%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : ALG269^4 : TPTP v6.1.0. Bugfixed v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n112.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 13:18:21 EDT 2014

% Result   : Timeout 300.07s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : ALG269^4 : TPTP v6.1.0. Bugfixed v5.2.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n112.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 09:03:01 CDT 2014
% % CPUTime  : 300.07 
% 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 0xec2cb0>, <kernel.Type object at 0xec2638>) of role type named term_type
% Using role type
% Declaring term:Type
% FOF formula (<kernel.Constant object at 0xf1fd40>, <kernel.Type object at 0xec23b0>) of role type named subst_type
% Using role type
% Declaring subst:Type
% FOF formula (<kernel.Constant object at 0xec2c20>, <kernel.Constant object at 0xec2cb0>) of role type named one_type
% Using role type
% Declaring one:term
% FOF formula (<kernel.Constant object at 0xec2368>, <kernel.DependentProduct object at 0xec25a8>) of role type named ap_type
% Using role type
% Declaring ap:(term->(term->term))
% FOF formula (<kernel.Constant object at 0x9b1488>, <kernel.DependentProduct object at 0xec2c68>) of role type named lam_type
% Using role type
% Declaring lam:(term->term)
% FOF formula (<kernel.Constant object at 0xec2cb0>, <kernel.DependentProduct object at 0xec24d0>) of role type named sub_type
% Using role type
% Declaring sub:(term->(subst->term))
% FOF formula (<kernel.Constant object at 0xec2368>, <kernel.Constant object at 0xec24d0>) of role type named id_type
% Using role type
% Declaring id:subst
% FOF formula (<kernel.Constant object at 0xec25a8>, <kernel.Constant object at 0xec24d0>) of role type named sh_type
% Using role type
% Declaring sh:subst
% FOF formula (<kernel.Constant object at 0xec2cb0>, <kernel.DependentProduct object at 0xec2368>) of role type named push_type
% Using role type
% Declaring push:(term->(subst->subst))
% FOF formula (<kernel.Constant object at 0xec24d0>, <kernel.DependentProduct object at 0xec3c68>) of role type named comp_type
% Using role type
% Declaring comp:(subst->(subst->subst))
% FOF formula (<kernel.Constant object at 0xec25a8>, <kernel.DependentProduct object at 0xec3b48>) of role type named var_type
% Using role type
% Declaring var:(term->Prop)
% FOF formula (<kernel.Constant object at 0xec2cb0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem1v2_type
% Using role type
% Declaring pushprop_lem1v2:Prop
% FOF formula (<kernel.Constant object at 0xec2368>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem1_gthm_type
% Using role type
% Declaring pushprop_lem1_gthm:Prop
% FOF formula (<kernel.Constant object at 0xec2cb0>, <kernel.Sort object at 0x9b1128>) of role type named axmap_type
% Using role type
% Declaring axmap:Prop
% FOF formula (<kernel.Constant object at 0xec2cb0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem0_gthm_type
% Using role type
% Declaring pushprop_lem0_gthm:Prop
% FOF formula (<kernel.Constant object at 0xec3c68>, <kernel.Sort object at 0x9b1128>) of role type named shinj_type
% Using role type
% Declaring shinj:Prop
% FOF formula (<kernel.Constant object at 0xec30e0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem1v2_type
% Using role type
% Declaring hoasinduction_lem1v2:Prop
% FOF formula (<kernel.Constant object at 0xec3b48>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem1v2_gthm_type
% Using role type
% Declaring hoasinduction_lem1v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xec30e0>, <kernel.DependentProduct object at 0xec3248>) of role type named hoasap_type
% Using role type
% Declaring hoasap:(subst->(term->(subst->(term->term))))
% FOF formula (<kernel.Constant object at 0xec3128>, <kernel.Sort object at 0x9b1128>) of role type named induction2lem_type
% Using role type
% Declaring induction2lem:Prop
% FOF formula (<kernel.Constant object at 0xec32d8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2_f_type
% Using role type
% Declaring hoasinduction_lem3v2_f:Prop
% FOF formula (<kernel.Constant object at 0xec3518>, <kernel.Sort object at 0x9b1128>) of role type named axvarshift_type
% Using role type
% Declaring axvarshift:Prop
% FOF formula (<kernel.Constant object at 0xec3d88>, <kernel.Sort object at 0x9b1128>) of role type named hoasapinj2_type
% Using role type
% Declaring hoasapinj2:Prop
% FOF formula (<kernel.Constant object at 0xec3368>, <kernel.Sort object at 0x9b1128>) of role type named hoasapnotvar_gthm_type
% Using role type
% Declaring hoasapnotvar_gthm:Prop
% FOF formula (<kernel.Constant object at 0xec3c68>, <kernel.Sort object at 0x9b1128>) of role type named hoasapinj1_type
% Using role type
% Declaring hoasapinj1:Prop
% FOF formula (<kernel.Constant object at 0xec3e18>, <kernel.Sort object at 0x9b1128>) of role type named ulamvar1_type
% Using role type
% Declaring ulamvar1:Prop
% FOF formula (<kernel.Constant object at 0xec3758>, <kernel.Sort object at 0x9b1128>) of role type named induction2lem_lthm_type
% Using role type
% Declaring induction2lem_lthm:Prop
% FOF formula (<kernel.Constant object at 0xec3200>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2_gthm_type
% Using role type
% Declaring hoasinduction_lem3v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xec3758>, <kernel.Sort object at 0x9b1128>) of role type named apnotvar_type
% Using role type
% Declaring apnotvar:Prop
% FOF formula (<kernel.Constant object at 0xec34d0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lthm_orig_type
% Using role type
% Declaring pushprop_lthm_orig:Prop
% FOF formula (<kernel.Constant object at 0xec3f38>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2_f_lthm_type
% Using role type
% Declaring hoasinduction_lem3v2_f_lthm:Prop
% FOF formula (<kernel.Constant object at 0xec34d0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lthm_type
% Using role type
% Declaring hoasinduction_lthm:Prop
% FOF formula (<kernel.Constant object at 0xec34d0>, <kernel.Sort object at 0x9b1128>) 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 0xec31b8>, <kernel.Sort object at 0x9b1128>) of role type named hoaslaminj_type
% Using role type
% Declaring hoaslaminj:Prop
% FOF formula (<kernel.Constant object at 0xec3e18>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3aaa_type
% Using role type
% Declaring hoasinduction_lem3aaa:Prop
% FOF formula (<kernel.Constant object at 0xec31b8>, <kernel.Sort object at 0x9b1128>) of role type named induction2lem_gthm_type
% Using role type
% Declaring induction2lem_gthm:Prop
% FOF formula (<kernel.Constant object at 0xec3e18>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3aa_lthm_type
% Using role type
% Declaring hoasinduction_lem3aa_lthm:Prop
% FOF formula (<kernel.Constant object at 0xec34d0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3_type
% Using role type
% Declaring hoasinduction_lem3:Prop
% FOF formula (<kernel.Constant object at 0xec34d0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem2_type
% Using role type
% Declaring hoasinduction_lem2:Prop
% FOF formula (<kernel.Constant object at 0xea41b8>, <kernel.Sort object at 0x9b1128>) of role type named termmset_lthm_type
% Using role type
% Declaring termmset_lthm:Prop
% FOF formula (<kernel.Constant object at 0xea4170>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem1_type
% Using role type
% Declaring hoasinduction_lem1:Prop
% FOF formula (<kernel.Constant object at 0xea4908>, <kernel.Sort object at 0x9b1128>) of role type named hoaslamnotap_lthm_type
% Using role type
% Declaring hoaslamnotap_lthm:Prop
% FOF formula (<kernel.Constant object at 0xea4170>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem1v2_lthm_type
% Using role type
% Declaring pushprop_lem1v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xea41b8>, <kernel.Sort object at 0x9b1128>) of role type named hoasapnotvar_type
% Using role type
% Declaring hoasapnotvar:Prop
% FOF formula (<kernel.Constant object at 0xea4170>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem0_type
% Using role type
% Declaring hoasinduction_lem0:Prop
% FOF formula (<kernel.Constant object at 0xea4170>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_type
% Using role type
% Declaring hoasinduction:Prop
% FOF formula (<kernel.Constant object at 0xacc908>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_gthm_type
% Using role type
% Declaring hoasinduction_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc950>, <kernel.Sort object at 0x9b1128>) of role type named axapp_type
% Using role type
% Declaring axapp:Prop
% FOF formula (<kernel.Constant object at 0xacc758>, <kernel.Sort object at 0x9b1128>) of role type named hoaslamnotvar_lthm_type
% Using role type
