%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : DAT056^1 : TPTP v6.1.0. Released v5.4.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n184.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 13:21:12 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : DAT056^1 : TPTP v6.1.0. Released v5.4.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:46:41 CDT 2014
% % CPUTime  : 300.09 
% 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 0x1939518>, <kernel.Type object at 0x1939f80>) of role type named ty_n_tc__Foo__Olst_It__J
% Using role type
% Declaring lst:Type
% FOF formula (<kernel.Constant object at 0x1d16320>, <kernel.Type object at 0x1939ea8>) of role type named ty_n_t_
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1939b48>, <kernel.DependentProduct object at 0x1939f80>) of role type named sy_c_Foo_Oap_001t_
% Using role type
% Declaring ap:(lst->(lst->lst))
% FOF formula (<kernel.Constant object at 0x1939758>, <kernel.DependentProduct object at 0x1939518>) of role type named sy_c_Foo_Olst_OCns_001t_
% Using role type
% Declaring cns:(a->(lst->lst))
% FOF formula (<kernel.Constant object at 0x1939950>, <kernel.Constant object at 0x1939518>) of role type named sy_c_Foo_Olst_ONl_001t_
% Using role type
% Declaring nl:lst
% FOF formula (<kernel.Constant object at 0x1939b48>, <kernel.Constant object at 0x1939758>) of role type named sy_v_xs
% Using role type
% Declaring xs:lst
% FOF formula (forall (Lst:lst), ((forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))->((forall (A:a) (Lst2:lst), ((forall (Ys3:lst) (Zs2:lst), (((eq lst) ((ap Lst2) ((ap Ys3) Zs2))) ((ap ((ap Lst2) Ys3)) Zs2)))->(forall (Ys:lst) (Zs:lst), (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)))))->(forall (Ys3:lst) (Zs2:lst), (((eq lst) ((ap Lst) ((ap Ys3) Zs2))) ((ap ((ap Lst) Ys3)) Zs2)))))) of role axiom named fact_0_lst_Oinduct_091where_AP_A_061_A_C_Fxs_O_AALL_Ays_Azs_O_Aap_Axs_A_Iap_Ays_Azs_J_A_061_Aap_A_Iap_Axs_Ays_J_Azs_C_093
% A new axiom: (forall (Lst:lst), ((forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))->((forall (A:a) (Lst2:lst), ((forall (Ys3:lst) (Zs2:lst), (((eq lst) ((ap Lst2) ((ap Ys3) Zs2))) ((ap ((ap Lst2) Ys3)) Zs2)))->(forall (Ys:lst) (Zs:lst), (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)))))->(forall (Ys3:lst) (Zs2:lst), (((eq lst) ((ap Lst) ((ap Ys3) Zs2))) ((ap ((ap Lst) Ys3)) Zs2))))))
% FOF formula (forall (Ys2:lst) (Xs:lst) (X:a), (((eq lst) ((ap ((cns X) Xs)) Ys2)) ((cns X) ((ap Xs) Ys2)))) of role axiom named fact_1p_Osimps_I2_J
% A new axiom: (forall (Ys2:lst) (Xs:lst) (X:a), (((eq lst) ((ap ((cns X) Xs)) Ys2)) ((cns X) ((ap Xs) Ys2))))
% FOF formula (forall (Ys2:lst), (((eq lst) ((ap nl) Ys2)) Ys2)) of role axiom named fact_2p_Osimps_I1_J
% A new axiom: (forall (Ys2:lst), (((eq lst) ((ap nl) Ys2)) Ys2))
% FOF formula (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap xs) ((ap Ys) Zs))) ((ap ((ap xs) Ys)) Zs))) of role conjecture named conj_0
% Conjecture to prove = (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap xs) ((ap Ys) Zs))) ((ap ((ap xs) Ys)) Zs))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Ys:lst) (Zs:lst), (((eq lst) ((ap xs) ((ap Ys) Zs))) ((ap ((ap xs) Ys)) Zs)))']
% Parameter lst:Type.
% Parameter a:Type.
% Parameter ap:(lst->(lst->lst)).
% Parameter cns:(a->(lst->lst)).
% Parameter nl:lst.
% Parameter xs:lst.
