cocATP---0.2.0.TMO-Non.f

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : ALG253^2 : TPTP v6.1.0. Bugfixed v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n180.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:18:18 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : ALG253^2 : TPTP v6.1.0. Bugfixed v5.2.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n180.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:54:11 CDT 2014
% % CPUTime  : 300.10 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/ALG003^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0xd45bd8>, <kernel.Type object at 0xd45050>) of role type named term_type
% Using role type
% Declaring term:Type
% FOF formula (<kernel.Constant object at 0xd454d0>, <kernel.Type object at 0xd25248>) of role type named subst_type
% Using role type
% Declaring subst:Type
% FOF formula (<kernel.Constant object at 0xd25488>, <kernel.Constant object at 0xd45bd8>) of role type named one_type
% Using role type
% Declaring one:term
% FOF formula (<kernel.Constant object at 0xd45488>, <kernel.DependentProduct object at 0xa300e0>) of role type named ap_type
% Using role type
% Declaring ap:(term->(term->term))
% FOF formula (<kernel.Constant object at 0xf9c200>, <kernel.DependentProduct object at 0xd45a70>) of role type named lam_type
% Using role type
% Declaring lam:(term->term)
% FOF formula (<kernel.Constant object at 0xf9cb48>, <kernel.DependentProduct object at 0xd45bd8>) of role type named sub_type
% Using role type
% Declaring sub:(term->(subst->term))
% FOF formula (<kernel.Constant object at 0xf9cb48>, <kernel.Constant object at 0xd454d0>) of role type named id_type
% Using role type
% Declaring id:subst
% FOF formula (<kernel.Constant object at 0xa30bd8>, <kernel.Constant object at 0xd454d0>) of role type named sh_type
% Using role type
% Declaring sh:subst
% FOF formula (<kernel.Constant object at 0xa30bd8>, <kernel.DependentProduct object at 0xd453b0>) of role type named push_type
% Using role type
% Declaring push:(term->(subst->subst))
% FOF formula (<kernel.Constant object at 0xd45a70>, <kernel.DependentProduct object at 0xd45a28>) of role type named comp_type
% Using role type
% Declaring comp:(subst->(subst->subst))
% FOF formula (<kernel.Constant object at 0xd453b0>, <kernel.DependentProduct object at 0xd44488>) of role type named var_type
% Using role type
% Declaring var:(term->Prop)
% FOF formula (<kernel.Constant object at 0xd454d0>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem1v2_type
% Using role type
% Declaring pushprop_lem1v2:Prop
% FOF formula (<kernel.Constant object at 0xd453b0>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem1_gthm_type
% Using role type
% Declaring pushprop_lem1_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd45a28>, <kernel.Sort object at 0xdf5c20>) of role type named axmap_type
% Using role type
% Declaring axmap:Prop
% FOF formula (<kernel.Constant object at 0xd453b0>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem0_gthm_type
% Using role type
% Declaring pushprop_lem0_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd453b0>, <kernel.Sort object at 0xdf5c20>) of role type named shinj_type
% Using role type
% Declaring shinj:Prop
% FOF formula (<kernel.Constant object at 0xd44320>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem1v2_type
% Using role type
% Declaring hoasinduction_lem1v2:Prop
% FOF formula (<kernel.Constant object at 0xd44830>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem1v2_gthm_type
% Using role type
% Declaring hoasinduction_lem1v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd44320>, <kernel.DependentProduct object at 0xd44cf8>) of role type named hoasap_type
% Using role type
% Declaring hoasap:(subst->(term->(subst->(term->term))))
% FOF formula (<kernel.Constant object at 0xd44b90>, <kernel.Sort object at 0xdf5c20>) of role type named induction2lem_type
% Using role type
% Declaring induction2lem:Prop
% FOF formula (<kernel.Constant object at 0xd44d40>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2_f_type
% Using role type
% Declaring hoasinduction_lem3v2_f:Prop
% FOF formula (<kernel.Constant object at 0xd44440>, <kernel.Sort object at 0xdf5c20>) of role type named axvarshift_type
% Using role type
% Declaring axvarshift:Prop
% FOF formula (<kernel.Constant object at 0xd44560>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapinj2_type
% Using role type
% Declaring hoasapinj2:Prop
% FOF formula (<kernel.Constant object at 0xd44bd8>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapnotvar_gthm_type
% Using role type
% Declaring hoasapnotvar_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd44f80>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapinj1_type
% Using role type
% Declaring hoasapinj1:Prop
% FOF formula (<kernel.Constant object at 0xd44200>, <kernel.Sort object at 0xdf5c20>) of role type named ulamvar1_type
% Using role type
% Declaring ulamvar1:Prop
% FOF formula (<kernel.Constant object at 0xd44b00>, <kernel.Sort object at 0xdf5c20>) of role type named induction2lem_lthm_type
% Using role type
% Declaring induction2lem_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd44488>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2_gthm_type
% Using role type
% Declaring hoasinduction_lem3v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd44b00>, <kernel.Sort object at 0xdf5c20>) of role type named apnotvar_type
% Using role type
% Declaring apnotvar:Prop
% FOF formula (<kernel.Constant object at 0xd44050>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lthm_orig_type
% Using role type
% Declaring pushprop_lthm_orig:Prop
% FOF formula (<kernel.Constant object at 0xd444d0>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2_f_lthm_type
% Using role type
% Declaring hoasinduction_lem3v2_f_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd44050>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lthm_type
% Using role type
% Declaring hoasinduction_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd44050>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_no_psi_cond_lthm_type
% Using role type
% Declaring hoasinduction_no_psi_cond_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd44098>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslaminj_type
% Using role type
% Declaring hoaslaminj:Prop
% FOF formula (<kernel.Constant object at 0xd44200>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3aaa_type
% Using role type
% Declaring hoasinduction_lem3aaa:Prop
% FOF formula (<kernel.Constant object at 0xd44098>, <kernel.Sort object at 0xdf5c20>) of role type named induction2lem_gthm_type
% Using role type
% Declaring induction2lem_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd44200>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3aa_lthm_type
% Using role type
% Declaring hoasinduction_lem3aa_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd44050>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3_type
% Using role type
% Declaring hoasinduction_lem3:Prop
% FOF formula (<kernel.Constant object at 0xd44050>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem2_type
% Using role type
% Declaring hoasinduction_lem2:Prop
% FOF formula (<kernel.Constant object at 0xf93050>, <kernel.Sort object at 0xdf5c20>) of role type named termmset_lthm_type
% Using role type
% Declaring termmset_lthm:Prop
% FOF formula (<kernel.Constant object at 0xf93368>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem1_type
% Using role type
% Declaring hoasinduction_lem1:Prop
% FOF formula (<kernel.Constant object at 0xf93878>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslamnotap_lthm_type
% Using role type
% Declaring hoaslamnotap_lthm:Prop
% FOF formula (<kernel.Constant object at 0xf93950>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem1v2_lthm_type
% Using role type
% Declaring pushprop_lem1v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xf93878>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapnotvar_type
% Using role type
% Declaring hoasapnotvar:Prop
% FOF formula (<kernel.Constant object at 0xf93368>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem0_type
% Using role type
% Declaring hoasinduction_lem0:Prop
% FOF formula (<kernel.Constant object at 0xf93878>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_type
% Using role type
% Declaring hoasinduction:Prop
% FOF formula (<kernel.Constant object at 0xf93878>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_gthm_type
% Using role type
% Declaring hoasinduction_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47908>, <kernel.Sort object at 0xdf5c20>) of role type named axapp_type
% Using role type
% Declaring axapp:Prop
% FOF formula (<kernel.Constant object at 0xb47950>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslamnotvar_lthm_type
% Using role type
% Declaring hoaslamnotvar_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47758>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem3v2_lthm_type
% Using role type
% Declaring pushprop_lem3v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47878>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3b_lthm_type
% Using role type
% Declaring hoasinduction_lem3b_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47758>, <kernel.Sort object at 0xdf5c20>) of role type named ulamvarind_type
% Using role type
% Declaring ulamvarind:Prop
% FOF formula (<kernel.Constant object at 0xb476c8>, <kernel.Sort object at 0xdf5c20>) of role type named induction_type
% Using role type
% Declaring induction:Prop
% FOF formula (<kernel.Constant object at 0xb477a0>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3a_lthm_type
% Using role type
% Declaring hoasinduction_lem3a_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb476c8>, <kernel.Sort object at 0xdf5c20>) of role type named termmset_gthm_type
% Using role type
% Declaring termmset_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47680>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3aa_type
% Using role type
% Declaring hoasinduction_lem3aa:Prop
% FOF formula (<kernel.Constant object at 0xb47758>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem1v2_gthm_type
% Using role type
% Declaring pushprop_lem1v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47518>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslamnotap_gthm_type
% Using role type
% Declaring hoaslamnotap_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47560>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslamnotvar_gthm_type
% Using role type
% Declaring hoaslamnotvar_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47950>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3b_gthm_type
% Using role type
% Declaring hoasinduction_lem3b_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47560>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem2v2_type
% Using role type
% Declaring pushprop_lem2v2:Prop
% FOF formula (<kernel.Constant object at 0xb47488>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3a_gthm_type
% Using role type
% Declaring hoasinduction_lem3a_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47560>, <kernel.Sort object at 0xdf5c20>) of role type named axclos_type
% Using role type
% Declaring axclos:Prop
% FOF formula (<kernel.Constant object at 0xb47518>, <kernel.Sort object at 0xdf5c20>) of role type named axassoc_type
% Using role type
% Declaring axassoc:Prop
% FOF formula (<kernel.Constant object at 0xb47440>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem2v2_type
% Using role type
% Declaring hoasinduction_lem2v2:Prop
% FOF formula (<kernel.Constant object at 0xb47a70>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lthm_type
% Using role type
% Declaring pushprop_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47ab8>, <kernel.Sort object at 0xdf5c20>) of role type named apinj2_type
% Using role type
% Declaring apinj2:Prop
% FOF formula (<kernel.Constant object at 0xb47b00>, <kernel.Sort object at 0xdf5c20>) of role type named apinj1_type
% Using role type
% Declaring apinj1:Prop
% FOF formula (<kernel.Constant object at 0xb47b48>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapinj2_lthm_type
% Using role type
% Declaring hoasapinj2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47b90>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2a_type
% Using role type
% Declaring hoasinduction_lem3v2a:Prop
% FOF formula (<kernel.Constant object at 0xb47bd8>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapinj1_lthm_type
% Using role type
% Declaring hoasapinj1_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47c20>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslaminj_lthm_type
% Using role type
% Declaring hoaslaminj_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47c68>, <kernel.Sort object at 0xdf5c20>) of role type named axvarcons_type
% Using role type
% Declaring axvarcons:Prop
% FOF formula (<kernel.Constant object at 0xb47cb0>, <kernel.DependentProduct object at 0xb47d40>) of role type named hoaslam_type
% Using role type
% Declaring hoaslam:(subst->((subst->(term->term))->term))
% FOF formula (<kernel.Constant object at 0xb47d88>, <kernel.Sort object at 0xdf5c20>) of role type named axscons_type
% Using role type
% Declaring axscons:Prop
% FOF formula (<kernel.Constant object at 0xb474d0>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem2v2_gthm_type
% Using role type