% Declaring hoaslamnotvar_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacc7a0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem3v2_lthm_type
% Using role type
% Declaring pushprop_lem3v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacc878>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3b_lthm_type
% Using role type
% Declaring hoasinduction_lem3b_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacc7a0>, <kernel.Sort object at 0x9b1128>) of role type named ulamvarind_type
% Using role type
% Declaring ulamvarind:Prop
% FOF formula (<kernel.Constant object at 0xacc710>, <kernel.Sort object at 0x9b1128>) of role type named induction_type
% Using role type
% Declaring induction:Prop
% FOF formula (<kernel.Constant object at 0xacc6c8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3a_lthm_type
% Using role type
% Declaring hoasinduction_lem3a_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacc710>, <kernel.Sort object at 0x9b1128>) of role type named termmset_gthm_type
% Using role type
% Declaring termmset_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc5a8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3aa_type
% Using role type
% Declaring hoasinduction_lem3aa:Prop
% FOF formula (<kernel.Constant object at 0xacc7a0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem1v2_gthm_type
% Using role type
% Declaring pushprop_lem1v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc560>, <kernel.Sort object at 0x9b1128>) of role type named hoaslamnotap_gthm_type
% Using role type
% Declaring hoaslamnotap_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc488>, <kernel.Sort object at 0x9b1128>) of role type named hoaslamnotvar_gthm_type
% Using role type
% Declaring hoaslamnotvar_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc758>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3b_gthm_type
% Using role type
% Declaring hoasinduction_lem3b_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc488>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem2v2_type
% Using role type
% Declaring pushprop_lem2v2:Prop
% FOF formula (<kernel.Constant object at 0xacc4d0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3a_gthm_type
% Using role type
% Declaring hoasinduction_lem3a_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacc488>, <kernel.Sort object at 0x9b1128>) of role type named axclos_type
% Using role type
% Declaring axclos:Prop
% FOF formula (<kernel.Constant object at 0xacc560>, <kernel.Sort object at 0x9b1128>) of role type named axassoc_type
% Using role type
% Declaring axassoc:Prop
% FOF formula (<kernel.Constant object at 0xacc9e0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem2v2_type
% Using role type
% Declaring hoasinduction_lem2v2:Prop
% FOF formula (<kernel.Constant object at 0xaccab8>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lthm_type
% Using role type
% Declaring pushprop_lthm:Prop
% FOF formula (<kernel.Constant object at 0xaccb00>, <kernel.Sort object at 0x9b1128>) of role type named apinj2_type
% Using role type
% Declaring apinj2:Prop
% FOF formula (<kernel.Constant object at 0xaccb48>, <kernel.Sort object at 0x9b1128>) of role type named apinj1_type
% Using role type
% Declaring apinj1:Prop
% FOF formula (<kernel.Constant object at 0xaccb90>, <kernel.Sort object at 0x9b1128>) of role type named hoasapinj2_lthm_type
% Using role type
% Declaring hoasapinj2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xaccbd8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2a_type
% Using role type
% Declaring hoasinduction_lem3v2a:Prop
% FOF formula (<kernel.Constant object at 0xaccc20>, <kernel.Sort object at 0x9b1128>) of role type named hoasapinj1_lthm_type
% Using role type
% Declaring hoasapinj1_lthm:Prop
% FOF formula (<kernel.Constant object at 0xaccc68>, <kernel.Sort object at 0x9b1128>) of role type named hoaslaminj_lthm_type
% Using role type
% Declaring hoaslaminj_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacccb0>, <kernel.Sort object at 0x9b1128>) of role type named axvarcons_type
% Using role type
% Declaring axvarcons:Prop
% FOF formula (<kernel.Constant object at 0xacccf8>, <kernel.DependentProduct object at 0xaccd88>) of role type named hoaslam_type
% Using role type
% Declaring hoaslam:(subst->((subst->(term->term))->term))
% FOF formula (<kernel.Constant object at 0xaccdd0>, <kernel.Sort object at 0x9b1128>) of role type named axscons_type
% Using role type
% Declaring axscons:Prop
% FOF formula (<kernel.Constant object at 0xacc3f8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem2v2_gthm_type
% Using role type
% Declaring hoasinduction_lem2v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xaccdd0>, <kernel.Sort object at 0x9b1128>) of role type named axidr_type
% Using role type
% Declaring axidr:Prop
% FOF formula (<kernel.Constant object at 0xacccb0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem1_type
% Using role type
% Declaring pushprop_lem1:Prop
% FOF formula (<kernel.Constant object at 0xacccf8>, <kernel.Sort object at 0x9b1128>) of role type named laminj_type
% Using role type
% Declaring laminj:Prop
% FOF formula (<kernel.Constant object at 0xaccd40>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3_lthm_type
% Using role type
% Declaring hoasinduction_lem3_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacccf8>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem0_type
% Using role type
% Declaring pushprop_lem0:Prop
% FOF formula (<kernel.Constant object at 0xaccea8>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_gthm_type
% Using role type
% Declaring pushprop_gthm:Prop
% FOF formula (<kernel.Constant object at 0xacccb0>, <kernel.Sort object at 0x9b1128>) of role type named axabs_type
% Using role type
% Declaring axabs:Prop
% FOF formula (<kernel.Constant object at 0xaccfc8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2a_lthm_type
% Using role type
% Declaring hoasinduction_lem3v2a_lthm:Prop
% FOF formula (<kernel.Constant object at 0xacce18>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem2_lthm_type
% Using role type
% Declaring hoasinduction_lem2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xaccea8>, <kernel.Sort object at 0x9b1128>) of role type named hoasapinj2_gthm_type
% Using role type
% Declaring hoasapinj2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xaccea8>, <kernel.DependentProduct object at 0xeb3248>) 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 0xeb3200>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem1_lthm_type
% Using role type
% Declaring hoasinduction_lem1_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3050>, <kernel.Sort object at 0x9b1128>) of role type named lamnotap_type
% Using role type
% Declaring lamnotap:Prop
% FOF formula (<kernel.Constant object at 0xeb3128>, <kernel.Sort object at 0x9b1128>) of role type named hoasapinj1_gthm_type
% Using role type
% Declaring hoasapinj1_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3170>, <kernel.Sort object at 0x9b1128>) of role type named hoaslamnotvar_type
% Using role type
% Declaring hoaslamnotvar:Prop
% FOF formula (<kernel.Constant object at 0xeb30e0>, <kernel.Sort object at 0x9b1128>) of role type named axidl_type
% Using role type
% Declaring axidl:Prop
% FOF formula (<kernel.Constant object at 0xeb3290>, <kernel.Sort object at 0x9b1128>) of role type named hoaslaminj_gthm_type
% Using role type
% Declaring hoaslaminj_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb31b8>, <kernel.Sort object at 0x9b1128>) of role type named induction2_lthm_type
% Using role type
% Declaring induction2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb32d8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem0_lthm_type
% Using role type
% Declaring hoasinduction_lem0_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb31b8>, <kernel.Sort object at 0x9b1128>) of role type named substmonoid_lthm_type
% Using role type
% Declaring substmonoid_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb33b0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_type
% Using role type
% Declaring pushprop:Prop
% FOF formula (<kernel.Constant object at 0xeb3320>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3_gthm_type
% Using role type
% Declaring hoasinduction_lem3_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3290>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem2_gthm_type
% Using role type
% Declaring hoasinduction_lem2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb34d0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3b_type
% Using role type
% Declaring hoasinduction_lem3b:Prop
% FOF formula (<kernel.Constant object at 0xeb3440>, <kernel.Sort object at 0x9b1128>) of role type named substmonoid_type
% Using role type
% Declaring substmonoid:Prop
