TSTP Solution File: SEV169^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV169^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n184.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:50 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 : SEV169^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.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 08:19:16 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 0xe7b488>, <kernel.Type object at 0xe7b1b8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((and ((and ((and (((eq ((a->(a->a))->a)) Xp) (fun (Xg:(a->(a->a)))=> ((Xg (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy)))))) (((eq ((a->(a->a))->a)) Xq) (fun (Xg:(a->(a->a)))=> ((Xg (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xx))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))->(((eq ((a->(a->a))->a)) Xp) Xq))) of role conjecture named cTHM188_pme
% Conjecture to prove = (forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((and ((and ((and (((eq ((a->(a->a))->a)) Xp) (fun (Xg:(a->(a->a)))=> ((Xg (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy)))))) (((eq ((a->(a->a))->a)) Xq) (fun (Xg:(a->(a->a)))=> ((Xg (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xx))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))->(((eq ((a->(a->a))->a)) Xp) Xq))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((and ((and ((and (((eq ((a->(a->a))->a)) Xp) (fun (Xg:(a->(a->a)))=> ((Xg (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy)))))) (((eq ((a->(a->a))->a)) Xq) (fun (Xg:(a->(a->a)))=> ((Xg (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xx))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))->(((eq ((a->(a->a))->a)) Xp) Xq)))']
% Parameter a:Type.
% Trying to prove (forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((and ((and ((and (((eq ((a->(a->a))->a)) Xp) (fun (Xg:(a->(a->a)))=> ((Xg (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy)))))) (((eq ((a->(a->a))->a)) Xq) (fun (Xg:(a->(a->a)))=> ((Xg (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xx))))) (((eq a) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xy))))->(((eq ((a->(a->a))->a)) Xp) Xq)))
% Found eq_ref000:=(eq_ref00 P):((P Xp)->(P Xp))
% Found (eq_ref00 P) as proof of (P0 Xp)
% Found ((eq_ref0 Xp) P) as proof of (P0 Xp)
% Found (((eq_ref ((a->(a->a))->a)) Xp) P) as proof of (P0 Xp)
% Found (((eq_ref ((a->(a->a))->a)) Xp) P) as proof of (P0 Xp)
% Found eq_ref000:=(eq_ref00 P):((P Xp)->(P Xp))
% Found (eq_ref00 P) as proof of (P0 Xp)
% Found ((eq_ref0 Xp) P) as proof of (P0 Xp)
% Found (((eq_ref ((a->(a->a))->a)) Xp) P) as proof of (P0 Xp)
% Found (((eq_ref ((a->(a->a))->a)) Xp) P) as proof of (P0 Xp)
% Found eq_ref000:=(eq_ref00 P):((P Xp)->(P Xp))
% Found (eq_ref00 P) as proof of (P0 Xp)
% Found ((eq_ref0 Xp) P) as proof of (P0 Xp)
% Found (((eq_ref ((a->(a->a))->a)) Xp) P) as proof of (P0 Xp)
% Found (((eq_ref ((a->(a->a))->a)) Xp) P) as proof of (P0 Xp)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xp):(((eq ((a->(a->a))->a)) Xp) (fun (x:(a->(a->a)))=> (Xp x)))
% Found (eta_expansion_dep00 Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->(a->a)))=> a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found (((eta_expansion_dep (a->(a->a))) (fun (x1:(a->(a->a)))=> a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found (((eta_expansion_dep (a->(a->a))) (fun (x1:(a->(a->a)))=> a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found (((eta_expansion_dep (a->(a->a))) (fun (x1:(a->(a->a)))=> a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found x10:=(x1 (fun (x2:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x1 (fun (x2:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x1 (fun (x2:a)=> (P Xp))) as proof of (P0 Xp)
% Found x10:=(x1 (fun (x2:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x1 (fun (x2:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x1 (fun (x2:a)=> (P Xp))) as proof of (P0 Xp)
% Found x10:=(x1 (fun (x2:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x1 (fun (x2:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x1 (fun (x2:a)=> (P Xp))) as proof of (P0 Xp)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of (P0 Xq)
% Found ((eq_ref0 Xq) P) as proof of (P0 Xq)
% Found (((eq_ref ((a->(a->a))->a)) Xq) P) as proof of (P0 Xq)