% Declaring hoasinduction_lem2v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47d88>, <kernel.Sort object at 0xdf5c20>) of role type named axidr_type
% Using role type
% Declaring axidr:Prop
% FOF formula (<kernel.Constant object at 0xb47c68>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem1_type
% Using role type
% Declaring pushprop_lem1:Prop
% FOF formula (<kernel.Constant object at 0xb47cb0>, <kernel.Sort object at 0xdf5c20>) of role type named laminj_type
% Using role type
% Declaring laminj:Prop
% FOF formula (<kernel.Constant object at 0xb47cf8>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3_lthm_type
% Using role type
% Declaring hoasinduction_lem3_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47cb0>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem0_type
% Using role type
% Declaring pushprop_lem0:Prop
% FOF formula (<kernel.Constant object at 0xb47e60>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_gthm_type
% Using role type
% Declaring pushprop_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47c68>, <kernel.Sort object at 0xdf5c20>) of role type named axabs_type
% Using role type
% Declaring axabs:Prop
% FOF formula (<kernel.Constant object at 0xb47f80>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2a_lthm_type
% Using role type
% Declaring hoasinduction_lem3v2a_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47dd0>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem2_lthm_type
% Using role type
% Declaring hoasinduction_lem2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47e60>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapinj2_gthm_type
% Using role type
% Declaring hoasapinj2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xb47f80>, <kernel.DependentProduct object at 0xd34200>) of role type named hoasinduction_p_and_p_prime_type
% Using role type
% Declaring hoasinduction_p_and_p_prime:((subst->(term->(subst->Prop)))->((term->Prop)->Prop))
% FOF formula (<kernel.Constant object at 0xb47f80>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem1_lthm_type
% Using role type
% Declaring hoasinduction_lem1_lthm:Prop
% FOF formula (<kernel.Constant object at 0xb47f80>, <kernel.Sort object at 0xdf5c20>) of role type named lamnotap_type
% Using role type
% Declaring lamnotap:Prop
% FOF formula (<kernel.Constant object at 0xd34290>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapinj1_gthm_type
% Using role type
% Declaring hoasapinj1_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd340e0>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslamnotvar_type
% Using role type
% Declaring hoaslamnotvar:Prop
% FOF formula (<kernel.Constant object at 0xd34050>, <kernel.Sort object at 0xdf5c20>) of role type named axidl_type
% Using role type
% Declaring axidl:Prop
% FOF formula (<kernel.Constant object at 0xd34248>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslaminj_gthm_type
% Using role type
% Declaring hoaslaminj_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd34170>, <kernel.Sort object at 0xdf5c20>) of role type named induction2_lthm_type
% Using role type
% Declaring induction2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd34200>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem0_lthm_type
% Using role type
% Declaring hoasinduction_lem0_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd34170>, <kernel.Sort object at 0xdf5c20>) of role type named substmonoid_lthm_type
% Using role type
% Declaring substmonoid_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd34368>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_type
% Using role type
% Declaring pushprop:Prop
% FOF formula (<kernel.Constant object at 0xd342d8>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3_gthm_type
% Using role type
% Declaring hoasinduction_lem3_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd34248>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem2_gthm_type
% Using role type
% Declaring hoasinduction_lem2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd34488>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3b_type
% Using role type
% Declaring hoasinduction_lem3b:Prop
% FOF formula (<kernel.Constant object at 0xd343f8>, <kernel.Sort object at 0xdf5c20>) of role type named substmonoid_type
% Using role type
% Declaring substmonoid:Prop
% FOF formula (<kernel.Constant object at 0xd342d8>, <kernel.Sort object at 0xdf5c20>) of role type named lamnotvar_type
% Using role type
% Declaring lamnotvar:Prop
% FOF formula (<kernel.Constant object at 0xd34518>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3a_type
% Using role type
% Declaring hoasinduction_lem3a:Prop
% FOF formula (<kernel.Constant object at 0xd34368>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem1_gthm_type
% Using role type
% Declaring hoasinduction_lem1_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd34560>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_no_psi_cond_type
% Using role type
% Declaring hoasinduction_no_psi_cond:Prop
% FOF formula (<kernel.Constant object at 0xd34638>, <kernel.Sort object at 0xdf5c20>) of role type named induction2_gthm_type
% Using role type
% Declaring induction2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd345a8>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem2v2_lthm_type
% Using role type
% Declaring pushprop_lem2v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd34368>, <kernel.DependentProduct object at 0xd346c8>) of role type named hoasvar_type
% Using role type
% Declaring hoasvar:(subst->(term->(subst->Prop)))
% FOF formula (<kernel.Constant object at 0xd34758>, <kernel.Sort object at 0xdf5c20>) of role type named hoaslamnotap_type
% Using role type
% Declaring hoaslamnotap:Prop
% FOF formula (<kernel.Constant object at 0xd34878>, <kernel.Sort object at 0xdf5c20>) of role type named substmonoid_gthm_type
% Using role type
% Declaring substmonoid_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd345a8>, <kernel.Sort object at 0xdf5c20>) of role type named ulamvarsh_type
% Using role type
% Declaring ulamvarsh:Prop
% FOF formula (<kernel.Constant object at 0xd34638>, <kernel.Sort object at 0xdf5c20>) of role type named induction2_type
% Using role type
% Declaring induction2:Prop
% FOF formula (<kernel.Constant object at 0xd34830>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem3v2_type
% Using role type
% Declaring pushprop_lem3v2:Prop
% FOF formula (<kernel.Constant object at 0xd34710>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem2v2_gthm_type
% Using role type
% Declaring pushprop_lem2v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xd347a0>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem1_lthm_type
% Using role type
% Declaring pushprop_lem1_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd34908>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2_type
% Using role type
% Declaring hoasinduction_lem3v2:Prop
% FOF formula (<kernel.Constant object at 0xd348c0>, <kernel.Sort object at 0xdf5c20>) of role type named axshiftcons_type
% Using role type
% Declaring axshiftcons:Prop
% FOF formula (<kernel.Constant object at 0xd34950>, <kernel.Sort object at 0xdf5c20>) of role type named termmset_type
% Using role type
% Declaring termmset:Prop
% FOF formula (<kernel.Constant object at 0xd34998>, <kernel.Sort object at 0xdf5c20>) of role type named pushprop_lem0_lthm_type
% Using role type
% Declaring pushprop_lem0_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd349e0>, <kernel.Sort object at 0xdf5c20>) of role type named hoasapnotvar_lthm_type
% Using role type
% Declaring hoasapnotvar_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd34518>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lem3v2_lthm_type
% Using role type
% Declaring hoasinduction_lem3v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xd349e0>, <kernel.DependentProduct object at 0xd34a70>) of role type named pushprop_p_and_p_prime_type
% Using role type
% Declaring pushprop_p_and_p_prime:(term->(subst->((term->Prop)->((term->Prop)->Prop))))
% FOF formula (<kernel.Constant object at 0xd34b48>, <kernel.Sort object at 0xdf5c20>) of role type named axvarid_type
% Using role type
% Declaring axvarid:Prop
% FOF formula (<kernel.Constant object at 0xd34c68>, <kernel.Sort object at 0xdf5c20>) of role type named hoasinduction_lthm_3_type
% Using role type
% Declaring hoasinduction_lthm_3:Prop
% FOF formula (((eq Prop) axapp) (forall (A:term) (B:term) (M:subst), (((eq term) ((sub ((ap A) B)) M)) ((ap ((sub A) M)) ((sub B) M))))) of role definition named axapp
% A new definition: (((eq Prop) axapp) (forall (A:term) (B:term) (M:subst), (((eq term) ((sub ((ap A) B)) M)) ((ap ((sub A) M)) ((sub B) M)))))
% Defined: axapp:=(forall (A:term) (B:term) (M:subst), (((eq term) ((sub ((ap A) B)) M)) ((ap ((sub A) M)) ((sub B) M))))
% FOF formula (((eq Prop) axvarcons) (forall (A:term) (M:subst), (((eq term) ((sub one) ((push A) M))) A))) of role definition named axvarcons
% A new definition: (((eq Prop) axvarcons) (forall (A:term) (M:subst), (((eq term) ((sub one) ((push A) M))) A)))
% Defined: axvarcons:=(forall (A:term) (M:subst), (((eq term) ((sub one) ((push A) M))) A))
% FOF formula (((eq Prop) axvarid) (forall (A:term), (((eq term) ((sub A) id)) A))) of role definition named axvarid
% A new definition: (((eq Prop) axvarid) (forall (A:term), (((eq term) ((sub A) id)) A)))
% Defined: axvarid:=(forall (A:term), (((eq term) ((sub A) id)) A))
% FOF formula (((eq Prop) axabs) (forall (A:term) (M:subst), (((eq term) ((sub (lam A)) M)) (lam ((sub A) ((push one) ((comp M) sh))))))) of role definition named axabs
% A new definition: (((eq Prop) axabs) (forall (A:term) (M:subst), (((eq term) ((sub (lam A)) M)) (lam ((sub A) ((push one) ((comp M) sh)))))))
% Defined: axabs:=(forall (A:term) (M:subst), (((eq term) ((sub (lam A)) M)) (lam ((sub A) ((push one) ((comp M) sh))))))
% FOF formula (((eq Prop) axclos) (forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N))))) of role definition named axclos
% A new definition: (((eq Prop) axclos) (forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N)))))
% Defined: axclos:=(forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N))))
% FOF formula (((eq Prop) axidl) (forall (M:subst), (((eq subst) ((comp id) M)) M))) of role definition named axidl
% A new definition: (((eq Prop) axidl) (forall (M:subst), (((eq subst) ((comp id) M)) M)))
% Defined: axidl:=(forall (M:subst), (((eq subst) ((comp id) M)) M))
% FOF formula (((eq Prop) axshiftcons) (forall (A:term) (M:subst), (((eq subst) ((comp sh) ((push A) M))) M))) of role definition named axshiftcons
% A new definition: (((eq Prop) axshiftcons) (forall (A:term) (M:subst), (((eq subst) ((comp sh) ((push A) M))) M)))
% Defined: axshiftcons:=(forall (A:term) (M:subst), (((eq subst) ((comp sh) ((push A) M))) M))
% FOF formula (((eq Prop) axassoc) (forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K))))) of role definition named axassoc
% A new definition: (((eq Prop) axassoc) (forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K)))))
% Defined: axassoc:=(forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K))))
% FOF formula (((eq Prop) axmap) (forall (A:term) (M:subst) (N:subst), (((eq subst) ((comp ((push A) M)) N)) ((push ((sub A) N)) ((comp M) N))))) of role definition named axmap
% A new definition: (((eq Prop) axmap) (forall (A:term) (M:subst) (N:subst), (((eq subst) ((comp ((push A) M)) N)) ((push ((sub A) N)) ((comp M) N)))))
% Defined: axmap:=(forall (A:term) (M:subst) (N:subst), (((eq subst) ((comp ((push A) M)) N)) ((push ((sub A) N)) ((comp M) N))))
% FOF formula (((eq Prop) axidr) (forall (M:subst), (((eq subst) ((comp M) id)) M))) of role definition named axidr
% A new definition: (((eq Prop) axidr) (forall (M:subst), (((eq subst) ((comp M) id)) M)))
% Defined: axidr:=(forall (M:subst), (((eq subst) ((comp M) id)) M))
% FOF formula (((eq Prop) axvarshift) (((eq subst) ((push one) sh)) id)) of role definition named axvarshift
% A new definition: (((eq Prop) axvarshift) (((eq subst) ((push one) sh)) id))
% Defined: axvarshift:=(((eq subst) ((push one) sh)) id)
% FOF formula (((eq Prop) axscons) (forall (M:subst), (((eq subst) ((push ((sub one) M)) ((comp sh) M))) M))) of role definition named axscons
% A new definition: (((eq Prop) axscons) (forall (M:subst), (((eq subst) ((push ((sub one) M)) ((comp sh) M))) M)))
% Defined: axscons:=(forall (M:subst), (((eq subst) ((push ((sub one) M)) ((comp sh) M))) M))
% FOF formula (((eq Prop) ulamvar1) (var one)) of role definition named ulamvar1
% A new definition: (((eq Prop) ulamvar1) (var one))
% Defined: ulamvar1:=(var one)
% FOF formula (((eq Prop) ulamvarsh) (forall (A:term), ((var A)->(var ((sub A) sh))))) of role definition named ulamvarsh
% A new definition: (((eq Prop) ulamvarsh) (forall (A:term), ((var A)->(var ((sub A) sh)))))
% Defined: ulamvarsh:=(forall (A:term), ((var A)->(var ((sub A) sh))))
% FOF formula (((eq Prop) ulamvarind) (forall (P:(term->Prop)), ((P one)->((forall (A:term), ((var A)->((P A)->(P ((sub A) sh)))))->(forall (A:term), ((var A)->(P A))))))) of role definition named ulamvarind
% A new definition: (((eq Prop) ulamvarind) (forall (P:(term->Prop)), ((P one)->((forall (A:term), ((var A)->((P A)->(P ((sub A) sh)))))->(forall (A:term), ((var A)->(P A)))))))
% Defined: ulamvarind:=(forall (P:(term->Prop)), ((P one)->((forall (A:term), ((var A)->((P A)->(P ((sub A) sh)))))->(forall (A:term), ((var A)->(P A))))))