% FOF formula (<kernel.Constant object at 0xeb3320>, <kernel.Sort object at 0x9b1128>) of role type named lamnotvar_type
% Using role type
% Declaring lamnotvar:Prop
% FOF formula (<kernel.Constant object at 0xeb3560>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3a_type
% Using role type
% Declaring hoasinduction_lem3a:Prop
% FOF formula (<kernel.Constant object at 0xeb33b0>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem1_gthm_type
% Using role type
% Declaring hoasinduction_lem1_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb35a8>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_no_psi_cond_type
% Using role type
% Declaring hoasinduction_no_psi_cond:Prop
% FOF formula (<kernel.Constant object at 0xeb3680>, <kernel.Sort object at 0x9b1128>) of role type named induction2_gthm_type
% Using role type
% Declaring induction2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb35f0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem2v2_lthm_type
% Using role type
% Declaring pushprop_lem2v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb33b0>, <kernel.DependentProduct object at 0xeb3710>) of role type named hoasvar_type
% Using role type
% Declaring hoasvar:(subst->(term->(subst->Prop)))
% FOF formula (<kernel.Constant object at 0xeb37a0>, <kernel.Sort object at 0x9b1128>) of role type named hoaslamnotap_type
% Using role type
% Declaring hoaslamnotap:Prop
% FOF formula (<kernel.Constant object at 0xeb38c0>, <kernel.Sort object at 0x9b1128>) of role type named substmonoid_gthm_type
% Using role type
% Declaring substmonoid_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb35f0>, <kernel.Sort object at 0x9b1128>) of role type named ulamvarsh_type
% Using role type
% Declaring ulamvarsh:Prop
% FOF formula (<kernel.Constant object at 0xeb3680>, <kernel.Sort object at 0x9b1128>) of role type named induction2_type
% Using role type
% Declaring induction2:Prop
% FOF formula (<kernel.Constant object at 0xeb3878>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem3v2_type
% Using role type
% Declaring pushprop_lem3v2:Prop
% FOF formula (<kernel.Constant object at 0xeb3758>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem2v2_gthm_type
% Using role type
% Declaring pushprop_lem2v2_gthm:Prop
% FOF formula (<kernel.Constant object at 0xeb37e8>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem1_lthm_type
% Using role type
% Declaring pushprop_lem1_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3950>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2_type
% Using role type
% Declaring hoasinduction_lem3v2:Prop
% FOF formula (<kernel.Constant object at 0xeb3908>, <kernel.Sort object at 0x9b1128>) of role type named axshiftcons_type
% Using role type
% Declaring axshiftcons:Prop
% FOF formula (<kernel.Constant object at 0xeb3998>, <kernel.Sort object at 0x9b1128>) of role type named termmset_type
% Using role type
% Declaring termmset:Prop
% FOF formula (<kernel.Constant object at 0xeb39e0>, <kernel.Sort object at 0x9b1128>) of role type named pushprop_lem0_lthm_type
% Using role type
% Declaring pushprop_lem0_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3a28>, <kernel.Sort object at 0x9b1128>) of role type named hoasapnotvar_lthm_type
% Using role type
% Declaring hoasapnotvar_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3560>, <kernel.Sort object at 0x9b1128>) of role type named hoasinduction_lem3v2_lthm_type
% Using role type
% Declaring hoasinduction_lem3v2_lthm:Prop
% FOF formula (<kernel.Constant object at 0xeb3a28>, <kernel.DependentProduct object at 0xeb3ab8>) 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 0xeb3b90>, <kernel.Sort object at 0x9b1128>) of role type named axvarid_type
% Using role type
% Declaring axvarid:Prop
% FOF formula (<kernel.Constant object at 0xeb3cb0>, <kernel.Sort object at 0x9b1128>) 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 hoasinduction_no_psi_cond_lthm of role conjecture named thm
% Conjecture to prove = hoasinduction_no_psi_cond_lthm:Prop
% We need to prove ['hoasinduction_no_psi_cond_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 hoasinduction_no_psi_cond_lthm
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:subst)=> (((P x5) A) x5))):((((P id) A) id)->(((P id) A) id))
% Found (eq_ref00 (fun (x5:subst)=> (((P x5) A) x5))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found ((eq_ref0 id) (fun (x5:subst)=> (((P x5) A) x5))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) A) x5))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) A) x5)))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) A) x5)))) as proof of (term->((((P id) A) id)->(((P id) A) id)))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x3:hoasinduction_no_psi_cond
% Found (fun (x3:hoasinduction_no_psi_cond)=> x3) as proof of hoasinduction_no_psi_cond
% Found (fun (B:term) (x3:hoasinduction_no_psi_cond)=> x3) as proof of (hoasinduction_no_psi_cond->hoasinduction_no_psi_cond)
% Found (fun (B:term) (x3:hoasinduction_no_psi_cond)=> x3) as proof of (term->(hoasinduction_no_psi_cond->hoasinduction_no_psi_cond))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))):(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id))))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))):(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id))))
% Found iff_refl0:=(iff_refl (((P0 id) A0) id)):((iff (((P0 id) A0) id)) (((P0 id) A0) id))
% Found (iff_refl (((P0 id) A0) id)) as proof of ((iff (((P0 id) A0) id)) (((P id) A) id))
% Found (iff_refl (((P0 id) A0) id)) as proof of ((iff (((P0 id) A0) id)) (((P id) A) id))
% Found (iff_refl (((P0 id) A0) id)) as proof of ((iff (((P0 id) A0) id)) (((P id) A) id))
% Found (iff_sym00 (iff_refl (((P0 id) A0) id))) as proof of ((iff (((P id) A) id)) (((P0 id) A0) id))
% Found ((iff_sym0 (((P id) A) id)) (iff_refl (((P0 id) A0) id))) as proof of ((iff (((P id) A) id)) (((P0 id) A0) id))
% Found (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id))) as proof of ((iff (((P id) A) id)) (((P0 id) A0) id))
% Found (fun (x5:(var A0))=> (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id)))) as proof of ((iff (((P id) A) id)) (((P0 id) A0) id))
% Found (fun (A0:term) (x5:(var A0))=> (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id)))) as proof of ((var A0)->((iff (((P id) A) id)) (((P0 id) A0) id)))
% Found (fun (A0:term) (x5:(var A0))=> (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id)))) as proof of (forall (A0:term), ((var A0)->((iff (((P id) A) id)) (((P0 id) A0) id))))
% Found iff_refl0:=(iff_refl (((P id) A) id)):((iff (((P id) A) id)) (((P id) A) id))
% Found (iff_refl (((P id) A) id)) as proof of ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))
% Found (fun (x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id)))))=> (iff_refl (((P id) A) id))) as proof of ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))
% Found (fun (A0:term) (x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id)))))=> (iff_refl (((P id) A) id))) as proof of ((forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (A0:term) (x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id)))))=> (iff_refl (((P id) A) id))) as proof of (forall (A0:term), ((forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))):(((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) B) id)))
% Found (eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id)))
% Found ((eq_ref0 ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id)))
% Found (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id)))
% Found (fun (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id)))
% Found (fun (B:term) (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) A0) id))->(((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id))))
% Found (fun (A0:term) (B:term) (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id))))) as proof of (forall (B:term), (((iff (((P id) A) id)) (((P0 id) A0) id))->(((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id)))))
% Found (fun (A0:term) (B:term) (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id))))) as proof of (forall (A0:term) (B:term), (((iff (((P id) A) id)) (((P0 id) A0) id))->(((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((ap A0) B)) id)))))
% Found (((x00 (fun (A0:term) (x5:(var A0))=> (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id))))) (fun (A0:term) (B:term) (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))))) (fun (A0:term) (x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id)))))=> (iff_refl (((P id) A) id)))) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))
% Found ((((x0 (fun (x9:term)=> ((iff (((P id) A) id)) (((P0 id) x9) id)))) (fun (A0:term) (x5:(var A0))=> (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id))))) (fun (A0:term) (B:term) (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))))) (fun (A0:term) (x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id)))))=> (iff_refl (((P id) A) id)))) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))