% Found (((eq_ref ((a->(a->a))->a)) Xq) P) as proof of (P0 Xq)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of (P0 Xq)
% Found ((eq_ref0 Xq) P) as proof of (P0 Xq)
% Found (((eq_ref ((a->(a->a))->a)) Xq) P) as proof of (P0 Xq)
% Found (((eq_ref ((a->(a->a))->a)) Xq) P) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found eta_expansion000:=(eta_expansion00 Xp):(((eq ((a->(a->a))->a)) Xp) (fun (x:(a->(a->a)))=> (Xp x)))
% Found (eta_expansion00 Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found ((eta_expansion0 a) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found (((eta_expansion (a->(a->a))) a) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found (((eta_expansion (a->(a->a))) a) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found (((eta_expansion (a->(a->a))) a) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found eq_ref000:=(eq_ref00 P):((P (Xp x0))->(P (Xp x0)))
% Found (eq_ref00 P) as proof of (P0 (Xp x0))
% Found ((eq_ref0 (Xp x0)) P) as proof of (P0 (Xp x0))
% Found (((eq_ref a) (Xp x0)) P) as proof of (P0 (Xp x0))
% Found (((eq_ref a) (Xp x0)) P) as proof of (P0 (Xp x0))
% Found eq_ref000:=(eq_ref00 P):((P (Xp x0))->(P (Xp x0)))
% Found (eq_ref00 P) as proof of (P0 (Xp x0))
% Found ((eq_ref0 (Xp x0)) P) as proof of (P0 (Xp x0))
% Found (((eq_ref a) (Xp x0)) P) as proof of (P0 (Xp x0))
% Found (((eq_ref a) (Xp x0)) P) as proof of (P0 (Xp x0))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion0 a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found x10:=(x1 (fun (x4:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x1 (fun (x4:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x1 (fun (x4:a)=> (P Xp))) as proof of (P0 Xp)
% Found eq_ref00:=(eq_ref0 (Xp x0)):(((eq a) (Xp x0)) (Xp x0))
% Found (eq_ref0 (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found eq_ref00:=(eq_ref0 (Xp x0)):(((eq a) (Xp x0)) (Xp x0))
% Found (eq_ref0 (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found x30:=(x3 (fun (x4:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x3 (fun (x4:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x3 (fun (x4:a)=> (P Xp))) as proof of (P0 Xp)
% Found x30:=(x3 (fun (x4:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x3 (fun (x4:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x3 (fun (x4:a)=> (P Xp))) as proof of (P0 Xp)
% Found x10:=(x1 (fun (x2:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found x10:=(x1 (fun (x2:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xq)
% Found eq_ref00:=(eq_ref0 Xp):(((eq ((a->(a->a))->a)) Xp) Xp)
% Found (eq_ref0 Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found ((eq_ref ((a->(a->a))->a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found ((eq_ref ((a->(a->a))->a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found ((eq_ref ((a->(a->a))->a)) Xp) as proof of (((eq ((a->(a->a))->a)) Xp) b)
% Found x10:=(x1 (fun (x3:a)=> (P (Xp x2)))):((P (Xp x2))->(P (Xp x2)))
% Found (x1 (fun (x3:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found (x1 (fun (x3:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found x10:=(x1 (fun (x3:a)=> (P (Xp x2)))):((P (Xp x2))->(P (Xp x2)))
% Found (x1 (fun (x3:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found (x1 (fun (x3:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found x10:=(x1 (fun (x2:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found x10:=(x1 (fun (x2:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x1 (fun (x2:a)=> (P Xq))) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion0 a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found eq_ref000:=(eq_ref00 P):((P (Xq x0))->(P (Xq x0)))
% Found (eq_ref00 P) as proof of (P0 (Xq x0))
% Found ((eq_ref0 (Xq x0)) P) as proof of (P0 (Xq x0))
% Found (((eq_ref a) (Xq x0)) P) as proof of (P0 (Xq x0))
% Found (((eq_ref a) (Xq x0)) P) as proof of (P0 (Xq x0))
% Found eq_ref000:=(eq_ref00 P):((P (Xq x0))->(P (Xq x0)))
% Found (eq_ref00 P) as proof of (P0 (Xq x0))
% Found ((eq_ref0 (Xq x0)) P) as proof of (P0 (Xq x0))
% Found (((eq_ref a) (Xq x0)) P) as proof of (P0 (Xq x0))
% Found (((eq_ref a) (Xq x0)) P) as proof of (P0 (Xq x0))
% Found x20:=(x2 (fun (x3:a)=> (P (Xp x0)))):((P (Xp x0))->(P (Xp x0)))
