%------------------------------------------------------------------------------ % 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 %------------------------------------------------------------------------------