% Found ((((x0 (fun (x9:term)=> ((iff (((P id) A) id)) (((P0 id) x9) id)))) (fun (A0:term) (x5:(var A0))=> (((iff_sym (((P0 id) A0) id)) (((P id) A) id)) (iff_refl (((P0 id) A0) id))))) (fun (A0:term) (B:term) (x5:((iff (((P id) A) id)) (((P0 id) A0) id)))=> (((eq_ref Prop) ((((P0 id) B) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) B) id)))))) (fun (A0:term) (x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id)))))=> (iff_refl (((P id) A) id)))) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))
% Found x5:(forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))
% Found x5 as proof of (forall (B:term), ((((P id) B) id)->(((P id) ((sub A) ((push B) id))) id)))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found X:term
% Found X as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))):(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id))))
% Found one:term
% Found one as proof of term
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))):(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(((P id) A) id))) (fun (x5:Prop)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) X) id))->((iff (((P id) A) id)) (((P0 id) X) id))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (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), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (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), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))
% Found eq_ref000:=(eq_ref00 (fun (x6:subst)=> (((P x6) A) x6))):((((P id) A) id)->(((P id) A) id))
% Found (eq_ref00 (fun (x6:subst)=> (((P x6) A) x6))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found ((eq_ref0 id) (fun (x6:subst)=> (((P x6) A) x6))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (((eq_ref subst) id) (fun (x6:subst)=> (((P x6) A) x6))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x6:subst)=> (((P x6) A) x6)))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x6:subst)=> (((P x6) A) x6)))) as proof of (term->((((P id) A) id)->(((P id) A) id)))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found iff_refl0:=(iff_refl (((P1 id) X0) id)):((iff (((P1 id) X0) id)) (((P1 id) X0) id))
% Found (iff_refl (((P1 id) X0) id)) as proof of ((iff (((P1 id) X0) id)) ((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (iff_refl (((P1 id) X0) id)) as proof of ((iff (((P1 id) X0) id)) ((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (iff_refl (((P1 id) X0) id)) as proof of ((iff (((P1 id) X0) id)) ((iff (((P id) A) id)) (((P0 id) X) id)))
% Found (iff_sym00 (iff_refl (((P1 id) X0) id))) as proof of ((iff ((fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))) X0)) (((P1 id) X0) id))
% Found ((iff_sym0 ((iff (((P id) A) id)) (((P0 id) X) id))) (iff_refl (((P1 id) X0) id))) as proof of ((iff ((fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))) X0)) (((P1 id) X0) id))
% Found (((iff_sym (((P1 id) X0) id)) ((iff (((P id) A) id)) (((P0 id) X) id))) (iff_refl (((P1 id) X0) id))) as proof of ((iff ((fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))) X0)) (((P1 id) X0) id))
% Found (fun (X0:term)=> (((iff_sym (((P1 id) X0) id)) ((iff (((P id) A) id)) (((P0 id) X) id))) (iff_refl (((P1 id) X0) id)))) as proof of ((iff ((fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))) X0)) (((P1 id) X0) id))
% Found (fun (X0:term)=> (((iff_sym (((P1 id) X0) id)) ((iff (((P id) A) id)) (((P0 id) X) id))) (iff_refl (((P1 id) X0) id)))) as proof of ((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x6:hoasinduction_no_psi_cond
% Found (fun (x6:hoasinduction_no_psi_cond)=> x6) as proof of hoasinduction_no_psi_cond
% Found (fun (B:term) (x6:hoasinduction_no_psi_cond)=> x6) as proof of (hoasinduction_no_psi_cond->hoasinduction_no_psi_cond)
% Found (fun (B:term) (x6:hoasinduction_no_psi_cond)=> x6) as proof of (term->(hoasinduction_no_psi_cond->hoasinduction_no_psi_cond))
% Found eq_ref000:=(eq_ref00 (fun (x8:subst)=> (((P0 x8) A) x8))):((((P0 id) A) id)->(((P0 id) A) id))
% Found (eq_ref00 (fun (x8:subst)=> (((P0 x8) A) x8))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found ((eq_ref0 id) (fun (x8:subst)=> (((P0 x8) A) x8))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found (((eq_ref subst) id) (fun (x8:subst)=> (((P0 x8) A) x8))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x8:subst)=> (((P0 x8) A) x8)))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x8:subst)=> (((P0 x8) A) x8)))) as proof of (term->((((P0 id) A) id)->(((P0 id) A) id)))
% Found x70:=(x7 F):((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (x7 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x7 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x7 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x7 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found (fun (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of ((forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of (term->((forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (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), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x8:subst)=> (((P x8) A) x8))):((((P id) A) id)->(((P id) A) id))
% Found (eq_ref00 (fun (x8:subst)=> (((P x8) A) x8))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found ((eq_ref0 id) (fun (x8:subst)=> (((P x8) A) x8))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (((eq_ref subst) id) (fun (x8:subst)=> (((P x8) A) x8))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x8:subst)=> (((P x8) A) x8)))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x8:subst)=> (((P x8) A) x8)))) as proof of (term->((((P id) A) id)->(((P id) A) id)))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found (fun (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of ((forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of (term->((forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found (fun (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of ((forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))=> x5) as proof of (term->((forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term) (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of (((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found (fun (B:term) (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of (term->(((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found A0:term
% Found A0 as proof of term
% Found A0:term
% Found A0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x7:subst)=> (((P x7) ((ap A) B)) x7))):((((P id) ((ap A) B)) id)->(((P id) ((ap A) B)) id))
% Found (eq_ref00 (fun (x7:subst)=> (((P x7) ((ap A) B)) x7))) as proof of ((((P id) ((ap A) B)) id)->(((P id) ((ap A) B)) id))
% Found ((eq_ref0 id) (fun (x7:subst)=> (((P x7) ((ap A) B)) x7))) as proof of ((((P id) ((ap A) B)) id)->(((P id) ((ap A) B)) id))
% Found (((eq_ref subst) id) (fun (x7:subst)=> (((P x7) ((ap A) B)) x7))) as proof of ((((P id) ((ap A) B)) id)->(((P id) ((ap A) B)) id))
% Found (fun (B0:term)=> (((eq_ref subst) id) (fun (x7:subst)=> (((P x7) ((ap A) B)) x7)))) as proof of ((((P id) ((ap A) B)) id)->(((P id) ((ap A) B)) id))
% Found (fun (B0:term)=> (((eq_ref subst) id) (fun (x7:subst)=> (((P x7) ((ap A) B)) x7)))) as proof of (term->((((P id) ((ap A) B)) id)->(((P id) ((ap A) B)) id)))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))):(((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found ((eq_ref0 ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F M) A)) N)) ((F ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (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), ((((P0 id) A) id)->(((P0 id) ((F id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))
% Found one:term
% Found one as proof of term
% Found iff_refl0:=(iff_refl (((P id) A) id)):((iff (((P id) A) id)) (((P id) A) id))
% Found (iff_refl (((P id) A) id)) as proof of ((iff (((P id) A) id)) hoasinduction_no_psi_cond)
% Found (iff_refl (((P id) A) id)) as proof of ((iff (((P id) A) id)) hoasinduction_no_psi_cond)
% Found (iff_refl (((P id) A) id)) as proof of ((iff (((P id) A) id)) hoasinduction_no_psi_cond)
% Found (iff_sym00 (iff_refl (((P id) A) id))) as proof of ((iff hoasinduction_no_psi_cond) (((P id) A) id))
% Found ((iff_sym0 hoasinduction_no_psi_cond) (iff_refl (((P id) A) id))) as proof of ((iff hoasinduction_no_psi_cond) (((P id) A) id))
% Found (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id))) as proof of ((iff hoasinduction_no_psi_cond) (((P id) A) id))
% Found (fun (x3:(var A))=> (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id)))) as proof of ((iff hoasinduction_no_psi_cond) (((P id) A) id))
