) Xz)->(P b)))
% Found (and_rect00 (fun (x1:((cQ Xx) Xy)) (x2:((cQ Xy) Xz))=> x1)) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x1:((cQ Xx) Xy)) (x2:((cQ Xy) Xz))=> x1)) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((cQ Xx) Xy)->(((cQ Xy) Xz)->P0)))=> (((((and_rect ((cQ Xx) Xy)) ((cQ Xy) Xz)) P0) x1) x0)) (P b)) (fun (x1:((cQ Xx) Xy)) (x2:((cQ Xy) Xz))=> x1)) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((cQ Xx) Xy)->(((cQ Xy) Xz)->P0)))=> (((((and_rect ((cQ Xx) Xy)) ((cQ Xy) Xz)) P0) x1) x0)) (P b)) (fun (x1:((cQ Xx) Xy)) (x2:((cQ Xy) Xz))=> x1)) as proof of (P b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x3:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found x3 as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x3:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found (fun (x4:((cQ Xy) Xz))=> x3) as proof of (P b)
% Found (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3) as proof of (((cQ Xy) Xz)->(P b))
% Found (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3) as proof of (((cQ Xx) Xy)->(((cQ Xy) Xz)->(P b)))
% Found (and_rect10 (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((cQ Xx) Xy)->(((cQ Xy) Xz)->P0)))=> (((((and_rect ((cQ Xx) Xy)) ((cQ Xy) Xz)) P0) x3) x2)) (P b)) (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((cQ Xx) Xy)->(((cQ Xy) Xz)->P0)))=> (((((and_rect ((cQ Xx) Xy)) ((cQ Xy) Xz)) P0) x3) x2)) (P b)) (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: a0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of a0
% Found x1:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P b)
% Found x3:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found x3 as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x3:((cQ Xx) Xy)
% Instantiate: b:=Xy:a
% Found (fun (x4:((cQ Xy) Xz))=> x3) as proof of (P b)
% Found (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3) as proof of (((cQ Xy) Xz)->(P b))
% Found (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3) as proof of (((cQ Xx) Xy)->(((cQ Xy) Xz)->(P b)))
% Found (and_rect10 (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((cQ Xx) Xy)->(((cQ Xy) Xz)->P0)))=> (((((and_rect ((cQ Xx) Xy)) ((cQ Xy) Xz)) P0) x3) x0)) (P b)) (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((cQ Xx) Xy)->(((cQ Xy) Xz)->P0)))=> (((((and_rect ((cQ Xx) Xy)) ((cQ Xy) Xz)) P0) x3) x0)) (P b)) (fun (x3:((cQ Xx) Xy)) (x4:((cQ Xy) Xz))=> x3)) as proof of (P b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: a0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of a0
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found conj10:=(conj1 (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))):((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((and (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))
% Found (conj1 (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of ((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->a0))
% Found ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of ((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->a0))
% Found ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of ((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->a0))
% Found ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of ((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->a0))
% Found (and_rect00 ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) as proof of a0
% Found ((and_rect0 a0) ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) as proof of a0
% Found (((fun (P0:Type) (x0:((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) P0) x0) x)) a0) ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) as proof of a0
% Found (((fun (P0:Type) (x0:((forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) P0) x0) x)) a0) ((conj (forall (Xp:(a->Prop)), (((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx:a), ((Xp Xx)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx) Xy))))))->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((and ((ex a) (fun (Xz:a)=> (Xp Xz)))) (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((cQ Xx0) Xy))))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((and ((ex a) (fun (Xz:a)=> (Xq Xz)))) (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((cQ Xx0) Xy))))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cQ Xx) Xy)) ((cQ Xy) Xz))->((cQ Xx) Xz)))) b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: a0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of a0
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))):(((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a), (((cQ Xx) Xy)->((cQ Xy) Xx))))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) A
% EOF
%------------------------------------------------------------------------------