%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEV113^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n091.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 13:33:45 EDT 2014 % Result : Theorem 2.47s % Output : Proof 2.47s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEV113^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n091.star.cs.uiowa.edu % % Model : x86_64 x86_64 % % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz % % Memory : 32286.75MB % % OS : Linux 2.6.32-431.20.3.el6.x86_64 % % CPULimit : 300 % % DateTime : Thu Jul 17 08:07:16 CDT 2014 % % CPUTime : 2.47 % 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 0x1dec5f0>, <kernel.Type object at 0x1deccb0>) of role type named a_type % Using role type % Declaring a:Type % FOF formula ((forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))->(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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))) of role conjecture named cTHM576_LFP_pme % Conjecture to prove = ((forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))->(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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))):Prop % Parameter a_DUMMY:a. % We need to prove ['((forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))->(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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))))'] % Parameter a:Type. % Trying to prove ((forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))->(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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))) % Found x000:((x0 Xa) Xb) % Instantiate: x0:=Xt:(a->(a->Prop)) % Found x000 as proof of ((Xt Xa) Xb) % Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Xt Xa) Xb) % Found (fun (Xb:a) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Xt Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb))) % Found (fun (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb))) % Found x00000:=(x0000 x00):((Xt Xa) Xb) % Found (x0000 x00) as proof of ((Xt Xa) Xb) % Found ((x000 Xt) x00) as proof of ((Xt Xa) Xb) % Found ((x000 Xt) x00) as proof of ((Xt Xa) Xb) % Found (fun (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)) as proof of ((Xt Xa) Xb) % Found (fun (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)) as proof of (((x0 Xa) Xb)->((Xt Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)) as proof of (forall (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb))) % Found (fun (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)) as proof of (forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb))) % Found (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)) as proof of (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))) % Found (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)) as proof of (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb))))) % Found x00:((Xr Xa) Xb) % Instantiate: x0:=Xr:(a->(a->Prop)) % Found (fun (x00:((Xr Xa) Xb))=> x00) as proof of ((x0 Xa) Xb) % Found (fun (Xb:a) (x00:((Xr Xa) Xb))=> x00) as proof of (((Xr Xa) Xb)->((x0 Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb))=> x00) as proof of (forall (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb))) % Found (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb))=> x00) as proof of (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb))) % Found x1000:=(x100 x00):((Xt Xa) Xb) % Found (x100 x00) as proof of ((Xt Xa) Xb) % Found ((x10 Xb) x00) as proof of ((Xt Xa) Xb) % Found (((x1 Xa) Xb) x00) as proof of ((Xt Xa) Xb) % Found (fun (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)) as proof of ((Xt Xa) Xb) % Found (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)) as proof of ((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->((Xt Xa) Xb)) % Found (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)) as proof of ((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->((Xt Xa) Xb))) % Found (and_rect00 (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))) as proof of ((Xt Xa) Xb) % Found (fun (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)))) as proof of ((Xt Xa) Xb) % Found (fun (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)))) as proof of (((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt Xa) Xb)) % Found (fun (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)))) as proof of ((x0 Xa) Xb) % Found (fun (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)))) as proof of (((Xr Xa) Xb)->((x0 Xa) Xb)) % Found (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)))) as proof of (forall (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb))) % Found (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00)))) as proof of (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb))) % Found x0010:=(x001 x000):((Xt Xx) Xy) % Found (x001 x000) as proof of ((Xt Xx) Xy) % Found ((x00 Xt) x000) as proof of ((Xt Xx) Xy) % Found ((x00 Xt) x000) as proof of ((Xt Xx) Xy) % Found (x200 ((x00 Xt) x000)) as proof of ((Xt Xy) Xx) % Found ((x20 Xy) ((x00 Xt) x000)) as proof of ((Xt Xy) Xx) % Found (((x2 Xx) Xy) ((x00 Xt) x000)) as proof of ((Xt Xy) Xx) % Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))) as proof of ((Xt Xy) Xx) % Found (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))) as proof of ((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->((Xt Xy) Xx)) % Found (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))) as proof of ((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->((Xt Xy) Xx))) % Found (and_rect00 (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))) as proof of ((Xt Xy) Xx) % Found ((and_rect0 ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))) as proof of ((Xt Xy) Xx) % Found (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))) as proof of ((Xt Xy) Xx) % Found (fun (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))) as proof of ((Xt Xy) Xx) % Found (fun (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))) as proof of (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((Xt Xy) Xx)) % Found (fun (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))) as proof of ((x0 Xy) Xx) % Found (fun (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))) as proof of (((x0 Xx) Xy)->((x0 Xy) Xx)) % Found (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))) as proof of (forall (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))) % Found (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))) as proof of (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))) % Found ((conj10 (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) % Found (((conj1 (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) % Found ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) % Found ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000)))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) % Found ((conj00 ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) % Found ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) % Found ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) % Found (ex_intro000 ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((x0 Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x0 Xa) Xb)))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy:a) (x00:((x0 Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:((x0 Xa) Xb))=> ((x000 Xt) x00)))) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00)))) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00)))) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00)))) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00))))) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) % Found (fun (x:(forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))) (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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00))))) 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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))) % Found (fun (x:(forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))) (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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00))))) as proof of ((forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))->(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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))) % Got proof (fun (x:(forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))) (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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00))))) % Time elapsed = 2.126819s % node=173 cost=1028.000000 depth=32 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (x:(forall (K:((a->(a->Prop))->(a->(a->Prop)))), ((forall (Xr1:(a->(a->Prop))) (Xr2:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), (((Xr1 Xx) Xy)->((Xr2 Xx) Xy)))->(forall (Xx:a) (Xy:a), ((((K Xr1) Xx) Xy)->(((K Xr2) Xx) Xy)))))->((ex (a->(a->Prop))) (fun (L:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((L Xx) Xy)->(((K L) Xx) Xy)))) (forall (Xx:a) (Xy:a), ((((K L) Xx) Xy)->((L Xx) Xy))))) (forall (T:(a->(a->Prop))), ((forall (Xx:a) (Xy:a), ((((K T) Xx) Xy)->((T Xx) Xy)))->(forall (Xx:a) (Xy:a), (((L Xx) Xy)->((T Xx) Xy))))))))))) (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) (Xy:a), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx))))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) 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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)->(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xy) Xx)))) (fun (Xa:a) (Xb:a) (x00:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x1) x000)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((x1 Xa) Xb) x00))))) (fun (Xx:a) (Xy: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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xx) Xy)) (Xt:(a->(a->Prop))) (x000:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x1) x000)) ((Xt Xy) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a) (Xy0:a), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x2 Xx) Xy) ((x00 Xt) x000))))))) (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (Xa:a) (Xb:a) (x000:(((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) (Xy:a), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((Xt a0) a1)))) Xa) Xb))=> ((x000 Xt) x00))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------