% Axiom fact_0_lst_Oinduct_091where_AP_A_061_A_C_Fxs_O_AALL_Ays_Azs_O_Aap_Axs_A_Iap_Ays_Azs_J_A_061_Aap_A_Iap_Axs_Ays_J_Azs_C_093:(forall (Lst:lst), ((forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))->((forall (A:a) (Lst2:lst), ((forall (Ys3:lst) (Zs2:lst), (((eq lst) ((ap Lst2) ((ap Ys3) Zs2))) ((ap ((ap Lst2) Ys3)) Zs2)))->(forall (Ys:lst) (Zs:lst), (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)))))->(forall (Ys3:lst) (Zs2:lst), (((eq lst) ((ap Lst) ((ap Ys3) Zs2))) ((ap ((ap Lst) Ys3)) Zs2)))))).
% Axiom fact_1p_Osimps_I2_J:(forall (Ys2:lst) (Xs:lst) (X:a), (((eq lst) ((ap ((cns X) Xs)) Ys2)) ((cns X) ((ap Xs) Ys2)))).
% Axiom fact_2p_Osimps_I1_J:(forall (Ys2:lst), (((eq lst) ((ap nl) Ys2)) Ys2)).
% Trying to prove (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap xs) ((ap Ys) Zs))) ((ap ((ap xs) Ys)) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys) Zs)):(((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap Ys) Zs))
% Found (fact_2p_Osimps_I1_J ((ap Ys) Zs)) as proof of (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap Ys) Zs))
% Found (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs))) as proof of (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap Ys) Zs))
% Found (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap Ys) Zs)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys) (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found (fun (Ys:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys) (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found (fun (Ys:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys) (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys) Zs))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys) Zs)))->(P ((ap ((cns A) Lst2)) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap xs) ((ap Ys) Zs))):(((eq lst) ((ap xs) ((ap Ys) Zs))) ((ap xs) ((ap Ys) Zs)))
% Found (eq_ref0 ((ap xs) ((ap Ys) Zs))) as proof of (((eq lst) ((ap xs) ((ap Ys) Zs))) b)
% Found ((eq_ref lst) ((ap xs) ((ap Ys) Zs))) as proof of (((eq lst) ((ap xs) ((ap Ys) Zs))) b)
% Found ((eq_ref lst) ((ap xs) ((ap Ys) Zs))) as proof of (((eq lst) ((ap xs) ((ap Ys) Zs))) b)
% Found ((eq_ref lst) ((ap xs) ((ap Ys) Zs))) as proof of (((eq lst) ((ap xs) ((ap Ys) Zs))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap xs) ((ap Ys) Zs)))->(P ((ap xs) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap xs) ((ap Ys) Zs))) P) as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found x:(P ((ap xs) ((ap Ys) Zs)))
% Found x as proof of (P0 ((ap xs) ((ap Ys) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs)):(((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap Ys0) Zs))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap Ys0) Zs))
% Found (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap Ys0) Zs))
% Found (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap Ys0) Zs)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap x0) Zs))))) (fun (Zs:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) ((cns A) ((ap Lst2) ((ap Ys) Zs))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap xs) ((ap Ys) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap ((ap xs) Ys)) Zs)):(((eq lst) ((ap ((ap xs) Ys)) Zs)) ((ap ((ap xs) Ys)) Zs))
% Found (eq_ref0 ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap xs) ((ap Ys) Zs))):(((eq lst) ((ap xs) ((ap Ys) Zs))) ((ap xs) ((ap Ys) Zs)))
% Found (eq_ref0 ((ap xs) ((ap Ys) Zs))) as proof of (P ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref lst) ((ap xs) ((ap Ys) Zs))) as proof of (P ((ap xs) ((ap Ys) Zs)))
% Found ((eq_ref lst) ((ap xs) ((ap Ys) Zs))) as proof of (P ((ap xs) ((ap Ys) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs))):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found (eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found ((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found ((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found ((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys) Zs)))->(P ((ap ((cns A) Lst2)) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs)):(((eq lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found (eq_ref0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) ((cns A) ((ap Lst2) ((ap Ys) Zs))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) b)
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) ((cns A) ((ap Lst2) ((ap Ys0) Zs0))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs0)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs0))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x1:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x1) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) Lst2)) ((ap Ys0) Zs)))