% Found (fun (A:term) (x3:(var A))=> (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id)))) as proof of ((var A)->((iff hoasinduction_no_psi_cond) (((P id) A) id)))
% Found (fun (A:term) (x3:(var A))=> (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id)))) as proof of (forall (A:term), ((var A)->((iff hoasinduction_no_psi_cond) (((P id) A) id))))
% Found iff_refl0:=(iff_refl hoasinduction_no_psi_cond):((iff hoasinduction_no_psi_cond) hoasinduction_no_psi_cond)
% Found (iff_refl hoasinduction_no_psi_cond) as proof of ((iff hoasinduction_no_psi_cond) (((P id) (lam A)) id))
% Found (fun (x3:(forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id)))))=> (iff_refl hoasinduction_no_psi_cond)) as proof of ((iff hoasinduction_no_psi_cond) (((P id) (lam A)) id))
% Found (fun (A:term) (x3:(forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id)))))=> (iff_refl hoasinduction_no_psi_cond)) as proof of ((forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id))))->((iff hoasinduction_no_psi_cond) (((P id) (lam A)) id)))
% Found (fun (A:term) (x3:(forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id)))))=> (iff_refl hoasinduction_no_psi_cond)) as proof of (forall (A:term), ((forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id))))->((iff hoasinduction_no_psi_cond) (((P id) (lam A)) id))))
% Found iff_sym00:=(iff_sym0 (((P id) B) id)):(((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff (((P id) B) id)) hoasinduction_no_psi_cond))
% Found (iff_sym0 (((P id) B) id)) as proof of (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((ap A) B)) id)))
% Found ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id)) as proof of (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((ap A) B)) id)))
% Found (fun (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id))) as proof of (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((ap A) B)) id)))
% Found (fun (B:term) (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id))) as proof of (((iff hoasinduction_no_psi_cond) (((P id) A) id))->(((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((ap A) B)) id))))
% Found (fun (A:term) (B:term) (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id))) as proof of (forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) A) id))->(((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((ap A) B)) id)))))
% Found (fun (A:term) (B:term) (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id))) as proof of (forall (A:term) (B:term), (((iff hoasinduction_no_psi_cond) (((P id) A) id))->(((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((ap A) B)) id)))))
% Found (((x00 (fun (A:term) (x3:(var A))=> (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id))))) (fun (A:term) (B:term) (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id)))) (fun (A:term) (x3:(forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id)))))=> (iff_refl hoasinduction_no_psi_cond))) as proof of ((hoasinduction_p_and_p_prime P) (fun (x6:term)=> hoasinduction_no_psi_cond))
% Found ((((x0 (fun (x7:term)=> ((iff hoasinduction_no_psi_cond) (((P id) x7) id)))) (fun (A:term) (x3:(var A))=> (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id))))) (fun (A:term) (B:term) (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id)))) (fun (A:term) (x3:(forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id)))))=> (iff_refl hoasinduction_no_psi_cond))) as proof of ((hoasinduction_p_and_p_prime P) (fun (x6:term)=> hoasinduction_no_psi_cond))
% Found ((((x0 (fun (x7:term)=> ((iff hoasinduction_no_psi_cond) (((P id) x7) id)))) (fun (A:term) (x3:(var A))=> (((iff_sym (((P id) A) id)) hoasinduction_no_psi_cond) (iff_refl (((P id) A) id))))) (fun (A:term) (B:term) (x3:((iff hoasinduction_no_psi_cond) (((P id) A) id)))=> ((iff_sym hoasinduction_no_psi_cond) (((P id) B) id)))) (fun (A:term) (x3:(forall (B:term), (((iff hoasinduction_no_psi_cond) (((P id) B) id))->((iff hoasinduction_no_psi_cond) (((P id) ((sub A) ((push B) id))) id)))))=> (iff_refl hoasinduction_no_psi_cond))) as proof of ((hoasinduction_p_and_p_prime P) (fun (x6:term)=> hoasinduction_no_psi_cond))
% Found x4:(forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))
% Found x4 as proof of (forall (B:term), ((((P id) B) id)->(((P id) ((sub ((F sh) one)) ((push B) id))) id)))
% Found A:term
% Found A as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x7:subst)=> (((P0 x7) A) x7))):((((P0 id) A) id)->(((P0 id) A) id))
% Found (eq_ref00 (fun (x7:subst)=> (((P0 x7) A) x7))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found ((eq_ref0 id) (fun (x7:subst)=> (((P0 x7) A) x7))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found (((eq_ref subst) id) (fun (x7:subst)=> (((P0 x7) A) x7))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x7:subst)=> (((P0 x7) A) x7)))) as proof of ((((P0 id) A) id)->(((P0 id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x7:subst)=> (((P0 x7) A) x7)))) as proof of (term->((((P0 id) A) id)->(((P0 id) A) id)))
% Found x60:=(x6 F0):((forall (M:subst) (A:term) (N:subst), (((eq term) ((sub ((F0 M) A)) N)) ((F0 ((comp M) N)) ((sub A) N))))->((forall (A:term), ((((P0 id) A) id)->(((P0 id) ((F0 id) A)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F0 M) A)))) id)))
% Found (x6 F0) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F0 M) A0)) N)) ((F0 ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F0 id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F0 M) A0)))) id)))
% Found (x6 F0) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F0 M) A0)) N)) ((F0 ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F0 id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F0 M) A0)))) id)))
% Found (fun (F0:(subst->(term->term)))=> (x6 F0)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F0 M) A0)) N)) ((F0 ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F0 id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F0 M) A0)))) id)))
% Found (fun (F0:(subst->(term->term)))=> (x6 F0)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:subst)=> (((P x5) ((hoaslam x5) (fun (M:subst) (A:term)=> ((F M) A)))) x5))):((((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))
% Found (eq_ref00 (fun (x5:subst)=> (((P x5) ((hoaslam x5) (fun (M:subst) (A:term)=> ((F M) A)))) x5))) as proof of ((((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))
% Found ((eq_ref0 id) (fun (x5:subst)=> (((P x5) ((hoaslam x5) (fun (M:subst) (A:term)=> ((F M) A)))) x5))) as proof of ((((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))
% Found (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) ((hoaslam x5) (fun (M:subst) (A:term)=> ((F M) A)))) x5))) as proof of ((((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) ((hoaslam x5) (fun (M:subst) (A:term)=> ((F M) A)))) x5)))) as proof of ((((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) ((hoaslam x5) (fun (M:subst) (A:term)=> ((F M) A)))) x5)))) as proof of (term->((((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found x3:((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)))
% Found (fun (x3:((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))))=> x3) as proof of ((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)))
% Found (fun (B:term) (x3:((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))))=> x3) as proof of (((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 (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))))
% Found (fun (B:term) (x3:((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))))=> x3) as proof of (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 (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)))))
% Found x3:((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) B) id)))
% Found (fun (x3:((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) B) id))))=> x3) as proof of ((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) ((sub ((F sh) one)) ((push B) id))) id)))
% Found (fun (B:term) (x3:((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) B) id))))=> x3) as proof of (((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) B) id)))->((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) ((sub ((F sh) one)) ((push B) id))) id))))
% Found (fun (B:term) (x3:((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) B) id))))=> x3) as proof of (forall (B: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) B) id)))->((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) ((sub ((F sh) one)) ((push B) id))) id)))))
% Found x4:((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))
% Found (fun (x4:((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)))=> x4) as proof of ((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))