% FOF formula (((eq Prop) apinj1) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) A) B)))) of role definition named apinj1
% A new definition: (((eq Prop) apinj1) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) A) B))))
% Defined: apinj1:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) A) B)))
% FOF formula (((eq Prop) apinj2) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) C) D)))) of role definition named apinj2
% A new definition: (((eq Prop) apinj2) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) C) D))))
% Defined: apinj2:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) C) D)))
% FOF formula (((eq Prop) laminj) (forall (A:term) (B:term), ((((eq term) (lam A)) (lam B))->(((eq term) A) B)))) of role definition named laminj
% A new definition: (((eq Prop) laminj) (forall (A:term) (B:term), ((((eq term) (lam A)) (lam B))->(((eq term) A) B))))
% Defined: laminj:=(forall (A:term) (B:term), ((((eq term) (lam A)) (lam B))->(((eq term) A) B)))
% FOF formula (((eq Prop) shinj) (forall (A:term) (B:term), ((((eq term) ((sub A) sh)) ((sub B) sh))->(((eq term) A) B)))) of role definition named shinj
% A new definition: (((eq Prop) shinj) (forall (A:term) (B:term), ((((eq term) ((sub A) sh)) ((sub B) sh))->(((eq term) A) B))))
% Defined: shinj:=(forall (A:term) (B:term), ((((eq term) ((sub A) sh)) ((sub B) sh))->(((eq term) A) B)))
% FOF formula (((eq Prop) lamnotap) (forall (A:term) (B:term) (C:term), (not (((eq term) (lam A)) ((ap B) C))))) of role definition named lamnotap
% A new definition: (((eq Prop) lamnotap) (forall (A:term) (B:term) (C:term), (not (((eq term) (lam A)) ((ap B) C)))))
% Defined: lamnotap:=(forall (A:term) (B:term) (C:term), (not (((eq term) (lam A)) ((ap B) C))))
% FOF formula (((eq Prop) apnotvar) (forall (A:term) (B:term), ((var ((ap A) B))->False))) of role definition named apnotvar
% A new definition: (((eq Prop) apnotvar) (forall (A:term) (B:term), ((var ((ap A) B))->False)))
% Defined: apnotvar:=(forall (A:term) (B:term), ((var ((ap A) B))->False))
% FOF formula (((eq Prop) lamnotvar) (forall (A:term), ((var (lam A))->False))) of role definition named lamnotvar
% A new definition: (((eq Prop) lamnotvar) (forall (A:term), ((var (lam A))->False)))
% Defined: lamnotvar:=(forall (A:term), ((var (lam A))->False))
% FOF formula (((eq Prop) induction) (forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((P A)->(P (lam A))))->(forall (A:term), (P A))))))) of role definition named induction
% A new definition: (((eq Prop) induction) (forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((P A)->(P (lam A))))->(forall (A:term), (P A)))))))
% Defined: induction:=(forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((P A)->(P (lam A))))->(forall (A:term), (P A))))))
% FOF formula (((eq (term->(subst->((term->Prop)->((term->Prop)->Prop))))) pushprop_p_and_p_prime) (fun (A:term) (M:subst) (P:(term->Prop)) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (P ((sub X) ((push A) M))))))) of role definition named pushprop_p_and_p_prime
% A new definition: (((eq (term->(subst->((term->Prop)->((term->Prop)->Prop))))) pushprop_p_and_p_prime) (fun (A:term) (M:subst) (P:(term->Prop)) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (P ((sub X) ((push A) M)))))))
% Defined: pushprop_p_and_p_prime:=(fun (A:term) (M:subst) (P:(term->Prop)) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (P ((sub X) ((push A) M))))))
% FOF formula (((eq Prop) pushprop_lem0) (forall (P:(term->Prop)) (A:term) (M:subst), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((((pushprop_p_and_p_prime A) M) P) Q))))) of role definition named pushprop_lem0
% A new definition: (((eq Prop) pushprop_lem0) (forall (P:(term->Prop)) (A:term) (M:subst), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((((pushprop_p_and_p_prime A) M) P) Q)))))
% Defined: pushprop_lem0:=(forall (P:(term->Prop)) (A:term) (M:subst), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((((pushprop_p_and_p_prime A) M) P) Q))))
% FOF formula (((eq Prop) pushprop_lem0_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem0)))))))))))))))))))))))) of role definition named pushprop_lem0_gthm
% A new definition: (((eq Prop) pushprop_lem0_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem0))))))))))))))))))))))))
% Defined: pushprop_lem0_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem0)))))))))))))))))))))))
% FOF formula (((eq Prop) pushprop_lem0_lthm) pushprop_lem0) of role definition named pushprop_lem0_lthm
% A new definition: (((eq Prop) pushprop_lem0_lthm) pushprop_lem0)
% Defined: pushprop_lem0_lthm:=pushprop_lem0
% FOF formula (((eq Prop) pushprop_lem1) (forall (P:(term->Prop)) (K:(term->Prop)) (A:term) (M:subst) (B:term), ((P A)->(K ((sub A) ((push B) M)))))) of role definition named pushprop_lem1
% A new definition: (((eq Prop) pushprop_lem1) (forall (P:(term->Prop)) (K:(term->Prop)) (A:term) (M:subst) (B:term), ((P A)->(K ((sub A) ((push B) M))))))
% Defined: pushprop_lem1:=(forall (P:(term->Prop)) (K:(term->Prop)) (A:term) (M:subst) (B:term), ((P A)->(K ((sub A) ((push B) M)))))
% FOF formula (((eq Prop) pushprop_lem1_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1)))))))))))))))))))))))) of role definition named pushprop_lem1_gthm
% A new definition: (((eq Prop) pushprop_lem1_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1))))))))))))))))))))))))
% Defined: pushprop_lem1_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1)))))))))))))))))))))))
% FOF formula (((eq Prop) pushprop_lem1_lthm) (axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop_lem1))))) of role definition named pushprop_lem1_lthm
% A new definition: (((eq Prop) pushprop_lem1_lthm) (axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop_lem1)))))
% Defined: pushprop_lem1_lthm:=(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop_lem1))))
% FOF formula (((eq Prop) pushprop_lem1v2) (forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), ((P A)->(((((pushprop_p_and_p_prime A) M) P) Q)->(Q one))))) of role definition named pushprop_lem1v2
% A new definition: (((eq Prop) pushprop_lem1v2) (forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), ((P A)->(((((pushprop_p_and_p_prime A) M) P) Q)->(Q one)))))
% Defined: pushprop_lem1v2:=(forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), ((P A)->(((((pushprop_p_and_p_prime A) M) P) Q)->(Q one))))
% FOF formula (((eq Prop) pushprop_lem1v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1v2)))))))))))))))))))))))) of role definition named pushprop_lem1v2_gthm
% A new definition: (((eq Prop) pushprop_lem1v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1v2))))))))))))))))))))))))
% Defined: pushprop_lem1v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1v2)))))))))))))))))))))))
% FOF formula (((eq Prop) pushprop_lem1v2_lthm) (axvarcons->pushprop_lem1v2)) of role definition named pushprop_lem1v2_lthm
% A new definition: (((eq Prop) pushprop_lem1v2_lthm) (axvarcons->pushprop_lem1v2))
% Defined: pushprop_lem1v2_lthm:=(axvarcons->pushprop_lem1v2)
% FOF formula (((eq Prop) pushprop_lem2v2) (forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(P ((sub B) M))))->(forall (C:term), ((var C)->((Q C)->(Q ((sub C) sh))))))))) of role definition named pushprop_lem2v2
% A new definition: (((eq Prop) pushprop_lem2v2) (forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(P ((sub B) M))))->(forall (C:term), ((var C)->((Q C)->(Q ((sub C) sh)))))))))
% Defined: pushprop_lem2v2:=(forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(P ((sub B) M))))->(forall (C:term), ((var C)->((Q C)->(Q ((sub C) sh))))))))
% FOF formula (((eq Prop) pushprop_lem2v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem2v2)))))))))))))))))))))))) of role definition named pushprop_lem2v2_gthm
% A new definition: (((eq Prop) pushprop_lem2v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem2v2))))))))))))))))))))))))
% Defined: pushprop_lem2v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem2v2)))))))))))))))))))))))
% FOF formula (((eq Prop) pushprop_lem2v2_lthm) (axclos->(axshiftcons->pushprop_lem2v2))) of role definition named pushprop_lem2v2_lthm
% A new definition: (((eq Prop) pushprop_lem2v2_lthm) (axclos->(axshiftcons->pushprop_lem2v2)))
% Defined: pushprop_lem2v2_lthm:=(axclos->(axshiftcons->pushprop_lem2v2))
% FOF formula (((eq Prop) pushprop_lem3v2) (forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(Q B)))->(forall (B:term), ((var B)->(P ((sub B) ((push A) M))))))))) of role definition named pushprop_lem3v2
% A new definition: (((eq Prop) pushprop_lem3v2) (forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(Q B)))->(forall (B:term), ((var B)->(P ((sub B) ((push A) M)))))))))
% Defined: pushprop_lem3v2:=(forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(Q B)))->(forall (B:term), ((var B)->(P ((sub B) ((push A) M))))))))
% FOF formula (((eq Prop) pushprop_lem3v2_lthm) pushprop_lem3v2) of role definition named pushprop_lem3v2_lthm
% A new definition: (((eq Prop) pushprop_lem3v2_lthm) pushprop_lem3v2)
% Defined: pushprop_lem3v2_lthm:=pushprop_lem3v2
% FOF formula (((eq Prop) pushprop) (forall (P:(term->Prop)) (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->((P A)->(forall (B:term), ((var B)->(P ((sub B) ((push A) M))))))))) of role definition named pushprop
% A new definition: (((eq Prop) pushprop) (forall (P:(term->Prop)) (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->((P A)->(forall (B:term), ((var B)->(P ((sub B) ((push A) M)))))))))
% Defined: pushprop:=(forall (P:(term->Prop)) (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->((P A)->(forall (B:term), ((var B)->(P ((sub B) ((push A) M))))))))
% FOF formula (((eq Prop) pushprop_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop)))))))))))))))))))))))) of role definition named pushprop_gthm
% A new definition: (((eq Prop) pushprop_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop))))))))))))))))))))))))
% Defined: pushprop_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop)))))))))))))))))))))))
% FOF formula (((eq Prop) pushprop_lthm_orig) (ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop)))))) of role definition named pushprop_lthm_orig
% A new definition: (((eq Prop) pushprop_lthm_orig) (ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop))))))
% Defined: pushprop_lthm_orig:=(ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop)))))
% FOF formula (((eq Prop) pushprop_lthm) (pushprop_lem0->(ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop))))))) of role definition named pushprop_lthm
% A new definition: (((eq Prop) pushprop_lthm) (pushprop_lem0->(ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop)))))))
% Defined: pushprop_lthm:=(pushprop_lem0->(ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop))))))
% FOF formula (((eq Prop) induction2lem) (forall (P:(term->Prop)), ((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->(P ((sub A) M)))))))) of role definition named induction2lem
% A new definition: (((eq Prop) induction2lem) (forall (P:(term->Prop)), ((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->(P ((sub A) M))))))))
% Defined: induction2lem:=(forall (P:(term->Prop)), ((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->(P ((sub A) M)))))))
% FOF formula (((eq Prop) induction2lem_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->induction2lem))))))))))))))))))))))))) of role definition named induction2lem_gthm
% A new definition: (((eq Prop) induction2lem_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->induction2lem)))))))))))))))))))))))))
% Defined: induction2lem_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->induction2lem))))))))))))))))))))))))
% FOF formula (((eq Prop) induction2lem_lthm) (axapp->(axvarcons->(axabs->(axclos->(axshiftcons->(axassoc->(axmap->(axidr->(induction->(pushprop->induction2lem))))))))))) of role definition named induction2lem_lthm
% A new definition: (((eq Prop) induction2lem_lthm) (axapp->(axvarcons->(axabs->(axclos->(axshiftcons->(axassoc->(axmap->(axidr->(induction->(pushprop->induction2lem)))))))))))
% Defined: induction2lem_lthm:=(axapp->(axvarcons->(axabs->(axclos->(axshiftcons->(axassoc->(axmap->(axidr->(induction->(pushprop->induction2lem))))))))))
% FOF formula (((eq Prop) induction2) (forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term), (P A))))))) of role definition named induction2
% A new definition: (((eq Prop) induction2) (forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term), (P A)))))))
% Defined: induction2:=(forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term), (P A))))))
% FOF formula (((eq Prop) induction2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->induction2)))))))))))))))))))))))))) of role definition named induction2_gthm
% A new definition: (((eq Prop) induction2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->induction2))))))))))))))))))))))))))
% Defined: induction2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->induction2)))))))))))))))))))))))))
% FOF formula (((eq Prop) induction2_lthm) (axvarid->(induction2lem->induction2))) of role definition named induction2_lthm
% A new definition: (((eq Prop) induction2_lthm) (axvarid->(induction2lem->induction2)))
% Defined: induction2_lthm:=(axvarid->(induction2lem->induction2))
% FOF formula (((eq Prop) substmonoid) ((and ((and (forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K))))) (forall (M:subst), (((eq subst) ((comp id) M)) M)))) (forall (M:subst), (((eq subst) ((comp M) id)) M)))) of role definition named substmonoid
