TSTP Solution File: SEV417^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV417^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n188.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 13:34:10 EDT 2014
% Result : Timeout 300.10s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV417^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n188.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 09:10:51 CDT 2014
% % CPUTime : 300.10
% 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 0x1daaf38>, <kernel.Type object at 0x21e2cf8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1daa4d0>, <kernel.DependentProduct object at 0x21e27e8>) of role type named cP
% Using role type
% Declaring cP:((a->Prop)->Prop)
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((and (((eq (a->Prop)) X) (fun (Xx:a)=> False))) (cP (fun (Xx:a)=> False))))) of role conjecture named cTHM502_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((and (((eq (a->Prop)) X) (fun (Xx:a)=> False))) (cP (fun (Xx:a)=> False))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((and (((eq (a->Prop)) X) (fun (Xx:a)=> False))) (cP (fun (Xx:a)=> False)))))']
% Parameter a:Type.
% Parameter cP:((a->Prop)->Prop).
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((and (((eq (a->Prop)) X) (fun (Xx:a)=> False))) (cP (fun (Xx:a)=> False)))))
% Found x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1 as proof of (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x300:=(x30 x1):(cP (fun (Xx:a)=> False))
% Found (x30 x1) as proof of (cP (fun (Xx:a)=> False))
% Found ((x3 cP) x1) as proof of (cP (fun (Xx:a)=> False))
% Found ((x3 cP) x1) as proof of (cP (fun (Xx:a)=> False))
% Found x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1 as proof of (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1 as proof of (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1 as proof of (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x1 as proof of (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x300:=(x30 x1):(cP (fun (Xx:a)=> False))
% Found (x30 x1) as proof of (cP (fun (Xx:a)=> False))
% Found ((x3 cP) x1) as proof of (cP (fun (Xx:a)=> False))
% Found ((x3 cP) x1) as proof of (cP (fun (Xx:a)=> False))
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x300:=(x30 x1):(cP (fun (Xx:a)=> False))
% Found (x30 x1) as proof of (cP (fun (Xx:a)=> False))
% Found ((x3 cP) x1) as proof of (cP (fun (Xx:a)=> False))
% Found (fun (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)) as proof of (cP (fun (Xx:a)=> False))
% Found (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)) as proof of ((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->(cP (fun (Xx:a)=> False)))
% Found (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)) as proof of (((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->(cP (fun (Xx:a)=> False))))
% Found (and_rect10 (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found ((and_rect1 (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P (X x2))->(P (X x2)))
% Found (eq_ref00 P) as proof of (P0 (X x2))
% Found ((eq_ref0 (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found eq_ref000:=(eq_ref00 P):((P (X x2))->(P (X x2)))
% Found (eq_ref00 P) as proof of (P0 (X x2))
% Found ((eq_ref0 (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 (cP (fun (Xx:a)=> False))):(((eq Prop) (cP (fun (Xx:a)=> False))) (cP (fun (Xx:a)=> False)))
% Found (eq_ref0 (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found ((eq_ref Prop) (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found ((eq_ref Prop) (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found ((eq_ref Prop) (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x300:=(x30 x1):(cP (fun (Xx:a)=> False))
% Found (x30 x1) as proof of (cP (fun (Xx:a)=> False))
% Found ((x3 cP) x1) as proof of (cP (fun (Xx:a)=> False))
% Found (fun (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)) as proof of (cP (fun (Xx:a)=> False))
% Found (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)) as proof of ((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->(cP (fun (Xx:a)=> False)))
% Found (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)) as proof of (((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->(cP (fun (Xx:a)=> False))))
% Found (and_rect10 (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found ((and_rect1 (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))) as proof of (cP (fun (Xx:a)=> False))
% Found (fun (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)))) as proof of (cP (fun (Xx:a)=> False))
% Found (fun (x0:((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)))) as proof of ((cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(cP (fun (Xx:a)=> False)))
% Found (fun (x0:((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1)))) as proof of (((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->((cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(cP (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x0:((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))))) as proof of (cP (fun (Xx:a)=> False))
% Found ((and_rect0 (cP (fun (Xx:a)=> False))) (fun (x0:((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))))) as proof of (cP (fun (Xx:a)=> False))
% Found (((fun (P:Type) (x0:(((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->((cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->P)))=> (((((and_rect ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))) P) x0) x)) (cP (fun (Xx:a)=> False))) (fun (x0:((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))))) as proof of (cP (fun (Xx:a)=> False))
% Found (((fun (P:Type) (x0:(((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->((cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->P)))=> (((((and_rect ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))) P) x0) x)) (cP (fun (Xx:a)=> False))) (fun (x0:((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) (x1:(cP (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> (((fun (P:Type) (x2:(((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))->((((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))->P)))=> (((((and_rect ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))) P) x2) x0)) (cP (fun (Xx:a)=> False))) (fun (x2:((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))=> ((x3 cP) x1))))) as proof of (cP (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (X x0))->(P (X x0)))
% Found (eq_ref00 P) as proof of (P0 (X x0))
% Found ((eq_ref0 (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found eq_ref000:=(eq_ref00 P):((P (X x0))->(P (X x0)))
% Found (eq_ref00 P) as proof of (P0 (X x0))
% Found ((eq_ref0 (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x4:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P (X x2))->(P (X x2)))
% Found (eq_ref00 P) as proof of (P0 (X x2))
% Found ((eq_ref0 (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found eq_ref000:=(eq_ref00 P):((P (X x2))->(P (X x2)))
% Found (eq_ref00 P) as proof of (P0 (X x2))
% Found ((eq_ref0 (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found (((eq_ref Prop) (X x2)) P) as proof of (P0 (X x2))
% Found eq_ref000:=(eq_ref00 P):((P False)->(P False))
% Found (eq_ref00 P) as proof of (P0 False)
% Found ((eq_ref0 False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P False)->(P False))
% Found (eq_ref00 P) as proof of (P0 False)
% Found ((eq_ref0 False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x3:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x3 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref000:=(eq_ref00 P):((P (X x0))->(P (X x0)))
% Found (eq_ref00 P) as proof of (P0 (X x0))
% Found ((eq_ref0 (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found eq_ref000:=(eq_ref00 P):((P (X x0))->(P (X x0)))
% Found (eq_ref00 P) as proof of (P0 (X x0))
% Found ((eq_ref0 (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 (cP (fun (Xx:a)=> False))):(((eq Prop) (cP (fun (Xx:a)=> False))) (cP (fun (Xx:a)=> False)))
% Found (eq_ref0 (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found ((eq_ref Prop) (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found ((eq_ref Prop) (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found ((eq_ref Prop) (cP (fun (Xx:a)=> False))) as proof of (((eq Prop) (cP (fun (Xx:a)=> False))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xx:a)=> False))
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x3 (fun (x4:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x3 (fun (x4:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found x30:=(x3 (fun (x4:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x3 (fun (x4:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x3 (fun (x4:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x30:=(x3 (fun (x6:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x3 (fun (x6:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% Found (((eq_ref (a->Prop)) X) P) as proof of (P0 X)
% F
% EOF
%------------------------------------------------------------------------------