TSTP Solution File: SEU873^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU873^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n089.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 13:33:19 EDT 2014
% Result : Timeout 300.05s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU873^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n089.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 11:38:56 CDT 2014
% % CPUTime : 300.05
% 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 0x2a10dd0>, <kernel.Type object at 0x2a10a28>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2a10b00>, <kernel.DependentProduct object at 0x29f18c0>) of role type named cC
% Using role type
% Declaring cC:(a->Prop)
% FOF formula (<kernel.Constant object at 0x2a6c8c0>, <kernel.DependentProduct object at 0x29f1320>) of role type named cB
% Using role type
% Declaring cB:(a->Prop)
% FOF formula (((and (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P Z))))->(P cB)))) (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P Z))))->(P cC))))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_22:a), ((iff (Z Xx_22)) ((or (Y Xx_22)) (((eq a) Xx_22) Xx)))))->(P Z))))->(P (fun (Xz:a)=> ((or (cB Xz)) (cC Xz))))))) of role conjecture named cTHM549_pme
% Conjecture to prove = (((and (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P Z))))->(P cB)))) (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P Z))))->(P cC))))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_22:a), ((iff (Z Xx_22)) ((or (Y Xx_22)) (((eq a) Xx_22) Xx)))))->(P Z))))->(P (fun (Xz:a)=> ((or (cB Xz)) (cC Xz))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((and (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P Z))))->(P cB)))) (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P Z))))->(P cC))))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_22:a), ((iff (Z Xx_22)) ((or (Y Xx_22)) (((eq a) Xx_22) Xx)))))->(P Z))))->(P (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))))))']
% Parameter a:Type.
% Parameter cC:(a->Prop).
% Parameter cB:(a->Prop).
% Trying to prove (((and (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P Z))))->(P cB)))) (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P Z))))->(P cC))))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_22:a), ((iff (Z Xx_22)) ((or (Y Xx_22)) (((eq a) Xx_22) Xx)))))->(P Z))))->(P (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) (fun (x:a)=> ((or (cB x)) (cC x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) (fun (x:a)=> ((or (cB x)) (cC x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (cB x1)) (cC x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) (fun (x:a)=> ((or (cB x)) (cC x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) (fun (x:a)=> ((or (cB x)) (cC x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cB Xz)) (cC Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and ((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y0:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P10 Y0)) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y0 Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz)))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->((forall (P10:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P10 E)))) (forall (Y:(a->Prop)) (Xx:a) (Z0:(a->Prop)), (((and (P10 Y)) (forall (Xx_21:a), ((iff (Z0 Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P10 Z0))))->(P10 cC)))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cB x3)) (cC x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cB x)) (cC x))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (Y Xz)) (cC Xz))))) (forall (Xx_20:a), ((iff (Z Xx_20)) ((or (Y Xx_20)) (((eq a) Xx_20) Xx)))))->(P (fun (Xz:a)=> ((or (Z Xz)) (cC Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx_21)) ((or (Y Xx_21)) (((eq a) Xx_21) Xx)))))->(P (fun (Xz:a)=> ((or (cB Xz)) (Z Xz))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P (fun (Xz:a)=> ((or (cB Xz)) (Y Xz))))) (forall (Xx_21:a), ((iff (Z Xx
% EOF
%------------------------------------------------------------------------------