%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEV302^5 : TPTP v6.2.0. Bugfixed v6.2.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n133.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 2.6.32-504.8.1.el6.x86_64 % CPULimit : 300s % DateTime : Tue Apr 21 16:51:42 EDT 2015 % Result : Theorem 16.28s % Output : Proof 16.38s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : SEV302^5 : TPTP v6.2.0. Bugfixed v6.2.0. % 0.00/0.03 % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % 0.01/1.07 % Computer : n133.star.cs.uiowa.edu % 0.01/1.07 % Model : x86_64 x86_64 % 0.01/1.07 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz % 0.01/1.07 % Memory : 32286.75MB % 0.01/1.07 % OS : Linux 2.6.32-504.8.1.el6.x86_64 % 0.01/1.07 % CPULimit : 300 % 0.01/1.07 % DateTime : Thu Apr 16 12:27:12 CDT 2015 % 0.01/1.08 % CPUTime : % 0.01/1.09 Python 2.7.5 % 0.26/1.43 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/'] % 0.26/1.43 FOF formula (<kernel.Constant object at 0x1b73f38>, <kernel.DependentProduct object at 0x1b73b90>) of role type named cNAT_type % 0.26/1.43 Using role type % 0.26/1.43 Declaring cNAT:(((fofType->Prop)->Prop)->Prop) % 0.26/1.43 FOF formula (<kernel.Constant object at 0x1b737a0>, <kernel.DependentProduct object at 0x1b733f8>) of role type named cSUCC_type % 0.26/1.43 Using role type % 0.26/1.43 Declaring cSUCC:(((fofType->Prop)->Prop)->((fofType->Prop)->Prop)) % 0.26/1.43 FOF formula (<kernel.Constant object at 0x1b73440>, <kernel.DependentProduct object at 0x1b73f38>) of role type named cZERO_type % 0.26/1.43 Using role type % 0.26/1.43 Declaring cZERO:((fofType->Prop)->Prop) % 0.26/1.43 FOF formula (((eq ((fofType->Prop)->Prop)) cZERO) (fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False))) of role definition named cZERO_def % 0.26/1.43 A new definition: (((eq ((fofType->Prop)->Prop)) cZERO) (fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False))) % 0.26/1.43 Defined: cZERO:=(fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False)) % 0.26/1.43 FOF formula (((eq (((fofType->Prop)->Prop)->((fofType->Prop)->Prop))) cSUCC) (fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt))))))))) of role definition named cSUCC_def % 0.26/1.43 A new definition: (((eq (((fofType->Prop)->Prop)->((fofType->Prop)->Prop))) cSUCC) (fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt))))))))) % 0.26/1.43 Defined: cSUCC:=(fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt)))))))) % 0.26/1.43 FOF formula (((eq (((fofType->Prop)->Prop)->Prop)) cNAT) (fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn))))) of role definition named cNAT_def % 0.26/1.43 A new definition: (((eq (((fofType->Prop)->Prop)->Prop)) cNAT) (fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn))))) % 0.26/1.43 Defined: cNAT:=(fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn)))) % 0.26/1.43 FOF formula (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))->(forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm))))) of role conjecture named cX6102_B % 0.26/1.43 Conjecture to prove = (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))->(forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm))))):Prop % 0.26/1.43 Parameter fofType_DUMMY:fofType. % 0.26/1.43 We need to prove ['(forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))->(forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm)))))'] % 0.26/1.43 Parameter fofType:Type. % 0.26/1.43 Definition cNAT:=(fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn)))):(((fofType->Prop)->Prop)->Prop). % 2.37/3.57 Definition cSUCC:=(fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt)))))))):(((fofType->Prop)->Prop)->((fofType->Prop)->Prop)). % 2.37/3.57 Definition cZERO:=(fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False)):((fofType->Prop)->Prop). % 2.37/3.57 Trying to prove (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))->(forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm))))) % 2.37/3.57 Found x4:(Xp Xm) % 2.37/3.57 Found (fun (x4:(Xp Xm))=> x4) as proof of (Xp Xm) % 2.37/3.57 Found (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4) as proof of ((Xp Xm)->(Xp Xm)) % 2.37/3.57 Found (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4) as proof of ((cNAT Xm)->((Xp Xm)->(Xp Xm))) % 2.37/3.57 Found (and_rect10 (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)) as proof of (Xp Xm) % 2.37/3.57 Found ((and_rect1 (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)) as proof of (Xp Xm) % 2.37/3.57 Found (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)) as proof of (Xp Xm) % 2.37/3.57 Found (fun (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))) as proof of (Xp Xm) % 2.37/3.57 Found (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xm)) % 2.37/3.57 Found (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xm))) % 2.37/3.57 Found (and_rect00 (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)))) as proof of (Xp Xm) % 2.37/3.57 Found ((and_rect0 (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)))) as proof of (Xp Xm) % 2.37/3.57 Found (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)))) as proof of (Xp Xm) % 2.37/3.57 Found (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) x00)) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)))) as proof of (Xp Xm) % 2.37/3.57 Found x1:(Xp cZERO) % 2.37/3.57 Found x1 as proof of (Xp cZERO) % 2.37/3.57 Found x1:(Xp cZERO) % 2.37/3.57 Found x1 as proof of (Xp cZERO) % 2.37/3.57 Found x1:(Xp cZERO) % 3.46/4.61 Found x1 as proof of (Xp cZERO) % 3.46/4.61 Found x1:(Xp cZERO) % 3.46/4.61 Found x1 as proof of (Xp