%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEV146^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n108.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 13:33:48 EDT 2014 % Result : Theorem 28.79s % Output : Proof 28.79s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEV146^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n108.star.cs.uiowa.edu % % Model : x86_64 x86_64 % % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz % % Memory : 32286.75MB % % OS : Linux 2.6.32-431.20.3.el6.x86_64 % % CPULimit : 300 % % DateTime : Thu Jul 17 08:13:36 CDT 2014 % % CPUTime : 28.79 % 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 0x1fff0e0>, <kernel.Type object at 0x1fffcb0>) of role type named a_type % Using role type % Declaring a:Type % FOF formula (forall (Xr:(a->(a->Prop))), ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))))) of role conjecture named cTHM525_pme % Conjecture to prove = (forall (Xr:(a->(a->Prop))), ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))))):Prop % Parameter a_DUMMY:a. % We need to prove ['(forall (Xr:(a->(a->Prop))), ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))))'] % Parameter a:Type. % Trying to prove (forall (Xr:(a->(a->Prop))), ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))))) % Found x100:=(x10 x):(Xq Xy) % Found (x10 x) as proof of (Xq Xy) % Found ((x1 Xy) x) as proof of (Xq Xy) % Found (fun (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)) as proof of (Xq Xy) % Found (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)) as proof of ((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->(Xq Xy)) % Found (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)) as proof of ((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->(Xq Xy))) % Found (and_rect00 (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))) as proof of (Xq Xy) % Found ((and_rect0 (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))) as proof of (Xq Xy) % Found (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))) as proof of (Xq Xy) % Found (fun (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)))) as proof of (Xq Xy) % Found (fun (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)))) as proof of (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)) % Found (fun (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)))) as proof of (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy))) % Found (fun (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)))) as proof of (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) % Found (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)))) as proof of (forall (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy))))) % Found (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x)))) as proof of (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy))))) % Found x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x2 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4:(Xq Xz) % Found (fun (x5:((Xr Xv) Xw))=> x4) as proof of (Xq Xz) % Found (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4) as proof of (((Xr Xv) Xw)->(Xq Xz)) % Found (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4) as proof of ((Xq Xz)->(((Xr Xv) Xw)->(Xq Xz))) % Found (and_rect10 (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4)) as proof of (Xq Xz) % Found ((and_rect1 (Xq Xz)) (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4)) as proof of (Xq Xz) % Found (((fun (P:Type) (x4:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x4) x3)) (Xq Xz)) (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4)) as proof of (Xq Xz) % Found (fun (x3:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x4:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x4) x3)) (Xq Xz)) (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4))) as proof of (Xq Xz) % Found (fun (Xw:a) (x3:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x4:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x4) x3)) (Xq Xz)) (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4))) as proof of (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz)) % Found (fun (Xv:a) (Xw:a) (x3:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x4:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x4) x3)) (Xq Xz)) (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4))) as proof of (forall (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found (fun (Xv:a) (Xw:a) (x3:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x4:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x4) x3)) (Xq Xz)) (fun (x4:(Xq Xz)) (x5:((Xr Xv) Xw))=> x4))) as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found x5:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x5 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x5:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x5 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x6:(Xq Xz) % Found (fun (x7:((Xr Xv) Xw))=> x6) as proof of (Xq Xz) % Found (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6) as proof of (((Xr Xv) Xw)->(Xq Xz)) % Found (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6) as proof of ((Xq Xz)->(((Xr Xv) Xw)->(Xq Xz))) % Found (and_rect20 (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found ((and_rect2 (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found (fun (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (Xq Xz) % Found (fun (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz)) % Found (fun (Xv:a) (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (forall (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found (fun (Xv:a) (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x2 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x6:(Xq Xz) % Found (fun (x7:((Xr Xv) Xw))=> x6) as proof of (Xq Xz) % Found (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6) as proof of (((Xr Xv) Xw)->(Xq Xz)) % Found (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6) as proof of ((Xq Xz)->(((Xr Xv) Xw)->(Xq Xz))) % Found (and_rect20 (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found ((and_rect2 (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found (fun (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (Xq Xz) % Found (fun (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz)) % Found (fun (Xv:a) (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (forall (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found (fun (Xv:a) (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found x4:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x5:((Xr Xy) Xw) % Instantiate: Xv:=Xy:a % Found x5 as proof of ((Xr Xv) Xw) % Found x6:(Xq Xz) % Found (fun (x7:((Xr Xv) Xw))=> x6) as proof of (Xq Xz) % Found (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6) as proof of (((Xr Xv) Xw)->(Xq Xz)) % Found (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6) as proof of ((Xq Xz)->(((Xr Xv) Xw)->(Xq Xz))) % Found (and_rect20 (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found ((and_rect2 (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6)) as proof of (Xq Xz) % Found (fun (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (Xq Xz) % Found (fun (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz)) % Found (fun (Xv:a) (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (forall (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found (fun (Xv:a) (Xw:a) (x5:((and (Xq Xz)) ((Xr Xv) Xw)))=> (((fun (P:Type) (x6:((Xq Xz)->(((Xr Xv) Xw)->P)))=> (((((and_rect (Xq Xz)) ((Xr Xv) Xw)) P) x6) x5)) (Xq Xz)) (fun (x6:(Xq Xz)) (x7:((Xr Xv) Xw))=> x6))) as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xz)) ((Xr Xv) Xw))->(Xq Xz))) % Found