% A new definition: (((eq Prop) substmonoid) ((and ((and (forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K))))) (forall (M:subst), (((eq subst) ((comp id) M)) M)))) (forall (M:subst), (((eq subst) ((comp M) id)) M))))
% Defined: substmonoid:=((and ((and (forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K))))) (forall (M:subst), (((eq subst) ((comp id) M)) M)))) (forall (M:subst), (((eq subst) ((comp M) id)) M)))
% FOF formula (((eq Prop) substmonoid_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->substmonoid))))))))))))))))))))))))))) of role definition named substmonoid_gthm
% A new definition: (((eq Prop) substmonoid_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->substmonoid)))))))))))))))))))))))))))
% Defined: substmonoid_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->substmonoid))))))))))))))))))))))))))
% FOF formula (((eq Prop) substmonoid_lthm) (axidl->(axassoc->(axidr->substmonoid)))) of role definition named substmonoid_lthm
% A new definition: (((eq Prop) substmonoid_lthm) (axidl->(axassoc->(axidr->substmonoid))))
% Defined: substmonoid_lthm:=(axidl->(axassoc->(axidr->substmonoid)))
% FOF formula (((eq Prop) termmset) ((and (forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N))))) (forall (A:term), (((eq term) ((sub A) id)) A)))) of role definition named termmset
% A new definition: (((eq Prop) termmset) ((and (forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N))))) (forall (A:term), (((eq term) ((sub A) id)) A))))
% Defined: termmset:=((and (forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N))))) (forall (A:term), (((eq term) ((sub A) id)) A)))
% FOF formula (((eq Prop) termmset_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->termmset)))))))))))))))))))))))))))) of role definition named termmset_gthm
% A new definition: (((eq Prop) termmset_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->termmset))))))))))))))))))))))))))))
% Defined: termmset_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->termmset)))))))))))))))))))))))))))
% FOF formula (((eq Prop) termmset_lthm) (axvarid->(axclos->termmset))) of role definition named termmset_lthm
% A new definition: (((eq Prop) termmset_lthm) (axvarid->(axclos->termmset)))
% Defined: termmset_lthm:=(axvarid->(axclos->termmset))
% FOF formula (((eq (subst->(term->(subst->(term->term))))) hoasap) (fun (M:subst) (A:term) (N:subst) (B:term)=> ((ap ((sub A) N)) B))) of role definition named hoasap
% A new definition: (((eq (subst->(term->(subst->(term->term))))) hoasap) (fun (M:subst) (A:term) (N:subst) (B:term)=> ((ap ((sub A) N)) B)))
% Defined: hoasap:=(fun (M:subst) (A:term) (N:subst) (B:term)=> ((ap ((sub A) N)) B))
% FOF formula (((eq (subst->((subst->(term->term))->term))) hoaslam) (fun (M:subst) (F:(subst->(term->term)))=> (lam ((F sh) one)))) of role definition named hoaslam
% A new definition: (((eq (subst->((subst->(term->term))->term))) hoaslam) (fun (M:subst) (F:(subst->(term->term)))=> (lam ((F sh) one))))
% Defined: hoaslam:=(fun (M:subst) (F:(subst->(term->term)))=> (lam ((F sh) one)))
% FOF formula (((eq (subst->(term->(subst->Prop)))) hoasvar) (fun (M:subst) (A:term) (N:subst)=> (var ((sub A) N)))) of role definition named hoasvar
% A new definition: (((eq (subst->(term->(subst->Prop)))) hoasvar) (fun (M:subst) (A:term) (N:subst)=> (var ((sub A) N))))
% Defined: hoasvar:=(fun (M:subst) (A:term) (N:subst)=> (var ((sub A) N)))
% FOF formula (((eq Prop) hoasapinj1) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) A) B)))) of role definition named hoasapinj1
% A new definition: (((eq Prop) hoasapinj1) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) A) B))))
% Defined: hoasapinj1:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) A) B)))
% FOF formula (((eq Prop) hoasapinj1_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->hoasapinj1))))))))))))))))))))))))))))) of role definition named hoasapinj1_gthm
% A new definition: (((eq Prop) hoasapinj1_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->hoasapinj1)))))))))))))))))))))))))))))
% Defined: hoasapinj1_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->hoasapinj1))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasapinj1_lthm) (axvarid->(apinj1->hoasapinj1))) of role definition named hoasapinj1_lthm
% A new definition: (((eq Prop) hoasapinj1_lthm) (axvarid->(apinj1->hoasapinj1)))
% Defined: hoasapinj1_lthm:=(axvarid->(apinj1->hoasapinj1))
% FOF formula (((eq Prop) hoasapinj2) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) C) D)))) of role definition named hoasapinj2
% A new definition: (((eq Prop) hoasapinj2) (forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) C) D))))
% Defined: hoasapinj2:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) C) D)))
% FOF formula (((eq Prop) hoasapinj2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->hoasapinj2)))))))))))))))))))))))))))))) of role definition named hoasapinj2_gthm
% A new definition: (((eq Prop) hoasapinj2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->hoasapinj2))))))))))))))))))))))))))))))
% Defined: hoasapinj2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->hoasapinj2)))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasapinj2_lthm) (apinj2->hoasapinj2)) of role definition named hoasapinj2_lthm
% A new definition: (((eq Prop) hoasapinj2_lthm) (apinj2->hoasapinj2))
% Defined: hoasapinj2_lthm:=(apinj2->hoasapinj2)
% FOF formula (((eq Prop) hoaslaminj) (forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (G:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((G M) A)) N)) ((G ((comp M) N)) ((sub A) N))))->((((eq term) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) ((hoaslam id) (fun (M:subst) (A:term)=> ((G M) A))))->(forall (M:subst) (A:term), (((eq term) ((F M) A)) ((G M) A))))))))) of role definition named hoaslaminj
% A new definition: (((eq Prop) hoaslaminj) (forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (G:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((G M) A)) N)) ((G ((comp M) N)) ((sub A) N))))->((((eq term) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) ((hoaslam id) (fun (M:subst) (A:term)=> ((G M) A))))->(forall (M:subst) (A:term), (((eq term) ((F M) A)) ((G M) A)))))))))
% Defined: hoaslaminj:=(forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (G:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((G M) A)) N)) ((G ((comp M) N)) ((sub A) N))))->((((eq term) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) ((hoaslam id) (fun (M:subst) (A:term)=> ((G M) A))))->(forall (M:subst) (A:term), (((eq term) ((F M) A)) ((G M) A))))))))
% FOF formula (((eq Prop) hoaslaminj_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->hoaslaminj))))))))))))))))))))))))))))))) of role definition named hoaslaminj_gthm
% A new definition: (((eq Prop) hoaslaminj_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->hoaslaminj)))))))))))))))))))))))))))))))
% Defined: hoaslaminj_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->hoaslaminj))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoaslaminj_lthm) (axvarcons->(axshiftcons->(laminj->hoaslaminj)))) of role definition named hoaslaminj_lthm
% A new definition: (((eq Prop) hoaslaminj_lthm) (axvarcons->(axshiftcons->(laminj->hoaslaminj))))
% Defined: hoaslaminj_lthm:=(axvarcons->(axshiftcons->(laminj->hoaslaminj)))
% FOF formula (((eq Prop) hoaslamnotap) (forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (A:term) (B:term), (not (((eq term) ((hoaslam id) (fun (M:subst) (C:term)=> ((F M) C)))) ((((hoasap id) A) id) B))))))) of role definition named hoaslamnotap
% A new definition: (((eq Prop) hoaslamnotap) (forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (A:term) (B:term), (not (((eq term) ((hoaslam id) (fun (M:subst) (C:term)=> ((F M) C)))) ((((hoasap id) A) id) B)))))))
% Defined: hoaslamnotap:=(forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (A:term) (B:term), (not (((eq term) ((hoaslam id) (fun (M:subst) (C:term)=> ((F M) C)))) ((((hoasap id) A) id) B))))))
% FOF formula (((eq Prop) hoaslamnotap_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->hoaslamnotap)))))))))))))))))))))))))))))))) of role definition named hoaslamnotap_gthm
% A new definition: (((eq Prop) hoaslamnotap_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->hoaslamnotap))))))))))))))))))))))))))))))))
% Defined: hoaslamnotap_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->hoaslamnotap)))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoaslamnotap_lthm) (lamnotap->hoaslamnotap)) of role definition named hoaslamnotap_lthm
% A new definition: (((eq Prop) hoaslamnotap_lthm) (lamnotap->hoaslamnotap))
% Defined: hoaslamnotap_lthm:=(lamnotap->hoaslamnotap)
% FOF formula (((eq Prop) hoaslamnotvar) (forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((((hoasvar id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->False)))) of role definition named hoaslamnotvar
% A new definition: (((eq Prop) hoaslamnotvar) (forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((((hoasvar id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->False))))
% Defined: hoaslamnotvar:=(forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((((hoasvar id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->False)))
% FOF formula (((eq Prop) hoaslamnotvar_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->hoaslamnotvar))))))))))))))))))))))))))))))))) of role definition named hoaslamnotvar_gthm
% A new definition: (((eq Prop) hoaslamnotvar_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->hoaslamnotvar)))))))))))))))))))))))))))))))))
% Defined: hoaslamnotvar_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->hoaslamnotvar))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoaslamnotvar_lthm) (axvarid->(lamnotvar->hoaslamnotvar))) of role definition named hoaslamnotvar_lthm
% A new definition: (((eq Prop) hoaslamnotvar_lthm) (axvarid->(lamnotvar->hoaslamnotvar)))
% Defined: hoaslamnotvar_lthm:=(axvarid->(lamnotvar->hoaslamnotvar))
% FOF formula (((eq Prop) hoasapnotvar) (forall (A:term) (B:term), ((((hoasvar id) ((((hoasap id) A) id) B)) id)->False))) of role definition named hoasapnotvar
% A new definition: (((eq Prop) hoasapnotvar) (forall (A:term) (B:term), ((((hoasvar id) ((((hoasap id) A) id) B)) id)->False)))
% Defined: hoasapnotvar:=(forall (A:term) (B:term), ((((hoasvar id) ((((hoasap id) A) id) B)) id)->False))
% FOF formula (((eq Prop) hoasapnotvar_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->hoasapnotvar)))))))))))))))))))))))))))))))))) of role definition named hoasapnotvar_gthm
% A new definition: (((eq Prop) hoasapnotvar_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->hoasapnotvar))))))))))))))))))))))))))))))))))
% Defined: hoasapnotvar_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->hoasapnotvar)))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasapnotvar_lthm) (axvarid->(apnotvar->hoasapnotvar))) of role definition named hoasapnotvar_lthm
% A new definition: (((eq Prop) hoasapnotvar_lthm) (axvarid->(apnotvar->hoasapnotvar)))
% Defined: hoasapnotvar_lthm:=(axvarid->(apnotvar->hoasapnotvar))
% FOF formula (((eq ((subst->(term->(subst->Prop)))->((term->Prop)->Prop))) hoasinduction_p_and_p_prime) (fun (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (((P id) X) id))))) of role definition named hoasinduction_p_and_p_prime
% A new definition: (((eq ((subst->(term->(subst->Prop)))->((term->Prop)->Prop))) hoasinduction_p_and_p_prime) (fun (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (((P id) X) id)))))
% Defined: hoasinduction_p_and_p_prime:=(fun (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (((P id) X) id))))
% FOF formula (((eq Prop) hoasinduction_lem0) (forall (P:(subst->(term->(subst->Prop)))), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((hoasinduction_p_and_p_prime P) Q))))) of role definition named hoasinduction_lem0
% A new definition: (((eq Prop) hoasinduction_lem0) (forall (P:(subst->(term->(subst->Prop)))), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((hoasinduction_p_and_p_prime P) Q)))))
% Defined: hoasinduction_lem0:=(forall (P:(subst->(term->(subst->Prop)))), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((hoasinduction_p_and_p_prime P) Q))))
% FOF formula (((eq Prop) hoasinduction_lem0_lthm) hoasinduction_lem0) of role definition named hoasinduction_lem0_lthm
% A new definition: (((eq Prop) hoasinduction_lem0_lthm) hoasinduction_lem0)
% Defined: hoasinduction_lem0_lthm:=hoasinduction_lem0
% FOF formula (((eq Prop) hoasinduction_lem1v2) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((var A)->(Q A))))))))) of role definition named hoasinduction_lem1v2
% A new definition: (((eq Prop) hoasinduction_lem1v2) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((var A)->(Q A)))))))))
% Defined: hoasinduction_lem1v2:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((var A)->(Q A))))))))
% FOF formula (((eq Prop) hoasinduction_lem1v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1v2))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem1v2_gthm
% A new definition: (((eq Prop) hoasinduction_lem1v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1v2)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem1v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1v2))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem2v2) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term) (B:term), ((Q A)->((Q B)->(Q ((ap A) B))))))))))) of role definition named hoasinduction_lem2v2