% Found (fun (B:term) (x4:((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)))=> x4) as proof of (((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)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)))
% Found (fun (B:term) (x4:((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)))=> x4) as proof of (term->(((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)->(((P id) ((F id) A)) id)))->(((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id))))
% Found x4:((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) B) id))
% Found (fun (x4:((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) B) id)))=> x4) as proof of ((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((sub ((F sh) one)) ((push B) id))) id))
% Found (fun (B:term) (x4:((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) B) id)))=> x4) as proof of (((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) B) id))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((sub ((F sh) one)) ((push B) id))) id)))
% Found (fun (B:term) (x4:((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) B) id)))=> x4) as proof of (forall (B:term), (((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) B) id))->((forall (A:term), ((((P id) A) id)->(((P id) ((F id) A)) id)))->(((P id) ((sub ((F sh) one)) ((push B) id))) id))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x5 as proof of (forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term) (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of (((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found (fun (B:term) (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of (term->(((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term) (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of (((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found (fun (B:term) (x5:((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))=> x5) as proof of (term->(((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))):(((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found ((eq_ref0 ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))):(((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (eq_ref00 (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found ((eq_ref0 ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) as proof of (((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) (lam A0)) id)->(((P id) A) id))) (fun (x6:Prop)=> ((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) as proof of (term->(((iff (((P id) A) id)) (((P0 id) (lam A0)) id))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))
% Found x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x5 as proof of (forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x5 as proof of (forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x5:(forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found x5 as proof of (forall (B:term), (((iff (((P id) A) id)) (((P0 id) B) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B) id))) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:subst)=> (((P x5) A) x5))):((((P id) A) id)->(((P id) A) id))
% Found (eq_ref00 (fun (x5:subst)=> (((P x5) A) x5))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found ((eq_ref0 id) (fun (x5:subst)=> (((P x5) A) x5))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) A) x5))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) A) x5)))) as proof of ((((P id) A) id)->(((P id) A) id))
% Found (fun (B:term)=> (((eq_ref subst) id) (fun (x5:subst)=> (((P x5) A) x5)))) as proof of (term->((((P id) A) id)->(((P id) A) id)))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x3:hoasinduction_no_psi_cond
% Found (fun (x3:hoasinduction_no_psi_cond)=> x3) as proof of hoasinduction_no_psi_cond
% Found (fun (B:term) (x3:hoasinduction_no_psi_cond)=> x3) as proof of (hoasinduction_no_psi_cond->hoasinduction_no_psi_cond)
% Found (fun (B:term) (x3:hoasinduction_no_psi_cond)=> x3) as proof of (term->(hoasinduction_no_psi_cond->hoasinduction_no_psi_cond))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found A0:term
% Found A0 as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x3:Prop)=> ((iff ((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)))) (((P0 id) X) id)))):(((iff ((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)))) (((P0 id) X) id))->((iff ((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)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x3:Prop)=> ((iff ((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)))) (((P0 id) X) id)))) as proof of (((iff ((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)))) (((P0 id) X) id))->((iff ((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)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((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))))) (fun (x3:Prop)=> ((iff ((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)))) (((P0 id) X) id)))) as proof of (((iff ((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)))) (((P0 id) X) id))->((iff ((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)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((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))))) (fun (x3:Prop)=> ((iff ((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)))) (((P0 id) X) id)))) as proof of (((iff ((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)))) (((P0 id) X) id))->((iff ((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)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((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))))) (fun (x3:Prop)=> ((iff ((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)))) (((P0 id) X) id))))) as proof of (((iff ((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)))) (((P0 id) X) id))->((iff ((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)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((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))))) (fun (x3:Prop)=> ((iff ((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)))) (((P0 id) X) id))))) as proof of (term->(((iff ((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)))) (((P0 id) X) id))->((iff ((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)))) (((P0 id) X) id))))
% Found A0:term
% Found A0 as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))))
% Found A0:term
% Found A0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))):(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (term->(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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)))))
% Found (fun (x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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))))))=> x3) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x60: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)))))
% Found (fun (B:term) (x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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))))))=> x3) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x60: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 P0) (fun (x60: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))))))
% Found (fun (B:term) (x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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))))))=> x3) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x60: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 P0) (fun (x60: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)))))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))) (((P0 id) X) id))))
% Found A0:term
% Found A0 as proof of term
% Found A0:term
% Found A0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))):(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (term->(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x3:Prop)=> ((iff ((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) X) id)))) (((P0 id) X) id)))):(((iff ((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) X) id)))) (((P0 id) X) id))->((iff ((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) X) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x3:Prop)=> ((iff ((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) X) id)))) (((P0 id) X) id)))) as proof of (((iff ((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) X) id)))) (((P0 id) X) id))->((iff ((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) X) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((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) X) id))))) (fun (x3:Prop)=> ((iff ((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) X) id)))) (((P0 id) X) id)))) as proof of (((iff ((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) X) id)))) (((P0 id) X) id))->((iff ((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) X) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((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) X) id))))) (fun (x3:Prop)=> ((iff ((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) X) id)))) (((P0 id) X) id)))) as proof of (((iff ((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) X) id)))) (((P0 id) X) id))->((iff ((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) X) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((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) X) id))))) (fun (x3:Prop)=> ((iff ((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) X) id)))) (((P0 id) X) id))))) as proof of (((iff ((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) X) id)))) (((P0 id) X) id))->((iff ((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) X) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((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) X) id))))) (fun (x3:Prop)=> ((iff ((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) X) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((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) X) id)))) (((P0 id) X) id))->((iff ((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) X) id)))) (((P0 id) X) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id))))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))))))))
