%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEV398^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n180.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 13:34:08 EDT 2014 % Result : Theorem 3.25s % Output : Proof 3.25s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEV398^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n180.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 09:06:26 CDT 2014 % % CPUTime : 3.25 % 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 0xf93dd0>, <kernel.Type object at 0xf93440>) of role type named a_type % Using role type % Declaring a:Type % FOF formula (<kernel.Constant object at 0x136b5f0>, <kernel.DependentProduct object at 0xf934d0>) of role type named cF % Using role type % Declaring cF:((a->Prop)->(a->Prop)) % FOF formula (<kernel.Constant object at 0xf937e8>, <kernel.DependentProduct object at 0xf93ab8>) of role type named cG % Using role type % Declaring cG:((a->Prop)->(a->Prop)) % FOF formula (((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))->(forall (S:(a->Prop)) (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)))) of role conjecture named cTHM67A_pme % Conjecture to prove = (((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))->(forall (S:(a->Prop)) (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)))):Prop % Parameter a_DUMMY:a. % We need to prove ['(((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))->(forall (S:(a->Prop)) (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))))'] % Parameter a:Type. % Parameter cF:((a->Prop)->(a->Prop)). % Parameter cG:((a->Prop)->(a->Prop)). % Trying to prove (((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))->(forall (S:(a->Prop)) (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)))) % Found x20:=(x2 Xx):((S0 Xx)->((cF (cG S0)) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found x3:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))) % Instantiate: S0:=S:(a->Prop) % Found x3 as proof of (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))) % Found (x0000 x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((x000 x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> (((x00 (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found x20:=(x2 Xx):((S0 Xx)->((cF (cG S0)) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found x20:=(x2 Xx):((S0 Xx)->((cF (cG S0)) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found x2:((cF (cG (cF S))) Xx) % Found x2 as proof of ((cF (cG (cF S))) Xx) % Found x2:((cF S) Xx) % Found x2 as proof of ((cF S) Xx) % Found x20:=(x2 Xx):((S0 Xx)->((cF (cG S0)) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found (x2 Xx) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found x3:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))) % Instantiate: S0:=S:(a->Prop) % Found x3 as proof of (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))) % Found (x0000 x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((x000 x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> (((x00 (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found x3:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))) % Instantiate: S0:=S:(a->Prop) % Found x3 as proof of (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))) % Found (x0000 x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((x000 x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> (((x00 (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found x3:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))) % Instantiate: S0:=S:(a->Prop) % Found x3 as proof of (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))) % Found (x0000 x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((x000 x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> (((x00 (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x4:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x4) Xx)) x3) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found x4:((cF (cG (cF S))) Xx) % Found x4 as proof of ((cF (cG (cF S))) Xx) % Found x2:((cF (cG (cF S))) Xx) % Found x2 as proof of ((cF (cG (cF S))) Xx) % Found x2:((cF S) Xx) % Found x2 as proof of ((cF S) Xx) % Found x300:=(x30 x2):((cF (cG S0)) Xx) % Found (x30 x2) as proof of ((cF (cG (cF S))) Xx) % Found ((x3 Xx) x2) as proof of ((cF (cG (cF S))) Xx) % Found ((x3 Xx) x2) as proof of ((cF (cG (cF S))) Xx) % Found (fun (x4:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))=> ((x3 Xx) x2)) as proof of ((cF (cG (cF S))) Xx) % Found (fun (x3:(forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (x4:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))=> ((x3 Xx) x2)) as proof of ((forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))->((cF (cG (cF S))) Xx)) % Found (fun (x3:(forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (x4:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))=> ((x3 Xx) x2)) as proof of ((forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))->((forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))->((cF (cG (cF S))) Xx))) % Found (and_rect10 (fun (x3:(forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (x4:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))=> ((x3 Xx) x2))) as proof of ((cF (cG (cF S))) Xx) % Found ((and_rect1 ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (x4:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))=> ((x3 Xx) x2))) as proof of ((cF (cG (cF S))) Xx) % Found (((fun (P:Type) (x3:((forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))->((forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))) P) x3) x10)) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (x4:(forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))=> ((x3 Xx) x2))) as proof of ((cF (cG (cF S))) Xx) % Found (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))) as proof of ((cF (cG (cF S))) Xx) % Found (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))) as proof of ((cF (cG (cF S))) Xx) % Found (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))) as proof of (((cF S) Xx)->((cF (cG (cF S))) Xx)) % Found x400:=(x40 x2):((cG (cF S0)) Xx0) % Found (x40 x2) as proof of ((cG (cF S)) Xx0) % Found ((x4 Xx0) x2) as proof of ((cG (cF