% A new definition: (((eq Prop) hoasinduction_lem2v2) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term) (B:term), ((Q A)->((Q B)->(Q ((ap A) B)))))))))))
% Defined: hoasinduction_lem2v2:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term) (B:term), ((Q A)->((Q B)->(Q ((ap A) B))))))))))
% FOF formula (((eq Prop) hoasinduction_lem2v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2v2))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem2v2_gthm
% A new definition: (((eq Prop) hoasinduction_lem2v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2v2)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem2v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2v2))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem3v2_f) (forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (forall (A:term) (M:subst), (((eq term) ((F M) A)) ((sub B) ((push A) M)))))))) of role definition named hoasinduction_lem3v2_f
% A new definition: (((eq Prop) hoasinduction_lem3v2_f) (forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (forall (A:term) (M:subst), (((eq term) ((F M) A)) ((sub B) ((push A) M))))))))
% Defined: hoasinduction_lem3v2_f:=(forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (forall (A:term) (M:subst), (((eq term) ((F M) A)) ((sub B) ((push A) M)))))))
% FOF formula (((eq Prop) hoasinduction_lem3v2_f_lthm) hoasinduction_lem3v2_f) of role definition named hoasinduction_lem3v2_f_lthm
% A new definition: (((eq Prop) hoasinduction_lem3v2_f_lthm) hoasinduction_lem3v2_f)
% Defined: hoasinduction_lem3v2_f_lthm:=hoasinduction_lem3v2_f
% FOF formula (((eq Prop) hoasinduction_lem3v2) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A)))))))))) of role definition named hoasinduction_lem3v2
% A new definition: (((eq Prop) hoasinduction_lem3v2) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A))))))))))
% Defined: hoasinduction_lem3v2:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A)))))))))
% FOF formula (((eq Prop) hoasinduction_lem3v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem3v2))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem3v2_gthm
% A new definition: (((eq Prop) hoasinduction_lem3v2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem3v2)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem3v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem3v2))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem3v2_lthm) (axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2))))) of role definition named hoasinduction_lem3v2_lthm
% A new definition: (((eq Prop) hoasinduction_lem3v2_lthm) (axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2)))))
% Defined: hoasinduction_lem3v2_lthm:=(axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2))))
% FOF formula (((eq Prop) hoasinduction_lem3v2a) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A)))))))) of role definition named hoasinduction_lem3v2a
% A new definition: (((eq Prop) hoasinduction_lem3v2a) (forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A))))))))
% Defined: hoasinduction_lem3v2a:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A)))))))
% FOF formula (((eq Prop) hoasinduction_lem3v2a_lthm) (hoasinduction_lem3v2_f->(axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2a)))))) of role definition named hoasinduction_lem3v2a_lthm
% A new definition: (((eq Prop) hoasinduction_lem3v2a_lthm) (hoasinduction_lem3v2_f->(axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2a))))))
% Defined: hoasinduction_lem3v2a_lthm:=(hoasinduction_lem3v2_f->(axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2a)))))
% FOF formula (((eq Prop) hoasinduction_lem1) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(forall (A:term), ((var A)->(((P id) A) id)))))))) of role definition named hoasinduction_lem1
% A new definition: (((eq Prop) hoasinduction_lem1) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(forall (A:term), ((var A)->(((P id) A) id))))))))
% Defined: hoasinduction_lem1:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(forall (A:term), ((var A)->(((P id) A) id)))))))
% FOF formula (((eq Prop) hoasinduction_lem1_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem1_gthm
% A new definition: (((eq Prop) hoasinduction_lem1_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem1_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem1_lthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem1_lthm
% A new definition: (((eq Prop) hoasinduction_lem1_lthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem1_lthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem2) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((ap A) B)) id))))))))) of role definition named hoasinduction_lem2
% A new definition: (((eq Prop) hoasinduction_lem2) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((ap A) B)) id)))))))))
% Defined: hoasinduction_lem2:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((ap A) B)) id))))))))
% FOF formula (((eq Prop) hoasinduction_lem2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem2_gthm
% A new definition: (((eq Prop) hoasinduction_lem2_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem2_lthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem2_lthm
% A new definition: (((eq Prop) hoasinduction_lem2_lthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2)))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem2_lthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem3aa) (forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id)))))) of role definition named hoasinduction_lem3aa
% A new definition: (((eq Prop) hoasinduction_lem3aa) (forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id))))))
% Defined: hoasinduction_lem3aa:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id)))))
% FOF formula (((eq Prop) hoasinduction_lem3aa_lthm) (axclos->(axmap->hoasinduction_lem3aa))) of role definition named hoasinduction_lem3aa_lthm
% A new definition: (((eq Prop) hoasinduction_lem3aa_lthm) (axclos->(axmap->hoasinduction_lem3aa)))
% Defined: hoasinduction_lem3aa_lthm:=(axclos->(axmap->hoasinduction_lem3aa))
% FOF formula (((eq Prop) hoasinduction_lem3aaa) (forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), (((ex term) (fun (C:term)=> (forall (M:subst) (A:term) (N:subst), ((and (((eq term) ((sub ((F M) A)) N)) ((sub ((sub C) ((push A) M))) N))) (((eq term) ((sub C) ((push ((sub A) N)) ((comp M) N)))) ((F ((comp M) N)) ((sub A) N)))))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id)))))) of role definition named hoasinduction_lem3aaa
% A new definition: (((eq Prop) hoasinduction_lem3aaa) (forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), (((ex term) (fun (C:term)=> (forall (M:subst) (A:term) (N:subst), ((and (((eq term) ((sub ((F M) A)) N)) ((sub ((sub C) ((push A) M))) N))) (((eq term) ((sub C) ((push ((sub A) N)) ((comp M) N)))) ((F ((comp M) N)) ((sub A) N)))))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id))))))
% Defined: hoasinduction_lem3aaa:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), (((ex term) (fun (C:term)=> (forall (M:subst) (A:term) (N:subst), ((and (((eq term) ((sub ((F M) A)) N)) ((sub ((sub C) ((push A) M))) N))) (((eq term) ((sub C) ((push ((sub A) N)) ((comp M) N)))) ((F ((comp M) N)) ((sub A) N)))))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id)))))
% FOF formula (((eq Prop) hoasinduction_lem3) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id)))))))) of role definition named hoasinduction_lem3
% A new definition: (((eq Prop) hoasinduction_lem3) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id))))))))
% Defined: hoasinduction_lem3:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id)))))))
% FOF formula (((eq Prop) hoasinduction_lem3_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3))))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem3_gthm
% A new definition: (((eq Prop) hoasinduction_lem3_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3)))))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem3_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3))))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem3_lthm) (axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3)))) of role definition named hoasinduction_lem3_lthm
% A new definition: (((eq Prop) hoasinduction_lem3_lthm) (axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3))))
% Defined: hoasinduction_lem3_lthm:=(axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3)))
% FOF formula (((eq Prop) hoasinduction_lem3a) (forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id)))))) of role definition named hoasinduction_lem3a
% A new definition: (((eq Prop) hoasinduction_lem3a) (forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id))))))
% Defined: hoasinduction_lem3a:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id)))))
% FOF formula (((eq Prop) hoasinduction_lem3a_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3a))))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem3a_gthm
% A new definition: (((eq Prop) hoasinduction_lem3a_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3a)))))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem3a_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3a))))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem3a_lthm) (axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3a)))) of role definition named hoasinduction_lem3a_lthm
% A new definition: (((eq Prop) hoasinduction_lem3a_lthm) (axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3a))))
% Defined: hoasinduction_lem3a_lthm:=(axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3a)))
% FOF formula (((eq Prop) hoasinduction_lem3b) (forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (((eq term) ((sub B) ((push one) sh))) ((F sh) one)))))) of role definition named hoasinduction_lem3b
% A new definition: (((eq Prop) hoasinduction_lem3b) (forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (((eq term) ((sub B) ((push one) sh))) ((F sh) one))))))
% Defined: hoasinduction_lem3b:=(forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (((eq term) ((sub B) ((push one) sh))) ((F sh) one)))))
% FOF formula (((eq Prop) hoasinduction_lem3b_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3b))))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_lem3b_gthm
% A new definition: (((eq Prop) hoasinduction_lem3b_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3b)))))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_lem3b_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3b))))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lem3b_lthm) hoasinduction_lem3b) of role definition named hoasinduction_lem3b_lthm
% A new definition: (((eq Prop) hoasinduction_lem3b_lthm) hoasinduction_lem3b)
% Defined: hoasinduction_lem3b_lthm:=hoasinduction_lem3b
% FOF formula (((eq Prop) hoasinduction) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id))))))))) of role definition named hoasinduction
% A new definition: (((eq Prop) hoasinduction) (forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id)))))))))
% Defined: hoasinduction:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id))))))))
% FOF formula (((eq Prop) hoasinduction_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction)))))))))))))))))))))))))))))))))))))) of role definition named hoasinduction_gthm
% A new definition: (((eq Prop) hoasinduction_gthm) (axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction))))))))))))))))))))))))))))))))))))))
% Defined: hoasinduction_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction)))))))))))))))))))))))))))))))))))))
% FOF formula (((eq Prop) hoasinduction_lthm) (induction2->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction))))) of role definition named hoasinduction_lthm
% A new definition: (((eq Prop) hoasinduction_lthm) (induction2->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction)))))
% Defined: hoasinduction_lthm:=(induction2->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction))))
% FOF formula (((eq Prop) hoasinduction_lthm_3) (hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction))))) of role definition named hoasinduction_lthm_3
% A new definition: (((eq Prop) hoasinduction_lthm_3) (hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction)))))
% Defined: hoasinduction_lthm_3:=(hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction))))
% FOF formula (((eq Prop) hoasinduction_no_psi_cond) (forall (P:(subst->(term->(subst->Prop)))), ((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id)))))) of role definition named hoasinduction_no_psi_cond
% A new definition: (((eq Prop) hoasinduction_no_psi_cond) (forall (P:(subst->(term->(subst->Prop)))), ((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id))))))
% Defined: hoasinduction_no_psi_cond:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id)))))
% FOF formula (((eq Prop) hoasinduction_no_psi_cond_lthm) (hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction_no_psi_cond))))) of role definition named hoasinduction_no_psi_cond_lthm
% A new definition: (((eq Prop) hoasinduction_no_psi_cond_lthm) (hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction_no_psi_cond)))))
% Defined: hoasinduction_no_psi_cond_lthm:=(hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction_no_psi_cond))))
% FOF formula induction2_lthm of role conjecture named thm
% Conjecture to prove = induction2_lthm:Prop
% We need to prove ['induction2_lthm']
% Parameter term:Type.
