TSTP Solution File: SET011^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET011^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n098.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 13:29:54 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET011^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n098.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:32:21 CDT 2014
% % CPUTime : 300.02
% 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 0x94b440>, <kernel.Type object at 0x94b878>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))) of role conjecture named cBOOL_PROP_82_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (x:a)=> ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% 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 (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) (fun (x:a)=> ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% 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 (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% 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 (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))):(((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))):(((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% 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 (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))):(((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))):(((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 (P ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 (P ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 (P ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found x01:(P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (fun (x01:(P ((and (X x)) (not ((and (X x)) (not (Y x)))))))=> x01) as proof of (P0 (P ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found eq_ref00:=(eq_ref0 (((and (X x)) ((Y x)->False))->False)):(((eq Prop) (((and (X x)) ((Y x)->False))->False)) (((and (X x)) ((Y x)->False))->False))
% Found (eq_ref0 (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found eq_ref00:=(eq_ref0 (((and (X x)) ((Y x)->False))->False)):(((eq Prop) (((and (X x)) ((Y x)->False))->False)) (((and (X x)) ((Y x)->False))->False))
% Found (eq_ref0 (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 (((and (X x)) ((Y x)->False))->False)):(((eq Prop) (((and (X x)) ((Y x)->False))->False)) (((and (X x)) ((Y x)->False))->False))
% Found (eq_ref0 (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found eq_ref00:=(eq_ref0 (((and (X x)) ((Y x)->False))->False)):(((eq Prop) (((and (X x)) ((Y x)->False))->False)) (((and (X x)) ((Y x)->False))->False))
% Found (eq_ref0 (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found ((eq_ref Prop) (((and (X x)) ((Y x)->False))->False)) as proof of (((eq Prop) (((and (X x)) ((Y x)->False))->False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found eq_ref00:=(eq_ref0 (not ((and (X x)) (not (Y x))))):(((eq Prop) (not ((and (X x)) (not (Y x))))) (not ((and (X x)) (not (Y x)))))
% Found (eq_ref0 (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found ((eq_ref Prop) (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found ((eq_ref Prop) (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found ((eq_ref Prop) (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x))
% Found eq_ref00:=(eq_ref0 (not ((and (X x)) (not (Y x))))):(((eq Prop) (not ((and (X x)) (not (Y x))))) (not ((and (X x)) (not (Y x)))))
% Found (eq_ref0 (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found ((eq_ref Prop) (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found ((eq_ref Prop) (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found ((eq_ref Prop) (not ((and (X x)) (not (Y x))))) as proof of (((eq Prop) (not ((and (X x)) (not (Y x))))) b)
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (x:a)=> ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) 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 (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((P ((and (X x)) (Y x)))->(P ((and (X x)) (Y x)))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (Y x))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 ((and (X x)) (Y x)))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found x01:(P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P ((and (X x)) (Y x)))
% Found (fun (x01:(P ((and (X x)) (Y x))))=> x01) as proof of (P0 (P ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))):(((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))):(((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (eq_ref0 ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found ((eq_ref Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) as proof of (((eq Prop) ((and (X x)) (((and (X x)) ((Y x)->False))->False))) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) (Y x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) (fun (x:a)=> ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) 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 (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))):(((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))):(((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) ((and (X x)) (not ((and (X x)) (not (Y x))))))
% Found (eq_ref0 ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found ((eq_ref Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) as proof of (((eq Prop) ((and (X x)) (not ((and (X x)) (not (Y x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) (Y x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))):(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x3) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))))=> x3) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (x:a)=> ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) 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 (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found x0:(P0 ((and (X x)) (Y x)))
% Found (fun (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P0 ((and (X x)) (Y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of ((P0 ((and (X x)) (Y x)))->(P0 ((and (X x)) (Y x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P (Y x))
% Found x0:(P0 ((and (X x)) (Y x)))
% Found (fun (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P0 ((and (X x)) (Y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of ((P0 ((and (X x)) (Y x)))->(P0 ((and (X x)) (Y x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P (Y x))
% Found x0:(P0 ((and (X x)) (Y x)))
% Found (fun (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P0 ((and (X x)) (Y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of ((P0 ((and (X x)) (Y x)))->(P0 ((and (X x)) (Y x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P (Y x))
% Found x0:(P0 ((and (X x)) (Y x)))
% Found (fun (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P0 ((and (X x)) (Y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of ((P0 ((and (X x)) (Y x)))->(P0 ((and (X x)) (Y x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (X x)) (Y x))))=> x0) as proof of (P (Y x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found x01:(P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P0 (Y x))
% Found x01:(P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P0 (Y x))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))):(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x3) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (fun (x3:(P (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))))=> x3) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx)))))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) 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 (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) (fun (x:a)=> ((and (X x)) (not ((and (X x)) (not (Y x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not ((and (X Xx)) (not (Y Xx))))))) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x as proof of (P0 f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) (fun (x:a)=> ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (X x)) ((Y x)->False))->False))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found x0:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Instantiate: b:=((and (X x)) (((and (X x)) ((Y x)->False))->False)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found x01:(P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P0 (Y x))
% Found x01:(P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P (Y x))
% Found (fun (x01:(P (Y x)))=> x01) as proof of (P0 (Y x))
% Found x0:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Instantiate: b:=((and (X x)) (((and (X x)) ((Y x)->False))->False)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (((eq Prop) ((and (X x)) (Y x))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) (Y x))):(((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x)))
% Found (eq_ref0 ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found ((eq_ref Prop) ((and (X x)) (Y x))) as proof of (P (((eq Prop) ((and (X x)) (Y x))) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 (Y x)):(((eq Prop) (Y x)) (Y x))
% Found (eq_ref0 (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found ((eq_ref Prop) (Y x)) as proof of (((eq Prop) (Y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not ((and (X x)) (not (Y 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 (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) 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 (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) 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 (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (((and (X Xx)) ((Y Xx)->False))->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 (P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False))))=> x01) as proof of (P0 ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found x01:(P ((and (X x)) (((and (X x)) ((Y x)->False))->False)))
% Found (fun (x01:(P ((and (X x)) (((an
% EOF
%------------------------------------------------------------------------------