%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEV414^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n100.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 13:34:10 EDT 2014 % Result : Theorem 1.06s % Output : Proof 1.06s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEV414^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n100.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 09:10:36 CDT 2014 % % CPUTime : 1.06 % 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 0x923cb0>, <kernel.DependentProduct object at 0x923c68>) of role type named cS % Using role type % Declaring cS:(fofType->fofType) % FOF formula (<kernel.Constant object at 0xb21290>, <kernel.Single object at 0x923b00>) of role type named c0 % Using role type % Declaring c0:fofType % FOF formula ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) of role conjecture named cTHM594_pme % Conjecture to prove = ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))):Prop % We need to prove ['((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))'] % Parameter fofType:Type. % Parameter cS:(fofType->fofType). % Parameter c0:fofType. % Trying to prove ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) % Found x00:(x Xx) % Instantiate: x:=Xp:(fofType->Prop) % Found x00 as proof of (Xp Xx) % Found (fun (x00:(x Xx))=> x00) as proof of (Xp Xx) % Found (fun (Xx:fofType) (x00:(x Xx))=> x00) as proof of ((x Xx)->(Xp Xx)) % Found (fun (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> x00) as proof of (forall (Xx:fofType), ((x Xx)->(Xp Xx))) % Found x0000:=(x000 x0):(Xp Xx) % Found (x000 x0) as proof of (Xp Xx) % Found ((x00 Xp) x0) as proof of (Xp Xx) % Found ((x00 Xp) x0) as proof of (Xp Xx) % Found (fun (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (Xp Xx) % Found (fun (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of ((x Xx)->(Xp Xx)) % Found (fun (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (forall (Xx:fofType), ((x Xx)->(Xp Xx))) % Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))) % Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx))))) % Found x1:(Xp c0) % Found (fun (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1) as proof of (Xp c0) % Found (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1) as proof of ((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->(Xp c0)) % Found (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1) as proof of ((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->(Xp c0))) % Found (and_rect00 (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)) as proof of (Xp c0) % Found ((and_rect0 (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)) as proof of (Xp c0) % Found (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)) as proof of (Xp c0) % Found (fun (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1))) as proof of (Xp c0) % Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1))) as proof of (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp c0)) % Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1))) as proof of (x c0) % Found x010:=(x01 x00):(Xp Xw) % Found (x01 x00) as proof of (Xp Xw) % Found ((x0 Xp) x00) as proof of (Xp Xw) % Found ((x0 Xp) x00) as proof of (Xp Xw) % Found (x20 ((x0 Xp) x00)) as proof of (Xp (cS Xw)) % Found ((x2 Xw) ((x0 Xp) x00)) as proof of (Xp (cS Xw)) % Found (fun (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))) as proof of (Xp (cS Xw)) % Found (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))) as proof of ((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->(Xp (cS Xw))) % Found (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))) as proof of ((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->(Xp (cS Xw)))) % Found (and_rect00 (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))) as proof of (Xp (cS Xw)) % Found ((and_rect0 (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))) as proof of (Xp (cS Xw)) % Found (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))) as proof of (Xp (cS Xw)) % Found (fun (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (Xp (cS Xw)) % Found (fun (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))->(Xp (cS Xw))) % Found (fun (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (x (cS Xw)) % Found (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of ((x Xw)->(x (cS Xw))) % Found (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))) % Found ((conj10 (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) % Found (((conj1 (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) % Found ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) % Found ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) % Found ((conj00 ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) % Found (((conj0 (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) % Found ((((conj ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) % Found ((((conj ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) % Found (ex_intro000 ((((conj ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) % Found ((ex_intro00 (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) % Found (((ex_intro0 (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) % Found ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) % Found ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) % Got proof ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) % Time elapsed = 0.738446s % node=108 cost=884.000000 depth=28 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------