% Parameter subst:Type.
% Parameter one:term.
% Parameter ap:(term->(term->term)).
% Parameter lam:(term->term).
% Parameter sub:(term->(subst->term)).
% Parameter id:subst.
% Parameter sh:subst.
% Parameter push:(term->(subst->subst)).
% Parameter comp:(subst->(subst->subst)).
% Parameter var:(term->Prop).
% Definition pushprop_lem1v2:=(forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), ((P A)->(((((pushprop_p_and_p_prime A) M) P) Q)->(Q one)))):Prop.
% Definition pushprop_lem1_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1))))))))))))))))))))))):Prop.
% Definition axmap:=(forall (A:term) (M:subst) (N:subst), (((eq subst) ((comp ((push A) M)) N)) ((push ((sub A) N)) ((comp M) N)))):Prop.
% Definition pushprop_lem0_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem0))))))))))))))))))))))):Prop.
% Definition shinj:=(forall (A:term) (B:term), ((((eq term) ((sub A) sh)) ((sub B) sh))->(((eq term) A) B))):Prop.
% Definition hoasinduction_lem1v2:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((var A)->(Q A)))))))):Prop.
% Definition hoasinduction_lem1v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1v2)))))))))))))))))))))))))))))))))):Prop.
% Definition hoasap:=(fun (M:subst) (A:term) (N:subst) (B:term)=> ((ap ((sub A) N)) B)):(subst->(term->(subst->(term->term)))).
% Definition induction2lem:=(forall (P:(term->Prop)), ((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->(P ((sub A) M))))))):Prop.
% Definition hoasinduction_lem3v2_f:=(forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (forall (A:term) (M:subst), (((eq term) ((F M) A)) ((sub B) ((push A) M))))))):Prop.
% Definition axvarshift:=(((eq subst) ((push one) sh)) id):Prop.
% Definition hoasapinj2:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) C) D))):Prop.
% Definition hoasapnotvar_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->hoasapnotvar))))))))))))))))))))))))))))))))):Prop.
% Definition hoasapinj1:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((((hoasap id) A) id) C)) ((((hoasap id) B) id) D))->(((eq term) A) B))):Prop.
% Definition ulamvar1:=(var one):Prop.
% Definition induction2lem_lthm:=(axapp->(axvarcons->(axabs->(axclos->(axshiftcons->(axassoc->(axmap->(axidr->(induction->(pushprop->induction2lem)))))))))):Prop.
% Definition hoasinduction_lem3v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem3v2)))))))))))))))))))))))))))))))))):Prop.
% Definition apnotvar:=(forall (A:term) (B:term), ((var ((ap A) B))->False)):Prop.
% Definition pushprop_lthm_orig:=(ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop))))):Prop.
% Definition hoasinduction_lem3v2_f_lthm:=hoasinduction_lem3v2_f:Prop.
% Definition hoasinduction_lthm:=(induction2->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction)))):Prop.
% Definition hoasinduction_no_psi_cond_lthm:=(hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction_no_psi_cond)))):Prop.
% Definition hoaslaminj:=(forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (G:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((G M) A)) N)) ((G ((comp M) N)) ((sub A) N))))->((((eq term) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) ((hoaslam id) (fun (M:subst) (A:term)=> ((G M) A))))->(forall (M:subst) (A:term), (((eq term) ((F M) A)) ((G M) A)))))))):Prop.
% Definition hoasinduction_lem3aaa:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), (((ex term) (fun (C:term)=> (forall (M:subst) (A:term) (N:subst), ((and (((eq term) ((sub ((F M) A)) N)) ((sub ((sub C) ((push A) M))) N))) (((eq term) ((sub C) ((push ((sub A) N)) ((comp M) N)))) ((F ((comp M) N)) ((sub A) N)))))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id))))):Prop.
% Definition induction2lem_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->induction2lem)))))))))))))))))))))))):Prop.
% Definition hoasinduction_lem3aa_lthm:=(axclos->(axmap->hoasinduction_lem3aa)):Prop.
% Definition hoasinduction_lem3:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id))))))):Prop.
% Definition hoasinduction_lem2:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((ap A) B)) id)))))))):Prop.
% Definition termmset_lthm:=(axvarid->(axclos->termmset)):Prop.
% Definition hoasinduction_lem1:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->(forall (A:term), ((var A)->(((P id) A) id))))))):Prop.
% Definition hoaslamnotap_lthm:=(lamnotap->hoaslamnotap):Prop.
% Definition pushprop_lem1v2_lthm:=(axvarcons->pushprop_lem1v2):Prop.
% Definition hoasapnotvar:=(forall (A:term) (B:term), ((((hoasvar id) ((((hoasap id) A) id) B)) id)->False)):Prop.
% Definition hoasinduction_lem0:=(forall (P:(subst->(term->(subst->Prop)))), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((hoasinduction_p_and_p_prime P) Q)))):Prop.
% Definition hoasinduction:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term), ((((hoasvar id) A) id)->(((P id) A) id)))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id)))))))):Prop.
% Definition hoasinduction_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->(hoasinduction_lem3->hoasinduction))))))))))))))))))))))))))))))))))))):Prop.
% Definition axapp:=(forall (A:term) (B:term) (M:subst), (((eq term) ((sub ((ap A) B)) M)) ((ap ((sub A) M)) ((sub B) M)))):Prop.
% Definition hoaslamnotvar_lthm:=(axvarid->(lamnotvar->hoaslamnotvar)):Prop.
% Definition pushprop_lem3v2_lthm:=pushprop_lem3v2:Prop.
% Definition hoasinduction_lem3b_lthm:=hoasinduction_lem3b:Prop.
% Definition ulamvarind:=(forall (P:(term->Prop)), ((P one)->((forall (A:term), ((var A)->((P A)->(P ((sub A) sh)))))->(forall (A:term), ((var A)->(P A)))))):Prop.
% Definition induction:=(forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((P A)->(P (lam A))))->(forall (A:term), (P A)))))):Prop.
% Definition hoasinduction_lem3a_lthm:=(axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3a))):Prop.
% Definition termmset_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->termmset))))))))))))))))))))))))))):Prop.
% Definition hoasinduction_lem3aa:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam ((sub A) ((push one) sh)))) id))))):Prop.
% Definition pushprop_lem1v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem1v2))))))))))))))))))))))):Prop.
% Definition hoaslamnotap_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->hoaslamnotap))))))))))))))))))))))))))))))):Prop.
% Definition hoaslamnotvar_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->hoaslamnotvar)))))))))))))))))))))))))))))))):Prop.
% Definition hoasinduction_lem3b_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3b)))))))))))))))))))))))))))))))))))):Prop.
% Definition pushprop_lem2v2:=(forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(P ((sub B) M))))->(forall (C:term), ((var C)->((Q C)->(Q ((sub C) sh)))))))):Prop.
% Definition hoasinduction_lem3a_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3a)))))))))))))))))))))))))))))))))))):Prop.
% Definition axclos:=(forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N)))):Prop.
% Definition axassoc:=(forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K)))):Prop.
% Definition hoasinduction_lem2v2:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term) (B:term), ((Q A)->((Q B)->(Q ((ap A) B)))))))))):Prop.
% Definition pushprop_lthm:=(pushprop_lem0->(ulamvar1->(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop)))))):Prop.
% Definition apinj2:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) C) D))):Prop.
% Definition apinj1:=(forall (A:term) (B:term) (C:term) (D:term), ((((eq term) ((ap A) C)) ((ap B) D))->(((eq term) A) B))):Prop.
% Definition hoasapinj2_lthm:=(apinj2->hoasapinj2):Prop.
% Definition hoasinduction_lem3v2a:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A))))))):Prop.
% Definition hoasapinj1_lthm:=(axvarid->(apinj1->hoasapinj1)):Prop.
% Definition hoaslaminj_lthm:=(axvarcons->(axshiftcons->(laminj->hoaslaminj))):Prop.
% Definition axvarcons:=(forall (A:term) (M:subst), (((eq term) ((sub one) ((push A) M))) A)):Prop.
% Definition hoaslam:=(fun (M:subst) (F:(subst->(term->term)))=> (lam ((F sh) one))):(subst->((subst->(term->term))->term)).
% Definition axscons:=(forall (M:subst), (((eq subst) ((push ((sub one) M)) ((comp sh) M))) M)):Prop.
% Definition hoasinduction_lem2v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2v2)))))))))))))))))))))))))))))))))):Prop.
% Definition axidr:=(forall (M:subst), (((eq subst) ((comp M) id)) M)):Prop.
