%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEV050^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n106.star.cs.uiowa.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz % Memory : 32286.75MB % OS : Linux 2.6.32-431.20.3.el6.x86_64 % CPULimit : 300s % DateTime : Thu Jul 17 13:33:40 EDT 2014 % Result : Theorem 0.85s % Output : Proof 0.85s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEV050^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n106.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 07:44:31 CDT 2014 % % CPUTime : 0.85 % 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 0x12e4b90>, <kernel.Type object at 0x12e4368>) of role type named a_type % Using role type % Declaring a:Type % FOF formula (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))) of role conjecture named cTHM599_pme % Conjecture to prove = (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))):Prop % Parameter a_DUMMY:a. % We need to prove ['(forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))'] % Parameter a:Type. % Trying to prove (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))) % Found x00:((x Xa) Xb) % Instantiate: x:=Xt:(a->(a->Prop)) % Found x00 as proof of ((Xt Xa) Xb) % Found (fun (x00:((x Xa) Xb))=> x00) as proof of ((Xt Xa) Xb) % Found (fun (Xb:a) (x00:((x Xa) Xb))=> x00) as proof of (((x Xa) Xb)->((Xt Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> x00) as proof of (forall (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))) % Found (fun (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> x00) as proof of (forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))) % Found x0000:=(x000 x0):((Xt Xa) Xb) % Found (x000 x0) as proof of ((Xt Xa) Xb) % Found ((x00 Xt) x0) as proof of ((Xt Xa) Xb) % Found ((x00 Xt) x0) as proof of ((Xt Xa) Xb) % Found (fun (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of ((Xt Xa) Xb) % Found (fun (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (((x Xa) Xb)->((Xt Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))) % Found (fun (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))) % Found (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))) % Found (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))))) % Found x0:((Xr Xa) Xb) % Instantiate: x:=Xr:(a->(a->Prop)) % Found (fun (x0:((Xr Xa) Xb))=> x0) as proof of ((x Xa) Xb) % Found (fun (Xb:a) (x0:((Xr Xa) Xb))=> x0) as proof of (((Xr Xa) Xb)->((x Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb))=> x0) as proof of (forall (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb))) % Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb))=> x0) as proof of (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb))) % Found x20:=(x2 Xx):((Xt Xx) Xx) % Found (x2 Xx) as proof of ((Xt Xx) Xx) % Found (fun (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)) as proof of ((Xt Xx) Xx) % Found (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)) as proof of ((forall (Xx0:a), ((Xt Xx0) Xx0))->((Xt Xx) Xx)) % Found (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)) as proof of ((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->((Xt Xx) Xx))) % Found (and_rect00 (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))) as proof of ((Xt Xx) Xx) % Found ((and_rect0 ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))) as proof of ((Xt Xx) Xx) % Found (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))) as proof of ((Xt Xx) Xx) % Found (fun (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of ((Xt Xx) Xx) % Found (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0)))->((Xt Xx) Xx)) % Found (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of ((x Xx) Xx) % Found (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of (forall (Xx:a), ((x Xx) Xx)) % Found x1000:=(x100 x0):((Xt Xa) Xb) % Found (x100 x0) as proof of ((Xt Xa) Xb) % Found ((x10 Xb) x0) as proof of ((Xt Xa) Xb) % Found (((x1 Xa) Xb) x0) as proof of ((Xt Xa) Xb) % Found (fun (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)) as proof of ((Xt Xa) Xb) % Found (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)) as proof of ((forall (Xx:a), ((Xt Xx) Xx))->((Xt Xa) Xb)) % Found (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)) as proof of ((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->((Xt Xa) Xb))) % Found (and_rect00 (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))) as proof of ((Xt Xa) Xb) % Found ((and_rect0 ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))) as proof of ((Xt Xa) Xb) % Found (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))) as proof of ((Xt Xa) Xb) % Found (fun (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of ((Xt Xa) Xb) % Found (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt Xa) Xb)) % Found (fun (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of ((x Xa) Xb) % Found (fun (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (((Xr Xa) Xb)->((x Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (forall (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb))) % Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb))) % Found ((conj10 (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) % Found (((conj1 (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) % Found ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) % Found ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) % Found ((conj00 ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) % Found (((conj0 (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) % Found ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) % Found ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) % Found (ex_intro000 ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) % Found ((ex_intro00 (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) % Found (((ex_intro0 (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) % Found ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) % Found (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0))))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) % Found (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0))))) as proof of (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))) % Got proof (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0))))) % Time elapsed = 0.533292s % node=67 cost=876.000000 depth=28 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------