cZERO) % 3.46/4.61 Found x4:(Xp0 cZERO) % 3.46/4.61 Found (fun (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4) as proof of (Xp0 cZERO) % 3.46/4.61 Found (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->(Xp0 cZERO)) % 3.46/4.61 Found (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4) as proof of ((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->(Xp0 cZERO))) % 3.46/4.61 Found (and_rect10 (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)) as proof of (Xp0 cZERO) % 3.46/4.61 Found ((and_rect1 (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)) as proof of (Xp0 cZERO) % 3.46/4.61 Found (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)) as proof of (Xp0 cZERO) % 3.46/4.61 Found (fun (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4))) as proof of (Xp0 cZERO) % 3.46/4.61 Found (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4))) as proof of (((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))->(Xp0 cZERO)) % 3.46/4.61 Found (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4))) as proof of (cNAT cZERO) % 3.46/4.61 Found ((conj10 (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)))) x1) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 3.46/4.61 Found (((conj1 (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)))) x1) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 3.46/4.61 Found ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)))) x1) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 6.06/7.25 Found ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x3:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x4:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x4) x3)) (Xp0 cZERO)) (fun (x4:(Xp0 cZERO)) (x5:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x4)))) x1) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 6.06/7.25 Found x20:=(x2 Xx):((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 6.06/7.25 Found (x2 Xx) as proof of ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 6.06/7.25 Found (x2 Xx) as proof of ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 6.06/7.25 Found (and_rect10 (x2 Xx)) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found ((and_rect1 (Xp (cSUCC Xx))) (x2 Xx)) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x3)) (Xp (cSUCC Xx))) (x2 Xx)) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x3)) (Xp (cSUCC Xx))) (x2 Xx)) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found x2000:=(x200 x5):(Xp (cSUCC Xx)) % 6.06/7.25 Found (x200 x5) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found ((x20 x4) x5) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found (((x2 Xx) x4) x5) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found (((x2 Xx) x4) x5) as proof of (Xp (cSUCC Xx)) % 6.06/7.25 Found x1:(Xp cZERO) % 6.06/7.25 Found (fun (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1) as proof of (Xp cZERO) % 6.06/7.25 Found (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp cZERO)) % 6.06/7.25 Found (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp cZERO))) % 6.06/7.25 Found (and_rect00 (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)) as proof of (Xp cZERO) % 6.06/7.25 Found ((and_rect0 (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)) as proof of (Xp cZERO) % 6.06/7.25 Found (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)) as proof of (Xp cZERO) % 6.06/7.25 Found (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)) as proof of (Xp cZERO) % 6.06/7.25 Found x2:(Xp0 cZERO) % 6.06/7.25 Found (fun (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2) as proof of (Xp0 cZERO) % 6.06/7.25 Found (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->(Xp0 cZERO)) % 6.06/7.25 Found (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2) as proof of ((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->(Xp0 cZERO))) % 6.06/7.26 Found (and_rect00 (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)) as proof of (Xp0 cZERO) % 6.06/7.26 Found ((and_rect0 (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)) as proof of (Xp0 cZERO) % 6.06/7.26 Found (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)) as proof of (Xp0 cZERO) % 6.06/7.26 Found (fun (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2))) as proof of (Xp0 cZERO) % 6.06/7.26 Found (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2))) as proof of (((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))->(Xp0 cZERO)) % 6.06/7.26 Found (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2))) as proof of (cNAT cZERO) % 6.06/7.26 Found ((conj10 (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1))) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 6.06/7.26 Found (((conj1 (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1))) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 6.06/7.26 Found ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1))) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 13.47/14.66 Found ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1))) as proof of ((and (cNAT cZERO)) (Xp cZERO)) % 13.47/14.66 Found x30:=(x3 Xx):((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 13.47/14.66 Found (x3 Xx) as proof of ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 13.47/14.66 Found (x3 Xx) as proof of ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 13.47/14.66 Found (and_rect10 (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found ((and_rect1 (Xp (cSUCC Xx))) (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found x3000:=(x300 x5):(Xp (cSUCC Xx)) % 13.47/14.66 Found (x300 