% Found eq_ref00:=(eq_ref0 ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)):(((eq lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs))
% Found (eq_ref0 ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) as proof of (((eq lst) ((ap ((ap ((cns A) Lst2)) Ys0)) Zs)) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys) Zs)))->(P ((ap ((cns A) Lst2)) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found eq_ref00:=(eq_ref0 b):(((eq lst) b) b)
% Found (eq_ref0 b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys0))) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys0))) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys0))) Zs))
% Found ((eq_ref lst) b) as proof of (((eq lst) b) ((ap ((cns A) ((ap Lst2) Ys0))) Zs))
% Found fact_1p_Osimps_I2_J000:=(fact_1p_Osimps_I2_J00 A):(((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) ((cns A) ((ap Lst2) ((ap Ys0) Zs))))
% Found (fact_1p_Osimps_I2_J00 A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found ((fact_1p_Osimps_I2_J0 Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found (((fact_1p_Osimps_I2_J ((ap Ys0) Zs)) Lst2) A) as proof of (((eq lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys) Zs)))->(P ((ap ((cns A) Lst2)) ((ap Ys) Zs))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys) Zs))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys) Zs)))
% Found x:(P ((ap xs) ((ap Ys) Zs)))
% Instantiate: b:=((ap xs) ((ap Ys) Zs)):lst
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ap ((ap xs) Ys)) Zs)):(((eq lst) ((ap ((ap xs) Ys)) Zs)) ((ap ((ap xs) Ys)) Zs))
% Found (eq_ref0 ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found ((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) as proof of (((eq lst) ((ap ((ap xs) Ys)) Zs)) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap ((cns A) Lst2)) Ys)) Zs))->(P ((ap ((ap ((cns A) Lst2)) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) P) as proof of (P0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) P) as proof of (P0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap ((cns A) Lst2)) Ys)) Zs)) P) as proof of (P0 ((ap ((ap ((cns A) Lst2)) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) ((ap Lst2) Ys))) Zs))->(P ((ap ((cns A) ((ap Lst2) Ys))) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found ((eq_ref0 ((ap ((cns A) ((ap Lst2) Ys))) Zs)) P) as proof of (P0 ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found (((eq_ref lst) ((ap ((cns A) ((ap Lst2) Ys))) Zs)) P) as proof of (P0 ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found (((eq_ref lst) ((ap ((cns A) ((ap Lst2) Ys))) Zs)) P) as proof of (P0 ((ap ((cns A) ((ap Lst2) Ys))) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((ap xs) Ys)) Zs))->(P ((ap ((ap xs) Ys)) Zs)))
% Found (eq_ref00 P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found ((eq_ref0 ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found (((eq_ref lst) ((ap ((ap xs) Ys)) Zs)) P) as proof of (P0 ((ap ((ap xs) Ys)) Zs))
% Found fact_2p_Osimps_I1_J0:=(fact_2p_Osimps_I1_J ((ap Ys0) Zs0)):(((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0))
% Found (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))) as proof of (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap Ys0) Zs0)))
% Found (fact_2p_Osimps_I1_J__eq_sym00 (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found ((fact_2p_Osimps_I1_J__eq_sym0 (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0)))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Zs:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs))) ((ap ((ap nl) Ys0)) Zs)))
% Found (fun (Ys0:lst)=> (((fact_2p_Osimps_I1_J__eq_sym Ys0) (fun (x0:lst)=> (forall (Zs0:lst), (((eq lst) ((ap nl) ((ap Ys0) Zs0))) ((ap x0) Zs0))))) (fun (Zs0:lst)=> (fact_2p_Osimps_I1_J ((ap Ys0) Zs0))))) as proof of (forall (Ys:lst) (Zs:lst), (((eq lst) ((ap nl) ((ap Ys) Zs))) ((ap ((ap nl) Ys)) Zs)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))->(P ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))))
% Found (eq_ref00 P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found ((eq_ref0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Zs0)))
% Found (((eq_ref lst) ((ap ((cns A) Lst2)) ((ap Ys0) Zs0))) P) as proof of (P0 ((ap ((cns A) Lst2)) ((ap Ys0) Z
% EOF
%------------------------------------------------------------------------------