% Definition pushprop_lem1:=(forall (P:(term->Prop)) (K:(term->Prop)) (A:term) (M:subst) (B:term), ((P A)->(K ((sub A) ((push B) M))))):Prop.
% Definition laminj:=(forall (A:term) (B:term), ((((eq term) (lam A)) (lam B))->(((eq term) A) B))):Prop.
% Definition hoasinduction_lem3_lthm:=(axvarid->(axvarshift->(hoasinduction_lem3aa->hoasinduction_lem3))):Prop.
% Definition pushprop_lem0:=(forall (P:(term->Prop)) (A:term) (M:subst), ((ex (term->Prop)) (fun (Q:(term->Prop))=> ((((pushprop_p_and_p_prime A) M) P) Q)))):Prop.
% Definition pushprop_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop))))))))))))))))))))))):Prop.
% Definition axabs:=(forall (A:term) (M:subst), (((eq term) ((sub (lam A)) M)) (lam ((sub A) ((push one) ((comp M) sh)))))):Prop.
% Definition hoasinduction_lem3v2a_lthm:=(hoasinduction_lem3v2_f->(axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2a))))):Prop.
% Definition hoasinduction_lem2_lthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2)))))))))))))))))))))))))))))))))):Prop.
% Definition hoasapinj2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->hoasapinj2))))))))))))))))))))))))))))):Prop.
% Definition hoasinduction_p_and_p_prime:=(fun (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (((P id) X) id)))):((subst->(term->(subst->Prop)))->((term->Prop)->Prop)).
% Definition hoasinduction_lem1_lthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1)))))))))))))))))))))))))))))))))):Prop.
% Definition lamnotap:=(forall (A:term) (B:term) (C:term), (not (((eq term) (lam A)) ((ap B) C)))):Prop.
% Definition hoasapinj1_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->hoasapinj1)))))))))))))))))))))))))))):Prop.
% Definition hoaslamnotvar:=(forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((((hoasvar id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->False))):Prop.
% Definition axidl:=(forall (M:subst), (((eq subst) ((comp id) M)) M)):Prop.
% Definition hoaslaminj_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->hoaslaminj)))))))))))))))))))))))))))))):Prop.
% Definition induction2_lthm:=(axvarid->(induction2lem->induction2)):Prop.
% Definition hoasinduction_lem0_lthm:=hoasinduction_lem0:Prop.
% Definition substmonoid_lthm:=(axidl->(axassoc->(axidr->substmonoid))):Prop.
% Definition pushprop:=(forall (P:(term->Prop)) (A:term) (M:subst), ((forall (B:term), ((var B)->(P ((sub B) M))))->((P A)->(forall (B:term), ((var B)->(P ((sub B) ((push A) M)))))))):Prop.
% Definition hoasinduction_lem3_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->(hoasinduction_lem1->(hoasinduction_lem2->hoasinduction_lem3)))))))))))))))))))))))))))))))))))):Prop.
% Definition hoasinduction_lem2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem2)))))))))))))))))))))))))))))))))):Prop.
% Definition hoasinduction_lem3b:=(forall (B:term), ((ex (subst->(term->term))) (fun (F:(subst->(term->term)))=> (((eq term) ((sub B) ((push one) sh))) ((F sh) one))))):Prop.
% Definition substmonoid:=((and ((and (forall (M:subst) (N:subst) (K:subst), (((eq subst) ((comp ((comp M) N)) K)) ((comp M) ((comp N) K))))) (forall (M:subst), (((eq subst) ((comp id) M)) M)))) (forall (M:subst), (((eq subst) ((comp M) id)) M))):Prop.
% Definition lamnotvar:=(forall (A:term), ((var (lam A))->False)):Prop.
% Definition hoasinduction_lem3a:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), ((forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))->(((P id) (lam A)) id))))):Prop.
% Definition hoasinduction_lem1_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->(substmonoid->(termmset->(hoasapinj1->(hoasapinj2->(hoaslaminj->(hoaslamnotap->(hoaslamnotvar->(hoasapnotvar->hoasinduction_lem1)))))))))))))))))))))))))))))))))):Prop.
% Definition hoasinduction_no_psi_cond:=(forall (P:(subst->(term->(subst->Prop)))), ((forall (A:term) (B:term), ((((P id) A) id)->((((P id) B) id)->(((P id) ((((hoasap id) A) id) B)) id))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(forall (A:term), (((P id) A) id))))):Prop.
% Definition induction2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->induction2))))))))))))))))))))))))):Prop.
% Definition pushprop_lem2v2_lthm:=(axclos->(axshiftcons->pushprop_lem2v2)):Prop.
% Definition hoasvar:=(fun (M:subst) (A:term) (N:subst)=> (var ((sub A) N))):(subst->(term->(subst->Prop))).
% Definition hoaslamnotap:=(forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->(forall (A:term) (B:term), (not (((eq term) ((hoaslam id) (fun (M:subst) (C:term)=> ((F M) C)))) ((((hoasap id) A) id) B)))))):Prop.
% Definition substmonoid_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->(pushprop->(induction2lem->(induction2->substmonoid)))))))))))))))))))))))))):Prop.
% Definition ulamvarsh:=(forall (A:term), ((var A)->(var ((sub A) sh)))):Prop.
% Definition induction2:=(forall (P:(term->Prop)), ((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))->((forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))->(forall (A:term), (P A)))))):Prop.
% Definition pushprop_lem3v2:=(forall (P:(term->Prop)) (Q:(term->Prop)) (A:term) (M:subst), (((((pushprop_p_and_p_prime A) M) P) Q)->((forall (B:term), ((var B)->(Q B)))->(forall (B:term), ((var B)->(P ((sub B) ((push A) M)))))))):Prop.
% Definition pushprop_lem2v2_gthm:=(axapp->(axvarcons->(axvarid->(axabs->(axclos->(axidl->(axshiftcons->(axassoc->(axmap->(axidr->(axvarshift->(axscons->(ulamvar1->(ulamvarsh->(ulamvarind->(apinj1->(apinj2->(laminj->(shinj->(lamnotap->(apnotvar->(lamnotvar->(induction->pushprop_lem2v2))))))))))))))))))))))):Prop.
% Definition pushprop_lem1_lthm:=(axvarcons->(axclos->(axshiftcons->(ulamvarind->pushprop_lem1)))):Prop.
% Definition hoasinduction_lem3v2:=(forall (P:(subst->(term->(subst->Prop)))) (Q:(term->Prop)), ((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P M) A) ((comp K) N))->(((P ((comp M) K)) ((sub A) K)) N)))->((forall (M:subst) (A:term) (N:subst) (K:subst), ((((P ((comp M) K)) ((sub A) K)) N)->(((P M) A) ((comp K) N))))->((forall (F:(subst->(term->term))), ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))->(((hoasinduction_p_and_p_prime P) Q)->(forall (A:term), ((forall (B:term), ((Q B)->(Q ((sub A) ((push B) id)))))->(Q (lam A))))))))):Prop.
% Definition axshiftcons:=(forall (A:term) (M:subst), (((eq subst) ((comp sh) ((push A) M))) M)):Prop.
% Definition termmset:=((and (forall (A:term) (M:subst) (N:subst), (((eq term) ((sub ((sub A) M)) N)) ((sub A) ((comp M) N))))) (forall (A:term), (((eq term) ((sub A) id)) A))):Prop.
% Definition pushprop_lem0_lthm:=pushprop_lem0:Prop.
% Definition hoasapnotvar_lthm:=(axvarid->(apnotvar->hoasapnotvar)):Prop.
% Definition hoasinduction_lem3v2_lthm:=(axvarid->(axvarshift->(axclos->(axmap->hoasinduction_lem3v2)))):Prop.
% Definition pushprop_p_and_p_prime:=(fun (A:term) (M:subst) (P:(term->Prop)) (Q:(term->Prop))=> (forall (X:term), ((iff (Q X)) (P ((sub X) ((push A) M)))))):(term->(subst->((term->Prop)->((term->Prop)->Prop)))).
% Definition axvarid:=(forall (A:term), (((eq term) ((sub A) id)) A)):Prop.
% Definition hoasinduction_lthm_3:=(hoasinduction_lem0->(induction2->(axvarid->(hoasinduction_lem3v2a->hoasinduction)))):Prop.