S)) Xx0) % Found ((x4 Xx0) x2) as proof of ((cG (cF S)) Xx0) % Found (fun (x4:(forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00))))=> ((x4 Xx0) x2)) as proof of ((cG (cF S)) Xx0) % Found (fun (x3:(forall (Xx00:a), ((S0 Xx00)->((cF (cG S0)) Xx00)))) (x4:(forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00))))=> ((x4 Xx0) x2)) as proof of ((forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00)))->((cG (cF S)) Xx0)) % Found (fun (x3:(forall (Xx00:a), ((S0 Xx00)->((cF (cG S0)) Xx00)))) (x4:(forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00))))=> ((x4 Xx0) x2)) as proof of ((forall (Xx00:a), ((S0 Xx00)->((cF (cG S0)) Xx00)))->((forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00)))->((cG (cF S)) Xx0))) % Found (and_rect10 (fun (x3:(forall (Xx00:a), ((S0 Xx00)->((cF (cG S0)) Xx00)))) (x4:(forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00))))=> ((x4 Xx0) x2))) as proof of ((cG (cF S)) Xx0) % Found ((and_rect1 ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S0 Xx00)->((cF (cG S0)) Xx00)))) (x4:(forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00))))=> ((x4 Xx0) x2))) as proof of ((cG (cF S)) Xx0) % Found (((fun (P:Type) (x3:((forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))->((forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))) P) x3) x10)) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S0 Xx00)->((cF (cG S0)) Xx00)))) (x4:(forall (Xx00:a), ((S0 Xx00)->((cG (cF S0)) Xx00))))=> ((x4 Xx0) x2))) as proof of ((cG (cF S)) Xx0) % Found (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2))) as proof of ((cG (cF S)) Xx0) % Found (fun (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))) as proof of ((cG (cF S)) Xx0) % Found (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))) as proof of ((S Xx0)->((cG (cF S)) Xx0)) % Found (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))) as proof of (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))) % Found (x0000 (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2))))) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((x000 x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2))))) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> (((x00 (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2))))) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2))))) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2))))) as proof of (((cF (cG (cF S))) Xx)->((cF S) Xx)) % Found ((conj00 ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found (((conj0 (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found (fun (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))) as proof of ((forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))))->((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) % Found (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))) as proof of ((forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))->((forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0)))))->((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)))) % Found (and_rect00 (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found ((and_rect0 ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2))))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found (fun (Xx:a)=> (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))))) as proof of ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)) % Found (fun (S:(a->Prop)) (Xx:a)=> (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))))) as proof of (forall (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) % Found (fun (x:((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))) (S:(a->Prop)) (Xx:a)=> (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))))) as proof of (forall (S:(a->Prop)) (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) % Found (fun (x:((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))) (S:(a->Prop)) (Xx:a)=> (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))))) as proof of (((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))->(forall (S:(a->Prop)) (Xx:a), ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx)))) % Got proof (fun (x:((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))) (S:(a->Prop)) (Xx:a)=> (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))))) % Time elapsed = 2.897709s % node=526 cost=1100.000000 depth=30 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (x:((and (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx))))))) (S:(a->Prop)) (Xx:a)=> (((fun (P:Type) (x0:((forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))->((forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))->P)))=> (((((and_rect (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((cF T) Xx)->((cF S) Xx)))))) (forall (S:(a->Prop)), ((and (forall (Xx:a), ((S Xx)->((cF (cG S)) Xx)))) (forall (Xx:a), ((S Xx)->((cG (cF S)) Xx)))))) P) x0) x)) ((iff ((cF (cG (cF S))) Xx)) ((cF S) Xx))) (fun (x0:(forall (S0:(a->Prop)) (T:(a->Prop)), ((forall (Xx0:a), ((S0 Xx0)->(T Xx0)))->(forall (Xx0:a), (((cF T) Xx0)->((cF S0) Xx0)))))) (x1:(forall (S0:(a->Prop)), ((and (forall (Xx0:a), ((S0 Xx0)->((cF (cG S0)) Xx0)))) (forall (Xx0:a), ((S0 Xx0)->((cG (cF S0)) Xx0))))))=> ((((conj (((cF (cG (cF S))) Xx)->((cF S) Xx))) (((cF S) Xx)->((cF (cG (cF S))) Xx))) ((fun (x2:(forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0))))=> ((((x0 S) (cG (cF S))) x2) Xx)) (fun (Xx0:a) (x2:(S Xx0))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))->((forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((S Xx0)->((cF (cG S)) Xx0)))) (forall (Xx0:a), ((S Xx0)->((cG (cF S)) Xx0)))) P) x3) (x1 S))) ((cG (cF S)) Xx0)) (fun (x3:(forall (Xx00:a), ((S Xx00)->((cF (cG S)) Xx00)))) (x4:(forall (Xx00:a), ((S Xx00)->((cG (cF S)) Xx00))))=> ((x4 Xx0) x2)))))) (fun (x2:((cF S) Xx))=> (((fun (P:Type) (x3:((forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))->((forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0)))) P) x3) (x1 (cF S)))) ((cF (cG (cF S))) Xx)) (fun (x3:(forall (Xx0:a), (((cF S) Xx0)->((cF (cG (cF S))) Xx0)))) (x4:(forall (Xx0:a), (((cF S) Xx0)->((cG (cF (cF S))) Xx0))))=> ((x3 Xx) x2)))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------