% Found (x2 (fun (x3:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found (x2 (fun (x3:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found x20:=(x2 (fun (x3:a)=> (P (Xp x0)))):((P (Xp x0))->(P (Xp x0)))
% Found (x2 (fun (x3:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found (x2 (fun (x3:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion0 a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found eq_ref00:=(eq_ref0 (Xp x2)):(((eq a) (Xp x2)) (Xp x2))
% Found (eq_ref0 (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found ((eq_ref a) (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found ((eq_ref a) (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found ((eq_ref a) (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x2))
% Found eq_ref00:=(eq_ref0 (Xp x2)):(((eq a) (Xp x2)) (Xp x2))
% Found (eq_ref0 (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found ((eq_ref a) (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found ((eq_ref a) (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found ((eq_ref a) (Xp x2)) as proof of (((eq a) (Xp x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x2))
% Found x30:=(x3 (fun (x6:a)=> (P Xp))):((P Xp)->(P Xp))
% Found (x3 (fun (x6:a)=> (P Xp))) as proof of (P0 Xp)
% Found (x3 (fun (x6:a)=> (P Xp))) as proof of (P0 Xp)
% Found eq_ref00:=(eq_ref0 (Xq x0)):(((eq a) (Xq x0)) (Xq x0))
% Found (eq_ref0 (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found eq_ref00:=(eq_ref0 (Xq x0)):(((eq a) (Xq x0)) (Xq x0))
% Found (eq_ref0 (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found eq_ref00:=(eq_ref0 (Xq x0)):(((eq a) (Xq x0)) (Xq x0))
% Found (eq_ref0 (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found eq_ref00:=(eq_ref0 (Xq x0)):(((eq a) (Xq x0)) (Xq x0))
% Found (eq_ref0 (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found ((eq_ref a) (Xq x0)) as proof of (((eq a) (Xq x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x0))
% Found x50:=(x5 (fun (x6:((a->(a->a))->a))=> (P Xp))):((P Xp)->(P Xp))
% Found (x5 (fun (x6:((a->(a->a))->a))=> (P Xp))) as proof of (P0 Xp)
% Found (x5 (fun (x6:((a->(a->a))->a))=> (P Xp))) as proof of (P0 Xp)
% Found x50:=(x5 (fun (x6:((a->(a->a))->a))=> (P Xp))):((P Xp)->(P Xp))
% Found (x5 (fun (x6:((a->(a->a))->a))=> (P Xp))) as proof of (P0 Xp)
% Found (x5 (fun (x6:((a->(a->a))->a))=> (P Xp))) as proof of (P0 Xp)
% Found eq_ref00:=(eq_ref0 (Xp x0)):(((eq a) (Xp x0)) (Xp x0))
% Found (eq_ref0 (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found eq_ref00:=(eq_ref0 (Xp x0)):(((eq a) (Xp x0)) (Xp x0))
% Found (eq_ref0 (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found ((eq_ref a) (Xp x0)) as proof of (((eq a) (Xp x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x0))
% Found x30:=(x3 (fun (x4:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found x30:=(x3 (fun (x4:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found x0:(P Xp)
% Instantiate: b:=Xp:((a->(a->a))->a)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->a)))=> a)) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found x30:=(x3 (fun (x5:a)=> (P (Xp x4)))):((P (Xp x4))->(P (Xp x4)))
% Found (x3 (fun (x5:a)=> (P (Xp x4)))) as proof of (P0 (Xp x4))
% Found (x3 (fun (x5:a)=> (P (Xp x4)))) as proof of (P0 (Xp x4))
% Found x10:=(x1 (fun (x5:a)=> (P (Xp x4)))):((P (Xp x4))->(P (Xp x4)))
% Found (x1 (fun (x5:a)=> (P (Xp x4)))) as proof of (P0 (Xp x4))
% Found (x1 (fun (x5:a)=> (P (Xp x4)))) as proof of (P0 (Xp x4))
% Found x30:=(x3 (fun (x4:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found x30:=(x3 (fun (x4:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found x30:=(x3 (fun (x4:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found x30:=(x3 (fun (x4:a)=> (P Xq))):((P Xq)->(P Xq))
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found (x3 (fun (x4:a)=> (P Xq))) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion0 a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found x10:=(x1 (fun (x3:a)=> (P (Xq x2)))):((P (Xq x2))->(P (Xq x2)))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found x10:=(x1 (fun (x3:a)=> (P (Xq x2)))):((P (Xq x2))->(P (Xq x2)))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found x4:(((eq ((a->(a->a))->a)) Xp) (fun (Xg:(a->(a->a)))=> ((Xg (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy)))))