% Trying to prove induction2_lthm
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x2:induction2
% Found (fun (x2:induction2)=> x2) as proof of induction2
% Found (fun (x1:induction2) (x2:induction2)=> x2) as proof of (induction2->induction2)
% Found (fun (B:term) (x1:induction2) (x2:induction2)=> x2) as proof of (induction2->(induction2->induction2))
% Found (fun (A:term) (B:term) (x1:induction2) (x2:induction2)=> x2) as proof of (term->(induction2->(induction2->induction2)))
% Found (fun (A:term) (B:term) (x1:induction2) (x2:induction2)=> x2) as proof of (term->(term->(induction2->(induction2->induction2))))
% Found x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))
% Found (fun (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of ((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found (fun (A:term) (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (term->(((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))))
% Found (fun (A:term) (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (term->(term->(((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))))
% Found x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))
% Found (fun (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of ((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))
% Found (fun (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (A:term) (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (term->(((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found (fun (A:term) (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (term->(term->(((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))))
% Found x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))
% Found (fun (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of ((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))
% Found (fun (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))
% Found (fun (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (A:term) (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (term->(((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (A:term) (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (term->(term->(((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x5:(forall (A:term), (P A))
% Found (fun (x5:(forall (A:term), (P A)))=> x5) as proof of (forall (A0:term), (P A0))
% Found (fun (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of ((forall (A:term), (P A))->(forall (A0:term), (P A0)))
% Found (fun (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of ((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0))))
% Found (fun (A:term) (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of (term->((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0)))))
% Found (fun (A:term) (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of (term->(term->((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0))))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x5:(P A)
% Found (fun (x5:(P A))=> x5) as proof of (P A)
% Found (fun (x4:(P A)) (x5:(P A))=> x5) as proof of ((P A)->(P A))
% Found (fun (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of ((P A)->((P A)->(P A)))
% Found (fun (A0:term) (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of (term->((P A)->((P A)->(P A))))
% Found (fun (A0:term) (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of (term->(term->((P A)->((P A)->(P A)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x5:(var A)
% Found (fun (x5:(var A))=> x5) as proof of (var A)
% Found (fun (x4:(var A)) (x5:(var A))=> x5) as proof of ((var A)->(var A))
% Found (fun (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of ((var A)->((var A)->(var A)))
% Found (fun (A0:term) (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of (term->((var A)->((var A)->(var A))))
% Found (fun (A0:term) (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of (term->(term->((var A)->((var A)->(var A)))))
% Found x6:(P ((sub A) id))
% Found (fun (x6:(P ((sub A) id)))=> x6) as proof of (P ((sub A) id))
% Found (fun (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of ((P ((sub A) id))->(P ((sub A) id)))
% Found (fun (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of ((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of (term->((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of (term->(term->((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id))))))
% Found x6:(var ((sub A) id))
% Found (fun (x6:(var ((sub A) id)))=> x6) as proof of (var ((sub A) id))
% Found (fun (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->(var ((sub A) id)))
% Found (fun (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->(term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))))
% Found x6:(var ((sub A) id))
% Found (fun (x6:(var ((sub A) id)))=> x6) as proof of (var ((sub A) id))
% Found (fun (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->(var ((sub A) id)))
% Found (fun (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->(term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x6:(P ((sub ((sub A) id)) id))
% Found (fun (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (P ((sub ((sub A) id)) id))
% Found (fun (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of ((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of ((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (term->((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x6:(P ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id))))))
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found x2:induction2
% Found (fun (x2:induction2)=> x2) as proof of induction2
% Found (fun (x1:induction2) (x2:induction2)=> x2) as proof of (induction2->induction2)
% Found (fun (B:term) (x1:induction2) (x2:induction2)=> x2) as proof of (induction2->(induction2->induction2))
% Found (fun (A:term) (B:term) (x1:induction2) (x2:induction2)=> x2) as proof of (term->(induction2->(induction2->induction2)))
% Found (fun (A:term) (B:term) (x1:induction2) (x2:induction2)=> x2) as proof of (term->(term->(induction2->(induction2->induction2))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))
% Found (fun (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of ((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found (fun (A:term) (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (term->(((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))))
% Found (fun (A:term) (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (term->(term->(((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))))
% Found x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))
% Found (fun (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of ((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))
% Found (fun (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (A:term) (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (term->(((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found (fun (A:term) (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (term->(term->(((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))))
% Found x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))
% Found (fun (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of ((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))
% Found (fun (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))
% Found (fun (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (A:term) (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (term->(((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (A:term) (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (term->(term->(((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found x5:(forall (A:term), (P A))
% Found (fun (x5:(forall (A:term), (P A)))=> x5) as proof of (forall (A0:term), (P A0))
% Found (fun (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of ((forall (A:term), (P A))->(forall (A0:term), (P A0)))
% Found (fun (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of ((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0))))
% Found (fun (A:term) (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of (term->((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0)))))
% Found (fun (A:term) (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of (term->(term->((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0))))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x5:(P A)
% Found (fun (x5:(P A))=> x5) as proof of (P A)
% Found (fun (x4:(P A)) (x5:(P A))=> x5) as proof of ((P A)->(P A))
% Found (fun (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of ((P A)->((P A)->(P A)))
% Found (fun (A0:term) (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of (term->((P A)->((P A)->(P A))))
% Found (fun (A0:term) (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of (term->(term->((P A)->((P A)->(P A)))))
% Found x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))
% Found (fun (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of (forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))
% Found (fun (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of ((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0)))))))
% Found (fun (B:term) (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of ((forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))->((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))))
% Found (fun (A:term) (B:term) (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of (term->((forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))->((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0)))))))))
% Found (fun (A:term) (B:term) (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of (term->(term->((forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))->((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))))))
% Found x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))
% Found (fun (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of ((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found (fun (A:term) (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (term->(((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))))
% Found (fun (A:term) (B:term) (x1:((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))) (x2:((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))))=> x2) as proof of (term->(term->(((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))->(((forall (A:term), ((var A)->(P A)))->((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))->((forall (A0:term), ((var A0)->(P A0)))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))))
% Found x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))
% Found (fun (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of ((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))
% Found (fun (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (A:term) (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (term->(((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found (fun (A:term) (B:term) (x2:((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))) (x3:((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))))=> x3) as proof of (term->(term->(((forall (A0:term) (B:term), ((P A0)->((P B)->(P ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))->(((forall (A:term) (B0:term), ((P A)->((P B0)->(P ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))->((forall (A0:term) (B0:term), ((P A0)->((P B0)->(P ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))))
% Found x5:(var A)
% Found (fun (x5:(var A))=> x5) as proof of (var A)
% Found (fun (x4:(var A)) (x5:(var A))=> x5) as proof of ((var A)->(var A))
% Found (fun (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of ((var A)->((var A)->(var A)))
% Found (fun (A0:term) (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of (term->((var A)->((var A)->(var A))))
% Found (fun (A0:term) (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of (term->(term->((var A)->((var A)->(var A)))))
% Found x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))
% Found (fun (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of ((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))
% Found (fun (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))
% Found (fun (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))
% Found (fun (A:term) (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (term->(((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0))))))
% Found (fun (A:term) (B:term) (x3:((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))) (x4:((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A))))=> x4) as proof of (term->(term->(((forall (A0:term), ((forall (B:term), ((P B)->(P ((sub A0) ((push B) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))->(((forall (A:term), ((forall (B0:term), ((P B0)->(P ((sub A) ((push B0) id)))))->(P (lam A))))->(forall (A:term), (P A)))->((forall (A0:term), ((forall (B0:term), ((P B0)->(P ((sub A0) ((push B0) id)))))->(P (lam A0))))->(forall (A0:term), (P A0)))))))
% Found x5:(forall (A:term), (P A))
% Found (fun (x5:(forall (A:term), (P A)))=> x5) as proof of (forall (A0:term), (P A0))
% Found (fun (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of ((forall (A:term), (P A))->(forall (A0:term), (P A0)))
% Found (fun (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of ((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0))))
% Found (fun (A:term) (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of (term->((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0)))))
% Found (fun (A:term) (B:term) (x4:(forall (A0:term), (P A0))) (x5:(forall (A:term), (P A)))=> x5) as proof of (term->(term->((forall (A0:term), (P A0))->((forall (A:term), (P A))->(forall (A0:term), (P A0))))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x5:(P A)
% Found (fun (x5:(P A))=> x5) as proof of (P A)
% Found (fun (x4:(P A)) (x5:(P A))=> x5) as proof of ((P A)->(P A))
% Found (fun (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of ((P A)->((P A)->(P A)))
% Found (fun (A0:term) (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of (term->((P A)->((P A)->(P A))))
% Found (fun (A0:term) (B:term) (x4:(P A)) (x5:(P A))=> x5) as proof of (term->(term->((P A)->((P A)->(P A)))))
% Found x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found x6:(P ((sub A) id))
% Found (fun (x6:(P ((sub A) id)))=> x6) as proof of (P ((sub A) id))
% Found (fun (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of ((P ((sub A) id))->(P ((sub A) id)))
% Found (fun (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of ((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of (term->((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of (term->(term->((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x5:(var A)
% Found (fun (x5:(var A))=> x5) as proof of (var A)
% Found (fun (x4:(var A)) (x5:(var A))=> x5) as proof of ((var A)->(var A))
% Found (fun (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of ((var A)->((var A)->(var A)))
% Found (fun (A0:term) (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of (term->((var A)->((var A)->(var A))))
% Found (fun (A0:term) (B:term) (x4:(var A)) (x5:(var A))=> x5) as proof of (term->(term->((var A)->((var A)->(var A)))))
% Found x6:(var ((sub A) id))
% Found (fun (x6:(var ((sub A) id)))=> x6) as proof of (var ((sub A) id))
% Found (fun (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->(var ((sub A) id)))
% Found (fun (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->(term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))))
% Found x6:(var ((sub A) id))
% Found (fun (x6:(var ((sub A) id)))=> x6) as proof of (var ((sub A) id))
% Found (fun (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->(var ((sub A) id)))
% Found (fun (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->(term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x6:(P ((sub A) id))
% Found (fun (x6:(P ((sub A) id)))=> x6) as proof of (P ((sub A) id))
% Found (fun (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of ((P ((sub A) id))->(P ((sub A) id)))
% Found (fun (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of ((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of (term->((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub A) id))) (x6:(P ((sub A) id)))=> x6) as proof of (term->(term->((P ((sub A) id))->((P ((sub A) id))->(P ((sub A) id))))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x6:(var ((sub A) id))
% Found (fun (x6:(var ((sub A) id)))=> x6) as proof of (var ((sub A) id))
% Found (fun (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->(var ((sub A) id)))
% Found (fun (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->(term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))))
% Found x6:(var ((sub A) id))
% Found (fun (x6:(var ((sub A) id)))=> x6) as proof of (var ((sub A) id))
% Found (fun (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->(var ((sub A) id)))
% Found (fun (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of ((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub A) id))) (x6:(var ((sub A) id)))=> x6) as proof of (term->(term->((var ((sub A) id))->((var ((sub A) id))->(var ((sub A) id))))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found x6:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x6:(P ((sub ((sub A) id)) id))
% Found (fun (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (P ((sub ((sub A) id)) id))
% Found (fun (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of ((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of ((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (term->((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id))))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x6:(P ((sub ((sub A) id)) id))
% Found (fun (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (P ((sub ((sub A) id)) id))
% Found (fun (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of ((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of ((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (term->((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub A) id)) id))) (x6:(P ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub A) id)) id))->((P ((sub ((sub A) id)) id))->(P ((sub ((sub A) id)) id))))))
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found x6:(var ((sub ((sub A) id)) id))
% Found (fun (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (var ((sub ((sub A) id)) id))
% Found (fun (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of ((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub A) id)) id))) (x6:(var ((sub ((sub A) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub A) id)) id))->((var ((sub ((sub A) id)) id))->(var ((sub ((sub A) id)) id))))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x6:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x6:(P ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id))))))
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))) (x6:(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->((var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))->(var ((sub ((sub ((sub ((sub ((sub ((sub A) id)) id)) id)) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(P ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub A) id)) id)) id))) (x6:(P ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub A) id)) id)) id))->((P ((sub ((sub ((sub A) id)) id)) id))->(P ((sub ((sub ((sub A) id)) id)) id))))))
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found id:subst
% Found id as proof of subst
% Found A:term
% Found A as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x6:(var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (var ((sub ((sub ((sub A) id)) id)) id))
% Found (fun (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))
% Found (fun (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of ((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(var ((sub ((sub ((sub A) id)) id)) id))) (x6:(var ((sub ((sub ((sub A) id)) id)) id)))=> x6) as proof of (term->(term->((var ((sub ((sub ((sub A) id)) id)) id))->((var ((sub ((sub ((sub A) id)) id)) id))->(var ((sub ((sub ((sub A) id)) id)) id))))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found one:term
% Found one as proof of term
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found id:subst
% Found id as proof of subst
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found id:subst
% Found id as proof of subst
% Found x3:(forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x3 as proof of (forall (A:term), ((forall (B:term), ((P B)->(P ((sub A) ((push B) id)))))->(P (lam A))))
% Found x2:(forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x2 as proof of (forall (A:term) (B:term), ((P A)->((P B)->(P ((ap A) B)))))
% Found x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))
% Found (fun (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))
% Found (fun (B:term) (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of ((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))))
% Found (fun (A0:term) (B:term) (x5:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))) (x6:(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id)))=> x6) as proof of (term->(term->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->((P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))->(P ((sub ((sub ((sub ((sub A) id)) id)) id)) id))))))
% Found sh:subst
% Found sh as proof of subst
% Found one:term
% Found one as proof of term
% Found x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))
% Found (fun (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of (forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))
% Found (fun (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of ((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0)))))))
% Found (fun (B:term) (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of ((forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))->((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))))
% Found (fun (A:term) (B:term) (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))) (x2:(forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A)))))))=> x2) as proof of (term->((forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0))))))->((forall (P0:(term->Prop)), ((forall (A:term), ((var A)->(P0 A)))->((forall (A:term) (B0:term), ((P0 A)->((P0 B0)->(P0 ((ap A) B0)))))->((forall (A:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A) ((push B0) id)))))->(P0 (lam A))))->(forall (A:term), (P0 A))))))->(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B0:term), ((P0 A0)->((P0 B0)->(P0 ((ap A0) B0)))))->((forall (A0:term), ((forall (B0:term), ((P0 B0)->(P0 ((sub A0) ((push B0) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0)))))))))
% Found (fun (A:term) (B:term) (x1:(forall (P0:(term->Prop)), ((forall (A0:term), ((var A0)->(P0 A0)))->((forall (A0:term) (B:term), ((P0 A0)->((P0 B)->(P0 ((ap A0) B)))))->((forall (A0:term), ((forall (B:term), ((P0 B)->(P0 ((sub A0) ((push B) id)))))->(P0 (lam A0))))->(forall (A0:term), (P0 A0)))))))
% EOF
%------------------------------------------------------------------------------