% Found A0:term
% Found A0 as proof of term
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))) (((P0 id) X0) id))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found X0:term
% Found X0 as proof of term
% Found x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id)))))
% Found (fun (x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id))))))=> x3) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id)))))
% Found (fun (B:term) (x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id))))))=> x3) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id))))))
% Found (fun (B:term) (x3:((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id))))))=> x3) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x60: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) x60) id)))))))
% Found A0:term
% Found A0 as proof of term
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found iff_refl0:=(iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)):((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))
% Found (iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) as proof of ((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) A) id))
% Found (iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) as proof of ((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) A) id))
% Found (fun (x5:(var A))=> (iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))) as proof of ((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) A) id))
% Found one:term
% Found one as proof of term
% Found x60:(((P id) X) id)
% Found x60 as proof of hoasinduction_no_psi_cond
% Found x60 as proof of hoasinduction_no_psi_cond
% Found one:term
% Found one as proof of term
% Found iff_refl0:=(iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)):((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))
% Found (iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) as proof of ((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) (lam A)) id))
% Found (iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) as proof of ((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) (lam A)) id))
% Found (fun (x5:(forall (B:term), (((iff (((P id) ((hoaslam id) (fun (M:subst) (A:term)=> ((F M) A)))) id)) (((P0 id) B) id))->((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) ((sub A) ((push B) id))) id)))))=> (iff_refl (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))) as proof of ((iff (((P id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)) (((P0 id) (lam A)) id))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:(forall (A0:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A0)) id))))
% Found (fun (x5:(forall (A0:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A0)) id)))))=> x5) as proof of (forall (A00:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A00)) id))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A0)) id)))))=> x5) as proof of ((forall (A0:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A00)) id)))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A0)) id)))))=> x5) as proof of (term->((forall (A0:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) B0) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) (lam A00)) id))))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found A0:term
% Found A0 as proof of term
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x700:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))))
% Found A:term
% Found A as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (((P id) A) id)))) (((P0 id) X) id))))))
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found A0:term
% Found A0 as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A0:term
% Found A0 as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found x60:(((P id) X) id)
% Found x60 as proof of hoasinduction_no_psi_cond
% Found (fun (x7:((((P id) X) id)->(x4 X)))=> x60) as proof of hoasinduction_no_psi_cond
% Found (fun (x7:((((P id) X) id)->(x4 X)))=> x60) as proof of (((((P id) X) id)->(x4 X))->hoasinduction_no_psi_cond)
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))):(((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))->((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))) as proof of (((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))->((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))
% Found ((eq_ref0 ((((P0 id) X1) id)->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) (fun (x5:Prop)=> ((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))) as proof of (((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))->((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))
% Found (((eq_ref Prop) ((((P0 id) X1) id)->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) (fun (x5:Prop)=> ((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))) as proof of (((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))->((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X1) id)->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) (fun (x5:Prop)=> ((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))))) as proof of (((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))->((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X1) id)->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) (fun (x5:Prop)=> ((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))))) as proof of (term->(((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))->((iff ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))) (((P0 id) X1) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found A0:term
% Found A0 as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found x51:=(x5 x50):(((P id) B) id)
% Found (x5 x50) as proof of (((P id) B) id)
% Found (x5 x50) as proof of (((P id) B) id)
% Found (x500 (x5 x50)) as proof of (((P id) ((sub A) ((push B) id))) id)
% Found ((x50 B) (x5 x50)) as proof of (((P id) ((sub A) ((push B) id))) id)
% Found (fun (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of (((P id) ((sub A) ((push B) id))) id)
% Found (fun (x5:((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))) (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) ((sub A) ((push B) id))) id))
% Found (fun (B:term) (x5:((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))) (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of (((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) ((sub A) ((push B) id))) id)))
% Found (fun (B:term) (x5:((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))) (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of (forall (B:term), (((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) ((sub A) ((push B) id))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))):(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))))
% Found x51:=(x5 x50):(((P id) B) id)
% Found (x5 x50) as proof of (((P id) B) id)
% Found (x5 x50) as proof of (((P id) B) id)
% Found (x500 (x5 x50)) as proof of (((P id) ((sub A) ((push B) id))) id)
% Found ((x50 B) (x5 x50)) as proof of (((P id) ((sub A) ((push B) id))) id)
% Found (fun (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of (((P id) ((sub A) ((push B) id))) id)
% Found (fun (x5:((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))) (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) ((sub A) ((push B) id))) id))
% Found (fun (B:term) (x5:((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))) (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of (((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) ((sub A) ((push B) id))) id)))