% Found x4 as proof of (((eq ((a->(a->a))->a)) Xp) (fun (Xg:(a->(a->a)))=> ((Xg (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy)))))
% Found x20:=(x2 (fun (x5:a)=> (P (Xp x0)))):((P (Xp x0))->(P (Xp x0)))
% Found (x2 (fun (x5:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found (x2 (fun (x5:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found x20:=(x2 (fun (x5:a)=> (P (Xp x0)))):((P (Xp x0))->(P (Xp x0)))
% Found (x2 (fun (x5:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found (x2 (fun (x5:a)=> (P (Xp x0)))) as proof of (P0 (Xp x0))
% Found x10:=(x1 (fun (x5:a)=> (P (Xp x2)))):((P (Xp x2))->(P (Xp x2)))
% Found (x1 (fun (x5:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found (x1 (fun (x5:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found x10:=(x1 (fun (x5:a)=> (P (Xp x2)))):((P (Xp x2))->(P (Xp x2)))
% Found (x1 (fun (x5:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found (x1 (fun (x5:a)=> (P (Xp x2)))) as proof of (P0 (Xp x2))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion0 a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->a))->a)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found ((eq_ref ((a->(a->a))->a)) b) as proof of (((eq ((a->(a->a))->a)) b) Xp)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq ((a->(a->a))->a)) Xq) (fun (x:(a->(a->a)))=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found ((eta_expansion0 a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found (((eta_expansion (a->(a->a))) a) Xq) as proof of (((eq ((a->(a->a))->a)) Xq) b)
% Found x10:=(x1 (fun (x3:a)=> (P (Xq x2)))):((P (Xq x2))->(P (Xq x2)))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found x10:=(x1 (fun (x3:a)=> (P (Xq x2)))):((P (Xq x2))->(P (Xq x2)))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found (x1 (fun (x3:a)=> (P (Xq x2)))) as proof of (P0 (Xq x2))
% Found eq_ref00:=(eq_ref0 (Xp x4)):(((eq a) (Xp x4)) (Xp x4))
% Found (eq_ref0 (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found ((eq_ref a) (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found ((eq_ref a) (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found ((eq_ref a) (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x4))
% Found eq_ref00:=(eq_ref0 (Xp x4)):(((eq a) (Xp x4)) (Xp x4))
% Found (eq_ref0 (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found ((eq_ref a) (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found ((eq_ref a) (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found ((eq_ref a) (Xp x4)) as proof of (((eq a) (Xp x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xq x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xq x4))
% Found x0:(P Xp)
% Instantiate: f:=Xp:((a->(a->a))->a)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq a) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq a) (f x1)) (Xq x1))
% Found ((eq_ref a) (f x1)) as proof of (((eq a) (f x1)) (Xq x1))
% Found ((eq_ref a) (f x1)) as proof of (((eq a) (f x1)) (Xq x1))
% Found (fun (x1:(a->(a->a)))=> ((eq_ref a) (f x1))) as proof of (((eq a) (f x1)) (Xq x1))
% Found (fun (x1:(a->(a->a)))=> ((eq_ref a) (f x1))) as proof of (forall (x:(a->(a->a))), (((eq a) (f x)) (Xq x)))
% Found x0:(P Xp)
% Instantiate: f:=Xp:((a->(a->a))->a)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq a) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq a) (f x1)) (Xq x1))
% Found ((eq_ref a) (f x1)) as proof of (((eq a) (f x1)) (Xq x1))
% Found ((eq_ref a) (f x1)) as proof of (((eq a) (f x1)) (Xq x1))
% Found (fun (x1:(a->(a->a)))=> ((eq_ref a) (f x1))) as proof of (((eq a) (f x1)) (Xq x1))
% Found (fun (x1:(a->(a->a)))=> ((eq_ref a) (f x1))) as proof of (forall (x:(a->(a->a))), (((eq a) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found eq_ref00:=(eq_ref0 (Xq x2)):(((eq a) (Xq x2)) (Xq x2))
% Found (eq_ref0 (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found eq_ref00:=(eq_ref0 (Xq x2)):(((eq a) (Xq x2)) (Xq x2))
% Found (eq_ref0 (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found eq_ref00:=(eq_ref0 (Xq x2)):(((eq a) (Xq x2)) (Xq x2))
% Found (eq_ref0 (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found ((eq_ref a) (Xq x2)) as proof of (((eq a) (Xq x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) (Xp x2))
% Found
% EOF
%------------------------------------------------------------------------------