x4:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x4 as proof of (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))) % Found x5:((Xr Xy) Xw) % Instantiate: Xv:=Xy:a % Found x5 as proof of ((Xr Xv) Xw) % Found x5:((Xr Xy) Xw) % Instantiate: Xv:=Xy:a % Found x5 as proof of ((Xr Xv) Xw) % Found x5:((Xr Xy) Xw) % Instantiate: Xv:=Xy:a % Found x5 as proof of ((Xr Xv) Xw) % Found x5:((Xr Xy) Xw) % Instantiate: Xv:=Xy:a % Found x5 as proof of ((Xr Xv) Xw) % Found x300:=(x30 x0):(Xq Xy) % Found (x30 x0) as proof of (Xq Xv) % Found ((x3 Xq) x0) as proof of (Xq Xv) % Found ((x3 Xq) x0) as proof of (Xq Xv) % Found ((conj20 ((x3 Xq) x0)) x5) as proof of ((and (Xq Xv)) ((Xr Xv) Xw)) % Found (((conj2 ((Xr Xv) Xw)) ((x3 Xq) x0)) x5) as proof of ((and (Xq Xv)) ((Xr Xv) Xw)) % Found ((((conj (Xq Xv)) ((Xr Xv) Xw)) ((x3 Xq) x0)) x5) as proof of ((and (Xq Xv)) ((Xr Xv) Xw)) % Found ((((conj (Xq Xv)) ((Xr Xv) Xw)) ((x3 Xq) x0)) x5) as proof of ((and (Xq Xv)) ((Xr Xv) Xw)) % Found (x200 ((((conj (Xq Xv)) ((Xr Xv) Xw)) ((x3 Xq) x0)) x5)) as proof of (Xq Xw) % Found ((x20 Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)) as proof of (Xq Xw) % Found (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)) as proof of (Xq Xw) % Found (fun (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5))) as proof of (Xq Xw) % Found (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5))) as proof of (((Xr Xy) Xw)->(Xq Xw)) % Found (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5))) as proof of (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw))) % Found ((conj10 (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2) as proof of ((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) % Found (((conj1 (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2) as proof of ((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) % Found ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2) as proof of ((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) % Found ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2) as proof of ((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) % Found (x40 ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)) as proof of (Xq Xz) % Found ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)) as proof of (Xq Xz) % Found (fun (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))) as proof of (Xq Xz) % Found (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))) as proof of ((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->(Xq Xz)) % Found (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))) as proof of ((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->(Xq Xz))) % Found (and_rect10 (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))) as proof of (Xq Xz) % Found ((and_rect1 (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))) as proof of (Xq Xz) % Found (((fun (P:Type) (x3:((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))->((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))->P)))=> (((((and_rect (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))) as proof of (Xq Xz) % Found (fun (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))->((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))->P)))=> (((((and_rect (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))) as proof of (Xq Xz) % Found (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))->((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))->P)))=> (((((and_rect (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))) as proof of ((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->(Xq Xz)) % Found (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))->((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))->P)))=> (((((and_rect (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))) as proof of ((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->(Xq Xz))) % Found (and_rect00 (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))->((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))->P)))=> (((((and_rect (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))))) as proof of (Xq Xz) % Found ((and_rect0 (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))))) as proof of (Xq Xz) % Found (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))))) as proof of (Xq Xz) % Found (fun (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (Xq Xz) % Found (fun (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)) % Found (fun (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))) % Found (fun (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))) % Found (fun (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (forall (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) % Found (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (forall (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) % Found (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) % Found ((conj00 (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))))))) as proof of ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) % Found (((conj0 (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))))))) as proof of ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) % Found ((((conj (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2)))))))) as proof of ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) % Found (fun (Xr:(a->(a->Prop)))=> ((((conj (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))))) as proof of ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) % Found (fun (Xr:(a->(a->Prop)))=> ((((conj (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))))) as proof of (forall (Xr:(a->(a->Prop))), ((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))))) % Got proof (fun (Xr:(a->(a->Prop)))=> ((((conj (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))))) % Time elapsed = 28.227213s % node=2668 cost=1606.000000 depth=42 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (Xr:(a->(a->Prop)))=> ((((conj (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz)))))) (fun (Xx:a) (Xy:a) (x:((Xr Xx) Xy)) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xy)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> ((x1 Xy) x))))) (fun (Xx:a) (Xy:a) (Xz:a) (x:((and (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xy)))) (forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))->(Xq Xz))))) (Xq:(a->Prop)) (x0:((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))))=> (((fun (P:Type) (x1:((forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))->((forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))->P)))=> (((((and_rect (forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) P) x1) x0)) (Xq Xz)) (fun (x1:(forall (Xw:a), (((Xr Xx) Xw)->(Xq Xw)))) (x2:(forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw))))=> (((fun (P:Type) (x3:((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))->((forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))->P)))=> (((((and_rect (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz)))) P) x3) x)) (Xq Xz)) (fun (x3:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xx) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xy)))) (x4:(forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((Xr Xy) Xw)->(Xq0 Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq0 Xv)) ((Xr Xv) Xw))->(Xq0 Xw))))->(Xq0 Xz))))=> ((x4 Xq) ((((conj (forall (Xw:a), (((Xr Xy) Xw)->(Xq Xw)))) (forall (Xv:a) (Xw:a), (((and (Xq Xv)) ((Xr Xv) Xw))->(Xq Xw)))) (fun (Xw:a) (x5:((Xr Xy) Xw))=> (((fun (Xv:a)=> ((x2 Xv) Xw)) Xy) ((((conj (Xq Xy)) ((Xr Xy) Xw)) ((x3 Xq) x0)) x5)))) x2))))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------