TSTP Solution File: SEV113^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% 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
%------------------------------------------------------------------------------