x5) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found ((x30 x4) x5) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((x3 Xx) x4) x5) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((x3 Xx) x4) x5) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found x5000:=(x500 x3):(Xp (cSUCC Xx)) % 13.47/14.66 Found (x500 x3) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found ((x50 x2) x3) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((x5 Xx) x2) x3) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((x5 Xx) x2) x3) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found x30:=(x3 Xx):((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 13.47/14.66 Found (x3 Xx) as proof of ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 13.47/14.66 Found (x3 Xx) as proof of ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))) % 13.47/14.66 Found (and_rect10 (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found ((and_rect1 (Xp (cSUCC Xx))) (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (fun (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))) as proof of (Xp (cSUCC Xx)) % 13.47/14.66 Found (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))) as proof of ((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->(Xp (cSUCC Xx))) % 13.47/14.66 Found (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))) as proof of ((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->(Xp (cSUCC Xx)))) % 16.07/17.25 Found (and_rect00 (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))) as proof of (Xp (cSUCC Xx)) % 16.07/17.25 Found ((and_rect0 (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))) as proof of (Xp (cSUCC Xx)) % 16.07/17.25 Found (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))) as proof of (Xp (cSUCC Xx)) % 16.07/17.25 Found (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))) as proof of (Xp (cSUCC Xx)) % 16.07/17.25 Found x200:=(x20 x4):(Xp0 Xx) % 16.07/17.25 Found (x20 x4) as proof of (Xp0 Xx) % 16.07/17.25 Found ((x2 Xp0) x4) as proof of (Xp0 Xx) % 16.07/17.25 Found ((x2 Xp0) x4) as proof of (Xp0 Xx) % 16.07/17.25 Found (x60 ((x2 Xp0) x4)) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found ((x6 Xx) ((x2 Xp0) x4)) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found (fun (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))) as proof of ((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->(Xp0 (cSUCC Xx))) % 16.07/17.25 Found (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))) as proof of ((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->(Xp0 (cSUCC Xx)))) % 16.07/17.25 Found (and_rect10 (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found ((and_rect1 (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found (fun (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))) as proof of (Xp0 (cSUCC Xx)) % 16.07/17.25 Found (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))) as proof of (((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))->(Xp0 (cSUCC Xx))) % 16.07/17.27 Found (fun (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))) as proof of (cNAT (cSUCC Xx)) % 16.07/17.27 Found (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))) as proof of ((Xp Xx)->(cNAT (cSUCC Xx))) % 16.07/17.27 Found (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))) as proof of ((cNAT Xx)->((Xp Xx)->(cNAT (cSUCC Xx)))) % 16.07/17.27 Found (and_rect00 (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))))) as proof of (cNAT (cSUCC Xx)) % 16.07/17.27 Found ((and_rect0 (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))))) as proof of (cNAT (cSUCC Xx)) % 16.07/17.27 Found (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))))) as proof of (cNAT (cSUCC Xx)) % 16.18/17.30 Found (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4)))))) as proof of (cNAT (cSUCC Xx)) % 16.18/17.30 Found ((conj10 (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp % Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))) as proof of ((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) % 16.18/17.30 Found (((conj1 (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall % (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))) as proof of ((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) % 16.18/17.30 Found ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall % (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))) as proof of ((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) % 16.18/17.32 Found (fun (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) % (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))) as proof of ((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) % 16.18/17.32 Found (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) % P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))) as proof of (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))) % 16.18/17.32 Found (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) % P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))) as proof of (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))) % 16.18/17.35 Found ((conj00 ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun % (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> % (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))))) as proof of ((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) % 16.18/17.35 Found (((conj0 (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun % (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC % Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))))) as proof of ((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) % 