TSTP Solution File: SEV074^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV074^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n112.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 13:33:42 EDT 2014
% Result : Theorem 3.71s
% Output : Proof 3.71s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV074^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n112.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 07:55:51 CDT 2014
% % CPUTime : 3.71
% 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 0x269eea8>, <kernel.Type object at 0x269ed40>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) of role conjecture named cTHM523_pme
% Conjecture to prove = (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))))
% Found or_introl00:=(or_introl0 (((eq a) Xx0) Xy0)):(((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))
% Found (or_introl0 (((eq a) Xx0) Xy0)) as proof of (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))
% Found ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)) as proof of (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))
% Found (fun (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))) as proof of (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))) as proof of (forall (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))) as proof of (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))
% Found x100:=(x10 Xy):(((Xr Xx) Xy)->((Xp Xx) Xy))
% Found (x10 Xy) as proof of (((Xr Xx) Xy)->((Xp Xx) Xy))
% Found ((x1 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp Xx) Xy))
% Found ((x1 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp Xx) Xy))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) Xx0)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) Xx0)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) Xx0)
% Found (or_intror00 ((eq_ref a) Xx0)) as proof of ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))
% Found ((or_intror0 (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)) as proof of ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))
% Found (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)) as proof of ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))
% Found (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))) as proof of ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))
% Found (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))) as proof of (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))
% Found ((conj10 (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))))
% Found (((conj1 (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))))
% Found ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))))
% Found ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0))))
% Found (x0 ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))) as proof of ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))
% Found ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))) as proof of ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))
% Found (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))))) as proof of ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))
% Found (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))))) as proof of ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))
% Found x20:=(x2 Xx):((Xp Xx) Xx)
% Found (x2 Xx) as proof of ((Xp Xx) Xx)
% Found (x2 Xx) as proof of ((Xp Xx) Xx)
% Found (x30 (x2 Xx)) as proof of ((Xp Xx) Xy)
% Found ((x3 (Xp Xx)) (x2 Xx)) as proof of ((Xp Xx) Xy)
% Found (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))) as proof of ((Xp Xx) Xy)
% Found (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))) as proof of ((((eq a) Xx) Xy)->((Xp Xx) Xy))
% Found ((or_ind00 ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))) as proof of ((Xp Xx) Xy)
% Found (((or_ind0 ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))) as proof of ((Xp Xx) Xy)
% Found ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))) as proof of ((Xp Xx) Xy)
% Found (fun (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))) as proof of ((Xp Xx) Xy)
% Found (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))) as proof of ((forall (Xx0:a), ((Xp Xx0) Xx0))->((Xp Xx) Xy))
% Found (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))) as proof of ((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->((Xp Xx) Xy)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))) as proof of ((Xp Xx) Xy)
% Found ((and_rect0 ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))) as proof of ((Xp Xx) Xy)
% Found (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))) as proof of ((Xp Xx) Xy)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))))) as proof of ((Xp Xx) Xy)
% Found (fun (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))
% Found (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))))) as proof of (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))
% Found (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx))))))) as proof of (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))
% Found ((conj00 (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))))) as proof of ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))
% Found (((conj0 ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))))) as proof of ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))
% Found ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))))) as proof of ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))
% Found (fun (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))))))) as proof of ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))
% Found (fun (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))))))) as proof of (forall (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))))
% Found (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))))))) as proof of (forall (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))))
% Found (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0)))))))) as proof of (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))))
% Got proof (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))))))
% Time elapsed = 3.367225s
% node=574 cost=495.000000 depth=25
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) (((eq a) Xx) Xy)))) (fun (x:((or ((Xr Xx) Xy)) (((eq a) Xx) Xy))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a), ((Xp Xx0) Xx0))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a), ((Xp Xx0) Xx0)))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:((((eq a) Xx) Xy)->P))=> ((((((or_ind ((Xr Xx) Xy)) (((eq a) Xx) Xy)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:(((eq a) Xx) Xy))=> ((x3 (Xp Xx)) (x2 Xx)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a), ((Xp Xx0) Xx0)))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) (((eq a) x2) x10)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0))))) (forall (Xx0:a), ((or ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) (((eq a) Xx0) Xy0)))) (fun (Xx0:a)=> (((or_intror ((Xr Xx0) Xx0)) (((eq a) Xx0) Xx0)) ((eq_ref a) Xx0))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------