TSTP Solution File: SEV072^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV072^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n180.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 1.57s
% Output : Proof 1.57s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV072^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n180.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 07:55:31 CDT 2014
% % CPUTime : 1.57
% 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 0xf13dd0>, <kernel.Type object at 0xf13518>) 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)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) of role conjecture named cTHM522_pme
% Conjecture to prove = (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) 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)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))))
% Found or_comm_i00:=(or_comm_i0 ((Xr Xy0) Xx0)):(((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))
% Found (or_comm_i0 ((Xr Xy0) Xx0)) as proof of (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))
% Found ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)) as proof of (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))
% Found (fun (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))) as proof of (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))) as proof of (forall (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))) as proof of (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))
% Found or_introl00:=(or_introl0 ((Xr Xy0) Xx0)):(((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))
% Found (or_introl0 ((Xr Xy0) Xx0)) as proof of (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))
% Found ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)) as proof of (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))
% Found (fun (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))) as proof of (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))) as proof of (forall (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))) as proof of (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))
% Found ((conj10 (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))))
% Found (((conj1 (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))))
% Found ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))))
% Found ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0)))))
% Found (x0 ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) as proof of ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))
% Found ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) as proof of ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))
% Found (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))))) as proof of ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))
% Found (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))))) as proof of ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))
% 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 x100:=(x10 Xx):(((Xr Xy) Xx)->((Xp Xy) Xx))
% Found (x10 Xx) as proof of (((Xr Xy) Xx)->((Xp Xy) Xx))
% Found ((x1 Xy) Xx) as proof of (((Xr Xy) Xx)->((Xp Xy) Xx))
% Found ((x1 Xy) Xx) as proof of (((Xr Xy) Xx)->((Xp Xy) Xx))
% Found x1000:=(x100 x3):((Xp Xy) Xx)
% Found (x100 x3) as proof of ((Xp Xy) Xx)
% Found ((x10 Xx) x3) as proof of ((Xp Xy) Xx)
% Found (((x1 Xy) Xx) x3) as proof of ((Xp Xy) Xx)
% Found (((x1 Xy) Xx) x3) as proof of ((Xp Xy) Xx)
% Found (x200 (((x1 Xy) Xx) x3)) as proof of ((Xp Xx) Xy)
% Found ((x20 Xx) (((x1 Xy) Xx) x3)) as proof of ((Xp Xx) Xy)
% Found (((x2 Xy) Xx) (((x1 Xy) Xx) x3)) as proof of ((Xp Xx) Xy)
% Found (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))) as proof of ((Xp Xx) Xy)
% Found (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))) as proof of (((Xr Xy) Xx)->((Xp Xx) Xy))
% Found ((or_ind00 ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))) as proof of ((Xp Xx) Xy)
% Found (((or_ind0 ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))) as proof of ((Xp Xx) Xy)
% Found ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))) as proof of ((Xp Xx) Xy)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))) as proof of ((Xp Xx) Xy)
% Found (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))) as proof of ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->((Xp Xx) Xy))
% Found (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))) as proof of ((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->((Xp Xx) Xy)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))) 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) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))) 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) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))) as proof of ((Xp Xx) Xy)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))))) 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) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))
% Found (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))))) as proof of (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))
% Found (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3))))))) as proof of (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))
% Found ((conj00 (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))))) as proof of ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) 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) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))))) as proof of ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))
% Found ((((conj (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))))) as proof of ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))
% Found (fun (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))))))) as proof of ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))
% Found (fun (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))))))) as proof of (forall (Xy:a), ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))))
% Found (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))))))) as proof of (forall (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))))
% Found (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))))))) as proof of (forall (Xr:(a->(a->Prop))) (Xx:a) (Xy:a), ((iff ((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))))
% Got proof (fun (Xr:(a->(a->Prop))) (Xx:a) (Xy:a)=> ((((conj (((or ((Xr Xx) Xy)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))))))
% Time elapsed = 1.238243s
% node=225 cost=719.000000 depth=28
% ::::::::::::::::::::::
% % 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)) ((Xr Xy) Xx))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))) ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy)))->((or ((Xr Xx) Xy)) ((Xr Xy) Xx)))) (fun (x:((or ((Xr Xx) Xy)) ((Xr Xy) Xx))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0)))) P) x1) x0)) ((Xp Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))=> ((((fun (P:Prop) (x3:(((Xr Xx) Xy)->P)) (x4:(((Xr Xy) Xx)->P))=> ((((((or_ind ((Xr Xx) Xy)) ((Xr Xy) Xx)) P) x3) x4) x)) ((Xp Xx) Xy)) ((x1 Xx) Xy)) (fun (x3:((Xr Xy) Xx))=> (((x2 Xy) Xx) (((x1 Xy) Xx) x3)))))))) (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->((Xp Xy0) Xx0))))->((Xp Xx) Xy))))=> ((x (fun (x2:a) (x10:a)=> ((or ((Xr x2) x10)) ((Xr x10) x2)))) ((((conj (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((or ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))->((or ((Xr Xy0) Xx0)) ((Xr Xx0) Xy0))))) (fun (Xx0:a) (Xy0:a)=> ((or_introl ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0)))) (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((Xr Xx0) Xy0)) ((Xr Xy0) Xx0))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------