16.18/17.37 Found ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp % (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC % Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))))) as proof of ((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) % 16.18/17.37 Found ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp % (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC % Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx))))))) as proof of ((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) % 16.18/17.38 Found (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 % cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall % (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4)))) as proof of (Xp Xm) % 16.28/17.42 Found (fun (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 % cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall % (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) as proof of (Xp Xm) % 16.28/17.42 Found (fun (Xm:((fofType->Prop)->Prop)) (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun % (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall % (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) as proof of ((((and ((and % (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm)) % 16.28/17.43 Found (fun (x:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))) (Xm:((fofType->Prop)->Prop)) (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) % (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) % (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC % Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) as proof of (forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm))) % 16.28/17.45 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))) (Xm:((fofType->Prop)->Prop)) (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun % (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp % Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT % Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))->(forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm)))) % 16.28/17.47 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))) (Xm:((fofType->Prop)->Prop)) (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun % (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp % Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT % Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) as proof of (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))->(forall (Xm:((fofType->Prop)->Prop)), ((((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm)))->(Xp Xm))))) % 16.28/17.49 Got proof (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))) (Xm:((fofType->Prop)->Prop)) (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) % (fun (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) % (Xp Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT % Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) % 16.28/17.49 Time elapsed = 16.031961s % 16.28/17.49 node=1519 cost=1437.000000 depth=35 % 16.28/17.49:::::::::::::::::::::: % 16.28/17.49 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 16.38/17.58 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % 16.38/17.58 (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))) (Xm:((fofType->Prop)->Prop)) (x0:(((and ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))))))->((and (cNAT Xm)) (Xp Xm))))=> (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp Xm)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x3:((cNAT Xm)->((Xp Xm)->P)))=> (((((and_rect (cNAT Xm)) (Xp Xm)) P) x3) (x0 ((((conj ((and (cNAT cZERO)) (Xp cZERO))) (forall (Xx:((fofType->Prop)->Prop)), (((and (cNAT Xx)) (Xp Xx))->((and (cNAT (cSUCC Xx))) (Xp (cSUCC Xx)))))) ((((conj (cNAT cZERO)) (Xp cZERO)) (fun % (Xp0:(((fofType->Prop)->Prop)->Prop)) (x1:((and (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))))=> (((fun (P:Type) (x2:((Xp0 cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx))))) P) x2) x1)) (Xp0 cZERO)) (fun (x2:(Xp0 cZERO)) (x3:(forall (Xx:((fofType->Prop)->Prop)), ((Xp0 Xx)->(Xp0 (cSUCC Xx)))))=> x2)))) (((fun (P:Type) (x1:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx)))))) P) x1) x)) (Xp cZERO)) (fun (x1:(Xp cZERO)) (x2:(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((Xp Xx)->(Xp (cSUCC Xx))))))=> x1)))) (fun (Xx:((fofType->Prop)->Prop)) (x1:((and (cNAT Xx)) (Xp Xx)))=> ((((conj (cNAT (cSUCC Xx))) (Xp (cSUCC Xx))) (((fun (P:Type) (x2:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT Xx)) (Xp % Xx)) P) x2) x1)) (cNAT (cSUCC Xx))) (fun (x2:(cNAT Xx)) (x3:(Xp Xx)) (Xp0:(((fofType->Prop)->Prop)->Prop)) (x4:((and (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))))=> (((fun (P:Type) (x5:((Xp0 cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))->P)))=> (((((and_rect (Xp0 cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0))))) P) x5) x4)) (Xp0 (cSUCC Xx))) (fun (x5:(Xp0 cZERO)) (x6:(forall (Xx0:((fofType->Prop)->Prop)), ((Xp0 Xx0)->(Xp0 (cSUCC Xx0)))))=> ((x6 Xx) ((x2 Xp0) x4))))))) (((fun (P:Type) (x2:((Xp cZERO)->((forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0)))))) P) x2) x)) (Xp (cSUCC Xx))) (fun (x2:(Xp cZERO)) (x3:(forall (Xx0:((fofType->Prop)->Prop)), ((cNAT Xx0)->((Xp Xx0)->(Xp (cSUCC Xx0))))))=> (((fun (P:Type) (x4:((cNAT Xx)->((Xp Xx)->P)))=> (((((and_rect (cNAT % Xx)) (Xp Xx)) P) x4) x1)) (Xp (cSUCC Xx))) (x3 Xx)))))))))) (Xp Xm)) (fun (x3:(cNAT Xm)) (x4:(Xp Xm))=> x4))))) % 16.38/17.58 % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p %------------------------------------------------------------------------------