% Found (fun (B:term) (x5:((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))) (x50:(forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id))))=> ((x50 B) (x5 x50))) as proof of (forall (B:term), (((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) B) id))->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) ((sub A) ((push B) id))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))):(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))))
% Found x40:=(x4 F):((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)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (x4 F) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id)))
% Found (fun (F:(subst->(term->term)))=> (x4 F)) as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x8) id))))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) x80) id))))))
% Found A0:term
% Found A0 as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id)))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))))
% Found one:term
% Found one as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found A:term
% Found A as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))))
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found X:term
% Found X as proof of term
% Found X:term
% Found X as proof of term
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))):(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))):(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))
% Found x60:(((P id) X) id)
% Instantiate: P:=P0:(subst->(term->(subst->Prop)));X:=A:term
% Found x60 as proof of (((P0 id) A) id)
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))):(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))):(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id)))) (fun (x5:Prop)=> ((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))->((iff ((forall (B0:term), ((((P id) B0) id)->(((P id) ((sub A) ((push B0) id))) id)))->(((P id) X) id))) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found A0:term
% Found A0 as proof of term
% Found A0:term
% Found A0 as proof of term
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found A0:term
% Found A0 as proof of term
% Found A0:term
% Found A0 as proof of term
% Found A0:term
% Found A0 as proof of term
% Found X:term
% Found X as proof of term
% Found X:term
% Found X as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P1 id) A0) id)->(((P1 id) ((F id) A0)) id)))->(((P1 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))))
% Found X:term
% Found X as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))):(((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))))) as proof of (((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))))
% Found X:term
% Found X as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))):(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (term->(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id))))
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x70:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))):(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (term->(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))):(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (term->(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))):(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id)))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->(forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))))) (fun (x5:Prop)=> ((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))))) as proof of (term->(((iff (forall (A0:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (((P0 id) X) id))->((iff (forall (A00:term), ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A00)) id))))) (((P0 id) X) id))))
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P0 id) X) id))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found one:term
% Found one as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id)))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))->((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> ((iff (((P id) A) id)) (((P0 id) X) id))))))
% Found X:term
% Found X as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found x5:((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x8:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x80:term)=> (((P id) A) id)))))))
% Found x60:=(x6 x70):(((P id) X) id)
% Found (x6 x70) as proof of hoasinduction_no_psi_cond
% Found (x6 x70) as proof of hoasinduction_no_psi_cond
% Found (x6 x70) as proof of hoasinduction_no_psi_cond
% Found x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of ((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))
% Found (fun (B:term) (x5:((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))))=> x5) as proof of (term->(((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))->((hoasinduction_p_and_p_prime P1) (fun (x80:term)=> ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))))))
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A0:term) (N:subst), (((eq term) ((sub ((F M) A0)) N)) ((F ((comp M) N)) ((sub A0) N))))->((forall (A0:term), ((((P0 id) A0) id)->(((P0 id) ((F id) A0)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A0:term)=> ((F M) A0)))) id))))
% Found A:term
% Found A as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))):(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found ((eq_ref0 ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P1 id) X) id)->((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id))))) (fun (x5:Prop)=> ((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))) as proof of (term->(((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))->((iff ((hoasinduction_p_and_p_prime P0) (fun (x800:term)=> (((P id) A) id)))) (((P1 id) X) id))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found X0:term
% Found X0 as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))):(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found ((eq_ref0 ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id)))
% Found (fun (B:term)=> (((eq_ref Prop) ((((P0 id) X0) id)->((iff (((P id) A) id)) (((P0 id) X) id)))) (fun (x5:Prop)=> ((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))) as proof of (term->(((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) X0) id))))
% Found x5:(forall (A0:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A0)) id))))
% Found (fun (x5:(forall (A0:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A0)) id)))))=> x5) as proof of (forall (A00:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A00)) id))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A0)) id)))))=> x5) as proof of ((forall (A0:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A00)) id)))))
% Found (fun (B:term) (x5:(forall (A0:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A0)) id)))))=> x5) as proof of (term->((forall (A0:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A0)) id))))->(forall (A00:term), ((forall (B0:term), (((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) B0) id))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) ((sub A00) ((push B0) id))) id))))->((iff ((iff (((P id) A) id)) (((P0 id) X) id))) (((P0 id) (lam A00)) id))))))
% Found A:term
% Found A as proof of term
% Found A:term
% Found A as proof of term
% Found x4:(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))))
% Found x4 as proof of (forall (F:(subst->(term->term))), ((forall (M:subst) (A1:term) (N:subst), (((eq term) ((sub ((F M) A1)) N)) ((F ((comp M) N)) ((sub A1) N))))->((forall (A1:term), ((((P0 id) A1) id)->(((P0 id) ((F id) A1)) id)))->(((P0 id) ((hoaslam id) (fun (M:subst) (A1:term)=> ((F M) A1)))) id))))
% Found one:term
% Found one as proof of term
% Found A:term
% Found A as proof of term
% Found x80:(((P id) X) id)
% Instantiate: P:=P0:(subst->(term->(subst->Prop)));X:=A:term
% Found x80 as proof of (((P0 id) A) id)
% Found x60:(((P id) X) id)
% Found x60 as proof of (forall (A:term), (((P0 id) A) id))
% Found x60 as proof of (forall (A:term), (((P0 id) A) id))
% Found one:term
% Found one as proof of term
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))):(((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found (eq_ref00 (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id))->((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))
% Found ((eq_ref0 ((((P0 id) X) id)->((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id))))) (fun (x5:Prop)=> ((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A) id)) (((P0 id) ((sub A0) ((push B0) id))) id))))->((iff (((P id) A) id)) (((P0 id) (lam A0)) id)))) (((P0 id) X) id)))) as proof of (((iff ((forall (B0:term), (((iff (((P id) A) id)) (((P0 id) B0) id))->((iff (((P id) A
% EOF
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