%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU948^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n099.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 13:33:25 EDT 2014

% Result   : Theorem 3.87s
% Output   : Proof 3.87s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU948^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n099.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 11:44:51 CDT 2014
% % CPUTime  : 3.87 
% 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 0x2345998>, <kernel.Type object at 0x2345830>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx)))))))) of role conjecture named cTHM135_pme
% Conjecture to prove = (forall (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))))
% Found x20:=(x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))):((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))) as proof of ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))) as proof of ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))) as proof of (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))
% Found x40:=(x4 (fun (Xx:a)=> (Xj (Xg2 Xx)))):((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (x4 (fun (Xx:a)=> (Xj (Xg2 Xx)))) as proof of ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (fun (Xj:(a->a))=> (x4 (fun (Xx:a)=> (Xj (Xg2 Xx))))) as proof of ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (fun (Xj:(a->a))=> (x4 (fun (Xx:a)=> (Xj (Xg2 Xx))))) as proof of (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))
% Found x40:=(x4 (fun (Xx:a)=> (Xj (Xg2 Xx)))):((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (x4 (fun (Xx:a)=> (Xj (Xg2 Xx)))) as proof of ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (fun (Xj:(a->a))=> (x4 (fun (Xx:a)=> (Xj (Xg2 Xx))))) as proof of ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))
% Found (fun (Xj:(a->a))=> (x4 (fun (Xx:a)=> (Xj (Xg2 Xx))))) as proof of (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))
% Found x1:(Xp (fun (Xu:a)=> Xu))
% Found x1 as proof of (Xp (fun (Xx:a)=> Xx))
% Found x20:=(x2 (fun (Xx:a)=> (Xj Xx))):((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))
% Found (x2 (fun (Xx:a)=> (Xj Xx))) as proof of ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))
% Found (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx)))) as proof of ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))
% Found (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx)))) as proof of (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))
% Found ((conj10 x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))) as proof of ((and (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))
% Found (((conj1 (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))) as proof of ((and (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))
% Found ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))) as proof of ((and (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))
% Found ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))) as proof of ((and (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))
% Found (x40 ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx)))))) as proof of (Xp (fun (Xx:a)=> (Xg2 Xx)))
% Found ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx)))))) as proof of (Xp (fun (Xx:a)=> (Xg2 Xx)))
% Found ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx)))))) as proof of (Xp (fun (Xx:a)=> (Xg2 Xx)))
% Found ((conj00 ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))) as proof of ((and (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))))
% Found (((conj0 (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))) as proof of ((and (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))))
% Found ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))) as proof of ((and (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))))
% Found ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))) as proof of ((and (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx))))))))
% Found (x30 ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (fun (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))) as proof of ((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx)))))
% Found (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))) as proof of ((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))
% Found (and_rect10 (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found ((and_rect1 (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (((fun (P:Type) (x3:((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))->((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))->P)))=> (((((and_rect (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (fun (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))->((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))->P)))=> (((((and_rect (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))->((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))->P)))=> (((((and_rect (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))) as proof of ((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx)))))
% Found (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))->((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))->P)))=> (((((and_rect (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))) as proof of ((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))
% Found (and_rect00 (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))->((forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))->P)))=> (((((and_rect (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found ((and_rect0 (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (fun (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))
% Found (fun (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx)))))
% Found (fun (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))
% Found (fun (Xg2:(a->a)) (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx)))))))
% Found (fun (Xg1:(a->a)) (Xg2:(a->a)) (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (forall (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))))
% Found (fun (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)) (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (forall (Xg1:(a->a)) (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))))
% Found (fun (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)) (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx)))))))))))) as proof of (forall (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)), (((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))->(forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))))))
% Got proof (fun (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)) (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))))))
% Time elapsed = 3.525543s
% node=559 cost=867.000000 depth=34
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (Xf:(a->a)) (Xg1:(a->a)) (Xg2:(a->a)) (x:((and (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg1)))) (forall (Xp:((a->a)->Prop)), (((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp Xg2))))) (Xp:((a->a)->Prop)) (x0:((and (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))))=> (((fun (P:Type) (x1:((Xp (fun (Xu:a)=> Xu))->((forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))->P)))=> (((((and_rect (Xp (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) P) x1) x0)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x1:(Xp (fun (Xu:a)=> Xu))) (x2:(forall (Xj:(a->a)), ((Xp Xj)->(Xp (fun (Xx:a)=> (Xf (Xj Xx)))))))=> (((fun (P:Type) (x3:((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))->((forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))->P)))=> (((((and_rect (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2)))) P) x3) x)) (Xp (fun (Xx:a)=> (Xg1 (Xg2 Xx))))) (fun (x3:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg1)))) (x4:(forall (Xp0:((a->a)->Prop)), (((and (Xp0 (fun (Xu:a)=> Xu))) (forall (Xj:(a->a)), ((Xp0 Xj)->(Xp0 (fun (Xx:a)=> (Xf (Xj Xx)))))))->(Xp0 Xg2))))=> ((x3 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 (Xg2 Xx)))))) ((((conj (Xp (fun (Xx:a)=> (Xg2 Xx)))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj (Xg2 Xx))))->(Xp (fun (Xx:a)=> (Xf (Xj (Xg2 Xx)))))))) ((x4 (fun (x6:(a->a))=> (Xp (fun (Xx:a)=> (x6 Xx))))) ((((conj (Xp (fun (Xx:a)=> Xx))) (forall (Xj:(a->a)), ((Xp (fun (Xx:a)=> (Xj Xx)))->(Xp (fun (Xx:a)=> (Xf (Xj Xx))))))) x1) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj Xx))))))) (fun (Xj:(a->a))=> (x2 (fun (Xx:a)=> (Xj (Xg2 Xx))))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------